Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Hongyan Sun Denote by Ḃα,ϕ(Ω) the Orlicz–Besov space, where α∈R, ϕ is a Young function and Ω⊂Rn is a domain. For α∈(−n,0) and optimal ϕ, in this paper we characterize domains supporting the imbedding Ḃα,ϕ(Ω) into Ln∕ α (Ω) via globally n-regular domains. This extends the known characterizations for domains supporting the Besov imbedding Ḃpps(Ω) into Lnp∕(n−sp)(Ω) with s∈(0,1) and 1≤p

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Shigeaki Koike, Andrzej Święch, Shota Tateyama The weak Harnack inequality for Lp-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of Lp-viscosity solutions is derived from the weak Harnack inequality for Lp-viscosity supersolutions. The local maximum principle for Lp-viscosity subsolutions and the Harnack inequality for Lp-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Huatao Chen, Juan Luis García Guirao, Dengqing Cao, Jingfei Jiang, Xiaoming Fan This paper concerns with the long time dynamical behavior of a stochastic Euler–Bernoulli beam driven by additive white noise. By verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam, the existence of global random attractors that attracts all bounded sets in phase space is proved. Furthermore, the finite Hausdorff dimension for the global random attractors is attained. In light of the relationship between global random attractor and random invariant probability measure, the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Marco Degiovanni, Marco Marzocchi We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of “natural constraint”.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Martin Dindoš, Luke Dyer We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form Lu=div(A∇u)−ut=0 in Lip(1,1∕2) time-varying cylinders, where the coefficient matrix A=aij(X,t) is uniformly elliptic and bounded.We show that if the Regularity problem (R)p for the equation Lu=0 is solvable for some 1

p
-Laplacian+in+

Abstract: Publication date: Available online 5 April 2019Source: Nonlinear AnalysisAuthor(s): Leandro M. Del Pezzo, Alexander Quaas In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional p−Laplacian with weight in RN. We also show a nonexistence result when the weight has positive integral.In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schrödinger type equation.

Abstract: Publication date: Available online 4 April 2019Source: Nonlinear AnalysisAuthor(s): Miroslav Bulíček, Piotr Gwiazda, Martin Kalousek, Agnieszka Świerczewska-Gwiazda We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic N-function. First, an existence result is shown under the assumption that the N-function or its convex conjugate satisfies Δ2-condition. The second result concerns the homogenization process for families of strongly nonlinear elliptic problems with the homogeneous Dirichlet boundary condition under above stated conditions on the elliptic operator, which is additionally assumed to be periodic in the spatial variable.

Abstract: Publication date: Available online 3 April 2019Source: Nonlinear AnalysisAuthor(s): Petru Mironescu, Emmanuel Russ, Yannick Sire Let Ω be a smooth bounded (simply connected) domain in Rn and let u be a complex-valued measurable function on Ω such that u(x) =1 a.e. Assume that u belongs to a Besov space Bp,qs(Ω;ℂ). We investigate whether there exists a real-valued function φ∈Bp,qs(Ω;R) such that u=eıφ. This complements the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The microscopic parameter q turns out to play an important role in some limiting situations. The analysis of this lifting problem relies on some interesting new properties of Besov spaces, in particular a non-restriction property when q>p.

Abstract: Publication date: Available online 28 March 2019Source: Nonlinear AnalysisAuthor(s): Luca Alasio, Maria Bruna, José Antonio Carrillo We show that solutions of nonlinear nonlocal Fokker–Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for L2 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.

Abstract: Publication date: Available online 26 March 2019Source: Nonlinear AnalysisAuthor(s): Paolo Marcellini We consider variational solutions to the Cauchy-Dirichlet problem ∂tu=divDξf(x,u,Du)−Duf(x,u,Du)inΩTu=u0on∂parΩTwhere the function f=fx,u,ξ, f:Rn×RN×RN×n→[0,∞), is convex with respect to u,ξ and coercive in ξ∈RN×n, but it not necessarily satisfies a growth condition from above. A motivation to consider a class of such energy functions f can be also easily found in the stationary case, where a large literature in the calculus of variations is devoted to the minimization of p,q-growth problems [45] and to double phase problems [23], [24], [4], [5], [6]. In the parabolic context the notion of variational solution (see the references from [8] to [15]) is compatible with the lack of the same polynomial growth from below and from above.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Eiji Yanagida This paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the Neumann or Dirichlet boundary conditions. Assuming that the nonlinearity satisfies a rather general growth condition, we consider the blow-up and global existence of sign-changing solutions. It is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution, the solution blows up in finite time if an initial value changes its sign at most k times, whereas there exist stationary solutions with more than k zeros. The proof is based on the intersection number principle combined with a topological method.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Sungchol Kim, Dukman Ri In this paper we study the integral functionals with the general nonstandard growth. We prove the boundedness and Hölder continuity of quasiminimizers of these functionals. Our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in “Direct Methods in the Calculus of Variations, 2003” by Giusti.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Marino Badiale, Stefano Greco, Sergio Rolando Given three measurable functions Vr≥0, Kr>0 and Qr≥0, r>0, we consider the bilaplacian equation Δ2u+V( x )u=K( x )f(u)+Q( x )inRNand we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Sitong Chen, Binlin Zhang, Xianhua Tang This paper is concerned with the following singularly perturbed problem in H1(R2) −ε2Δu+V(x)u+A0(u(x))u+∑j=12Aj2(u(x))u=f(u),ε(∂1A2(u(x))−∂2A1(u(x)))=−12u2,∂1A1(u(x))+∂2A2(u(x))=0,εΔA0(u)=∂1(A2 u 2)−∂2(A1 u 2),where ε is a small parameter, V∈C(R2,R) and f∈C(R,R). By using some new variational and analytic techniques joined with the manifold of Pohoz̆aev–Nehari type, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem admits a semiclassical ground state solution

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Verena Bögelein, Pekka Lehtelä, Stefan Sturm We investigate the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions um∈Lloc2(0,T;Hloc1(Ω)) and um+12∈Lloc2(0,T;Hloc1(Ω)) in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Zhaoxia Liu Asymptotic behavior of the higher order spacial derivatives of the strong solution to the incompressible non-stationary magneto-hydrodynamic (MHD) equations is given in a half-space, which is a long-time difficult question. The main tools employed in this article are the non-stationary Stokes solution formula, and some a priori estimates of the steady Stokes system in the half-space.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Claudio Arancibia-Ibarra I analyse a modified May–Holling–Tanner predator–prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator–prey population. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle–node, Hopf, and Bogdanov–Takens bifurcations. We use simulations to illustrate the behaviour of the model.

Abstract: Publication date: August 2019Source: Nonlinear Analysis, Volume 185Author(s): Brian Allen, Edward Bryden We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the Hölder sense. This is done by establishing a general trace inequality on Riemannian manifolds which is an interesting result on its own. We provide examples demonstrating how each of our hypotheses are necessary. In the Appendix by the first author with Christina Sormani, we prove that sequences of compact integral current spaces without boundary (including Riemannian manifolds) that have uniform Hölder bounds on their distance functions have subsequences converging in the Gromov–Hausdorff (GH) sense. If in addition they have a uniform upper bound on mass (volume) then they converge in the Sormani–Wenger Intrinsic Flat (SWIF) sense to a limit whose metric completion is the GH limit. We provide an example of a sequence developing a cusp demonstrating why the SWIF and GH limits may not agree.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Italo Capuzzo Dolcetta, Antonio Vitolo We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains Ω⊂Rn of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of Ω, with possible degeneracy along the unbounded directions.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Sujin Khomrutai We investigate a nonlocal equation ∂tu=∫RnJ(x−y)u(y,t)dy−‖J‖L1u(x,t)+a(x,t)up in Rn, where a is unbounded and J belongs to a weighted space. Crucial weighted Lp and interpolation estimates for the Green operator are established by a new method based on the sharp Young’s inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions Γ=(1+ x 2∕η)b∕2 that −Γ∕η≲J∗Γ−Γ≲Γ∕η and η−b+∕2≲Γ∕xb≲η−b−∕2. Blow-up behaviors are investigated by employing test functions ϕR=Γ (η=R) instead of principal eigenfunctions. Global well-posedness in weighted Lp spaces for the Cauchy problem is proved. When a∼xσ the Fujita exponent is shown to be 1+(σ+2)∕n. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Jiecheng Chen, Dashan Fan, Fayou Zhao We establish Littlewood–Paley characterizations of the Sobolev spaces Wα,p in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner–Riesz means and some other well known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Setenay Akduman, Alexander Pankov The paper deals with nonlinear Schrödinger equations on infinite metric graphs. We assume that the linear potential is infinitely growing. We prove an existence and multiplicity result that covers both self-focusing and defocusing cases. Furthermore, under some additional assumptions we show that solutions obtained bifurcate from trivial ones. We prove that these solutions are superexponentially localized. Our approach is variational and based on generalized Nehari manifold.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Wei Lian, Md Salik Ahmed, Runzhang Xu In this paper we consider the semilinear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with logarithmic Sobolev inequality, we derive the global existence and infinite time blow up of the solution at low energy level E(0)0.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Dominic Breit, Prince Romeo Mensah We prove the existence of a unique local strong solution to the stochastic compressible Euler system with nonlinear multiplicative noise. This solution exists up to a positive stopping time and is strong in both the PDE and probabilistic sense. Based on this existence result, we study the inviscid limit of the stochastic compressible Navier–Stokes system. As the viscosity tends to zero, any sequence of finite energy weak martingale solutions converges to the compressible Euler system.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): E. Lokharu, E. Wahlén We consider steady three-dimensional gravity–capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Wei Ding, Guozhen Lu, YuePing Zhu Though multi-parameter Hardy space theory has been well developed in the past half century, not much has been studied for a local Hardy space theory in the multi-parameter settings. Such multi-parameter local Hardy spaces can play an important role in studying the boundedness of multi-parameter pseudo-differential operators, multi-parameter singular integrals of non-convolution type, and applications to partial differential equations, etc. By establishing a bi-parameter local reproducing formula, bi-parameter local Hardy space hp(Rn1×Rn2) is introduced in this paper. This space coincides with Lp(Rn1+n2) when p>1. While p≤1, this space is substantially different from the classical bi-parameter Hardy spaces Hp(Rn1×Rn2). We will establish its atomic decomposition of the bi-parameter local Hardy spaces and as an application, we prove the boundedness from hp(Rn1×Rn2) to Lp(Rn1+n2) of the bi-parameter singular integral operators in inhomogeneous Journé class for p near 1. For the simplicity, we have chosen to present all the results in the bi-parameter setting. Nevertheless, all of them hold for arbitrary number of parameters. The multi-parameter local theory developed in this paper can serve as a model case for similar theory in other multi-parameter settings.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Xueke Pu, Min Li This paper concerns the existence of global smooth solutions to the initial boundary value problem for a three-dimensional compressible quantum hydrodynamic model with damping and heat diffusion in a bounded domain in R3. Based on the continuation argument and the uniform a priori estimates with respect to the time, we obtain the existence of global solutions in a bounded smooth domain provided that the initial perturbation around a constant state is small enough. The key difficulty is to deal with the higher order quantum terms, which do play an essential role in establishing the a priori estimates. The boundary conditions finally adopted are the insulating boundary conditions.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): João Vitor da Silva In this survey we establish geometric C1+α regularity estimates for bounded solutions of a number of nonlinear evolution models in divergence and non-divergence form. The main insights to obtain such estimates are based on geometric tangential methods, and make use of systematic oscillation mechanisms combined with intrinsic scaling techniques.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Shuxing Zhang In this paper, we study the low Mach number limit for the full compressible Navier–Stokes equations with Cattaneo’s heat transfer law. It is rigorously justified that, in the framework of classical solutions with small density, temperature and heat flux variations, the solutions of the full compressible Navier–Stokes equations with Cattaneo’s heat transfer law converge to that of the incompressible Navier–Stokes equations as the Mach number tends to zero.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska Regularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set Γ⊂Rn. Such a regularity result was obtained by the second author and L. Rifford in Cannarsa and Rifford (2008) when Γ is a singleton. Furthermore, we derive sensitivity relations for time optimal control problems with general target sets Γ, that is, without imposing any geometric assumptions on Γ.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Stefano Biagi, Alessandro Calamai, Francesca Papalini We obtain existence results for strongly nonlinear BVPs of type (P)(Φ(k(t)x′(t)))′=f(t,x(t),x′(t))a.e. on [0,∞),x(0)=ν1,x(∞)=ν2where Φ:R→R is a strictly increasing homeomorphism such that Φ(0)=0 (the Φ-Laplacian operator), k:[0,∞)→R is a non-negative continuous function which may vanish on a subset of [0,∞) of measure zero, f is a Carathéodory function and ν1,ν2∈R are fixed. Under mild assumptions, including a weak form of a Nagumo–Wintner growth condition, we prove the existence of heteroclinic solutions of (P) in the Sobolev space Wloc1,p([0,∞)). Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Humberto Rafeiro, Stefan Samko We prove the boundedness of the fractional integration operator of variable order α(x) in the limiting Sobolev case α(x)p(x)=n−λ(x) from variable exponent Morrey spaces Lp⋅,λ⋅Ω into BMO (Ω), where Ω is a bounded open set. In the case α(x)≡ const, we also show the boundedness from variable exponent vanishing Morrey spaces VLp⋅,λ⋅Ω into VMO (Ω). The results seem to be new even when p and λ are constant.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Biswajit Basu We investigate the properties of velocity field for steady periodic water waves over a flat bed for a generalized class of C1 vorticity functions γ. Results are proved by exploiting the maximum principles and are based on Dubreil–Jacotin transformation of the fluid domain. Some properties of velocities interior to the fluid domain and the locations of maximum/minimum horizontal fluid velocities are investigated and proved for rotational water waves without any restriction to amplitude. For u

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Zuzana Došlá, Serena Matucci We investigate the existence of positive radial solutions for a nonlinear elliptic equation with p-Laplace operator and sign-changing weight, both in superlinear and sublinear case. We prove the existence of solutions u, which are globally defined and positive outside a ball of radius R, satisfy fixed initial conditions u(R)=c>0, u′(R)=0 and tend to zero at infinity. Our method is based on a fixed point result for boundary value problems on noncompact intervals and on asymptotic properties of suitable auxiliary half-linear differential equations. The results are new also for the classical Laplace operator and may be used for proving the existence of ground state solutions and decaying solutions with exactly k-zeros which are defined in the whole space. Some examples illustrate our results.

Abstract: Publication date: July 2019Source: Nonlinear Analysis, Volume 184Author(s): Marta Menci, Marco Papi In this paper we propose local and global existence results for the solution of systems characterized by the coupling of ODEs and PDEs. The coexistence of distinct mathematical formalisms represents the main feature of hybrid approaches, in which the dynamics of interacting agents are driven by second-order ODEs, while reaction–diffusion equations are used to model the time evolution of a signal influencing them. We first present an existence result of the solution, locally in time. In particular, we generalize the framework of recent works, presented in the literature with a modeling and numerical approach, which have not been investigated from an analytical point of view so far. Then, the previous result is extended in order to obtain a global solution.

ω
-minima+of+

Abstract: Publication date: Available online 21 March 2019Source: Nonlinear AnalysisAuthor(s): Cristiana De Filippis We focus on some regularity properties of ω-minima of variational integrals with φ-growth and provide an upper bound on the Hausdorff dimension of their singular set.

Abstract: Publication date: Available online 20 March 2019Source: Nonlinear AnalysisAuthor(s): Irena Lasiecka, Michael Pokojovy, Xiang Wan We consider an initial–boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell–Cattaneo–Vernotte law giving rise to a ‘second sound’ effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev Hk-spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small – not in the full topology of our solution class, but in a lower topology corresponding to weak solutions we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.

Abstract: Publication date: Available online 19 March 2019Source: Nonlinear AnalysisAuthor(s): Francescantonio Oliva, Francesco Petitta We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form ut−Δpu=h(u)f+μinΩ×(0,T),u=0on∂Ω×(0,T),u=u0inΩ×{0}, where Ω is an open bounded subset of RN (N≥2), u0 is a nonnegative integrable function, Δp is the p-Laplace operator, μ is a nonnegative bounded Radon measure on Ω×(0,T) and f is a nonnegative function of L1(Ω×(0,T)). The term h is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing h.

Abstract: Publication date: Available online 19 March 2019Source: Nonlinear AnalysisAuthor(s): Pietro Celada, Jihoon Ok We study partial C1,α – regularity of minimizers of quasi-convex variational integrals with non-standard growth. We assume in particular that the relevant integrands satisfy an Orlicz’s type growth condition, i.e. a so-called general growth condition. Moreover, the functionals are supposed to be non-autonomous and possibly degenerate.

Abstract: Publication date: Available online 15 March 2019Source: Nonlinear AnalysisAuthor(s): Marcone C. Pereira, Julio D. Rossi, Nicolas Saintier In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂Rn and V⊂Rm, we show that the solution uε to Δxsuε(x,y)+Δytuε(x,y)=f(x,ε−1y)inU×εV with uε(x,y)=0 if x⁄∈U and y∈εV, verifies that ũε(x,y)≔uε(x,εy)→u0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u0 and estimate the rate of convergence in the uniform norm. Here Δxsu and Δytu are the fractional Laplacian in the 1st variable x (with a Dirichlet condition, u(x)=0 if x⁄∈U) and in the 2nd variable y (with a Neumann condition, integrating only inside V), respectively, that is, Δxsu(x,y)=∫Rnu(x,y)−u(w,y) x

Abstract: Publication date: Available online 9 March 2019Source: Nonlinear AnalysisAuthor(s): Mark Allen, Mariana Smit Vega Garcia We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation (−Δ)su=χ{u>c}for 01∕2 these solutions are not stable and therefore not minimizers of the corresponding functional.

Abstract: Publication date: Available online 9 March 2019Source: Nonlinear AnalysisAuthor(s): Nestor Guillen, Russell W. Schwab An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min–max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min–max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.

Abstract: Publication date: Available online 8 March 2019Source: Nonlinear AnalysisAuthor(s): Paolo Antonelli, Pierangelo Marcati, Hao Zheng This paper is concerned with an existence and stability result on the nonlinear derivative Schrödinger equation in 1-D, which is originated by the study of the stability of nontrivial steady states in Quantum Hydrodynamics. The problem is equivalent to a compressible Euler fluid system with a very specific Korteweg–Kirchhoff stress K(ρ)=ħ4ρ. As a simple, but significative, example we consider the nonlinear derivative Schrödinger equation obtained via a complex Cole–Hopf type transformation, applied to the 1-D free Schrödinger equation. The resulting problem (possibly unstable) is investigated for small solutions around the null steady state. The stability is proved to be valid for long time intervals of order O(ϵ−4∕5), where ϵ is the order of smallness of the initial data. This result brought back to the QHD system provides the stability of the steady state ρ=1,J=v=0. The validity in time of this result is far beyond what can be obtained via classical linearization analysis or via higher order energy estimates. Indeed in our analysis the nonlinear structure plays a crucial role in the corresponding iteration procedure, the use of local smoothing and the Schrödinger maximal operator provides the control of the potential lost of regularity.

Abstract: Publication date: Available online 8 March 2019Source: Nonlinear AnalysisAuthor(s): Daisuke Naimen, Masataka Shibata We investigate the Kirchhoff type elliptic problem with critical nonlinearity; −1+α∫Ω ∇u 2dxΔu=λuq+u2∗−1,u>0inΩ,u=0on∂Ω,where N≥5, Ω⊂RN is a bounded domain with smooth boundary ∂Ω, α>0, λ∈R, 2∗=2N∕(N−2), and q∈[1,2∗−1). We prove the existence of two solutions of it via the variational method. Since N≥5 and α>0, the uniqueness assertion for the associated limiting problem may fail. This causes serious difficulties in controlling concentrating Palais–Smale sequences. We overcome these by introducing new techniques. For a mountain pass type solution, we utilize the limit function of the fibering maps of the concentrating Palais–Smale sequence. This tool is based on our careful setting of Nehari type sets. On the other hand, a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution. This is the first work which proves the multiplicity of positive solutions of the Kirchhoff type critical problem in high dimension.

Abstract: Publication date: Available online 5 March 2019Source: Nonlinear AnalysisAuthor(s): Giuseppe Mingione, Giampiero Palatucci Nonlinear Potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered. In recent years there has been a substantial development of this subject, mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials. Here we give a brief account of such developments and outline the connections with different fields.

Abstract: Publication date: Available online 1 March 2019Source: Nonlinear AnalysisAuthor(s): Lucio Boccardo, Luigi Orsina In this paper we are going to prove existence and regularity results for positive solutions of the following elliptic system: −div(M(x)∇u)+rφur−1=f+φr,−div(M(x)∇φ)+ruφr−1=ur.where Ω is a bounded open subset of RN, M is a bounded, uniformly elliptic matrix, r>1, and f≥0 belongs to some Lebesgue space Lm(Ω), with m≥1. We will also prove the relationships of the solutions of the system with saddle points of the integral functional J(v,ψ)=12∫ΩM(x)∇v⋅∇v−12∫ΩM(x)∇ψ⋅∇ψ+∫Ω v rψ−∫Ω ψ rv−∫Ωfv

Abstract: Publication date: Available online 26 February 2019Source: Nonlinear AnalysisAuthor(s): Divya Goel, K. Sreenadh We consider the following Kirchhoff–Choquard equation −M(‖∇u‖L22)Δu=λf(x) u q−2u+∫Ω u(y) 2μ∗ x−y μdy u 2μ∗−2uinΩ,u=0on∂Ω, where Ω is a bounded domain in RN(N≥3) with C2 boundary, 2μ∗=2N−μN−2, 1

Abstract: Publication date: Available online 22 February 2019Source: Nonlinear AnalysisAuthor(s): Emilio Acerbi, Chao-Nien Chen, Yung-Sze Choi A singular limit of a FitzHugh–Nagumo system leads to a nonlocal geometric variational problem with periodic boundary conditions. We study the stationary lamellar set and give a criterion to select out the one with the lowest energy. Such an optimal structure is called a minimal lamella. While the empty set or the full torus is a global minimizer for appropriate parameter regimes, the minimal lamellae beat both in other circumstances. The concept of minimal lamella points out that a preferred 1D mesh size is universal.

Abstract: Publication date: Available online 21 February 2019Source: Nonlinear AnalysisAuthor(s): Weiwei Ao, Hardy Chan, María del Mar González, Juncheng Wei We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation (−Δ)su+V(x)u−up=0inRn, with u(x)→0 as x →+∞, where p>n+2sn−2s for s∈(0,1),n>2s. We show that if V(x)=o( x −2s) as x →+∞, then for p>n+2s−1n−2s−1, this problem admits a continuum of solutions. More generally, for p>n+2sn−2s, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of (−Δ)sw=wpinRn, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf’s argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.

Abstract: Publication date: Available online 16 February 2019Source: Nonlinear AnalysisAuthor(s): Daniele Cassani, Zhisu Liu, Cristina Tarsi, Jianjun Zhang We are concerned with sign-changing solutions to nonlocal perturbation of the nonlinear Schrödinger equation in the whole R3. By using the method of invariant sets of descending flow, we prove the existence of multiple sign-changing solutions provided the perturbation is sufficiently small.

Abstract: Publication date: Available online 16 February 2019Source: Nonlinear AnalysisAuthor(s): Michela Eleuteri, Antonia Passarelli di Napoli In this paper we prove the higher differentiability in the scale of Besov spaces of the solutions to a class of obstacle problems of the type min∫ΩF(x,z,Dz):z∈Kψ(Ω). Here Ω is an open bounded set of Rn, n≥2, ψ is a fixed function called obstacle and Kψ(Ω) is set of admissible functions z∈W1,p(Ω) such that z≥ψ a.e. in Ω. We assume that the gradient of the obstacle belongs to a suitable Besov space. The main novelty here is that we are not assuming any differentiability on the partial maps x↦F(x,z,Dz) and z↦F(x,z,Dz), but only their Hölder continuity.

Abstract: Publication date: Available online 16 February 2019Source: Nonlinear AnalysisAuthor(s): Youchan Kim, Yeonghun Youn We consider elliptic equations with measurable nonlinearities in a half cube Q2R∩{(x1,x′)∈Rn:x1>0} where the boundary data is given on Q2R∩{(x1,x′)∈Rn:x1=0}. We obtain a point-wise estimate of the gradient in terms of Riesz potential of the right-hand side data and the oscillation of the gradient of the boundary data under the assumption that the nonlinearity is only allowed to be measurable in x1-variable.

Abstract: Publication date: Available online 7 February 2019Source: Nonlinear AnalysisAuthor(s): Zhuoran Du, Changfeng Gui We obtain some existence theorems for periodic solutions to several linear and nonlinear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.

Abstract: Publication date: Available online 30 January 2019Source: Nonlinear AnalysisAuthor(s): R. Arora, J. Giacomoni, T. Mukherjee, K. Sreenadh This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser–Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex–concave problem (Pλ,M) below.

Abstract: Publication date: Available online 30 January 2019Source: Nonlinear AnalysisAuthor(s): Giovanni Covi This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity equation is put into relation with a long jump random walk with weights.

Abstract: Publication date: Available online 21 January 2019Source: Nonlinear AnalysisAuthor(s): Juan C. Pozo, Vicente Vergara In this work, we derive a non-local in time telegraph equation. Our model includes as particular cases the classical telegraph equation and the fractional in time telegraph equation among others. Further, we define the fundamental solution of the problem and we prove that it can be interpreted as a probability density function. Finally, using versions of the Karamata–Feller Tauberian theorem, we study the temporal behavior of the variance of the distribution process associated with the solution of the equation in large times as well as in short times.

Abstract: Publication date: Available online 17 January 2019Source: Nonlinear AnalysisAuthor(s): Youchan Kim, Seungjin Ryu, Pilsoo Shin We prove a boundary higher integrability result for nonlinear elliptic systems with conormal boundary conditions in locally uniform domains. To do it, we derive a boundary version of Gehring–Giaquinta–Modica Lemma and Sobolev–Poincaré type inequality in locally uniform domains. Our result plays a key role for handling the coefficients when obtaining Calderó n–Zygmund type estimates with conormal boundary conditions in nonsmooth domains.

Abstract: Publication date: Available online 16 January 2019Source: Nonlinear AnalysisAuthor(s): Pilsoo Shin We derive Calderón–Zygmund type estimates for the gradient of bounded distributional solutions to the problem divb(x,u) Du p−2Du+a(x) Du q−2Du=div F p−2F+a(x) F q−2Fwhere 1

Abstract: Publication date: Available online 4 January 2019Source: Nonlinear AnalysisAuthor(s): Yuhua Li, Xiaocui Hao, Junping Shi In this paper, the existence and nonexistence of energy minimizer of the Kirchhoff–Schrödinger energy function with prescribed L2-norm in dimension four are considered. The energy infimum values are completely classified in terms of coefficient and exponent of the nonlinearity. The sharp existence results of global constraint minimizers for both the subcritical and critical exponent cases are obtained, and the criticality is in the sense of both Sobolev embedding and Gagliardo–Nirenberg inequality. Our results also show the delicate difference between the case without a trapping potential function and the one with potential function.

Abstract: Publication date: Available online 21 December 2018Source: Nonlinear AnalysisAuthor(s): Peter Lindqvist, Mikko Parviainen We study the possibility of prescribing infinite initial values for solutions of the Evolutionary p-Laplace Equation in the fast diffusion case p>2. This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space–time cylinder.

Abstract: Publication date: Available online 7 December 2018Source: Nonlinear AnalysisAuthor(s): Salvatore Leonardi, Francesco Leonetti, Cristina Pignotti, Eugénio Rocha, Vasile Staicu We give maximum principles for solutions u:Ω→RN to a class of quasilinear elliptic systems whose prototype is −∑i=1n∂∂xi∑β=1N∑j=1nai,jα,βx,u(x)∂uβ∂xj(x)=0,x∈Ω,where α∈{1,…,N} is the equation index and Ω is an open, bounded subset of Rn. We assume that coefficients ai,jα,β(x,y) are measurable with respect to x, continuous with respect to y∈RN, bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi’s counterexample by means of some additional structure assumptions on the coefficients ai,jα,β(x,y). In this paper, we assume that off-diagonal coefficients ai,jα,β, α≠β, have support in some staircase set along the diagonal in the yα,yβ plane.

Abstract: Publication date: Available online 1 December 2018Source: Nonlinear AnalysisAuthor(s): Mingqi Xiang, Binlin Zhang, Dušan Repovš We study the existence and multiplicity of solutions for a class of fractional Schrödinger–Kirchhoff type equations with the Trudinger–Moser nonlinearity. More precisely, we consider M(‖u‖N∕s)(−Δ)N∕ssu+V(x) u Ns−1u=f(x,u)+λh(x) u p−2uinRN,‖u‖=∬R2N u(x)−u(y) N∕s x−y 2Ndxdy+∫RNV(x) u N∕sdxs∕N,where M:[0,∞]→[0,∞) is a continuous function, s∈(0,1), N≥2, λ>0 is a parameter, 1

Abstract: Publication date: Available online 27 November 2018Source: Nonlinear AnalysisAuthor(s): Enno Lenzmann, Armin Schikorra We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations.The commutators we investigate are Jacobians, more generally Coifman–Rochberg–Weiss commutators, Chanillo’s commutator with the Riesz potential, Coifman–McIntosh–Meyer, Kato–Ponce–Vega type commutators, the fractional Leibniz rule, and the Da Lio–Rivière three-term commutators. We also give a new limiting L1-estimate for a double commutator of Coifman–Rochberg–Weiss-type, and several intermediate estimates.The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general para-product arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the “cancellation effect” responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.

Abstract: Publication date: Available online 27 November 2018Source: Nonlinear AnalysisAuthor(s): Giovanni Molica Bisci In this paper we study the existence of (weak) solutions for some Kirchhoff-type problems whose simple prototype is given by −a+b∫B ∇Hu(σ) 2dμΔHu=λf(u)inBRu=0on∂BR,where ΔH denotes the Laplace–Beltrami operator on the ball model of the Hyperbolic space BN (with N≥3), a,b and λ are real parameters, BR⊂BN is a geodesic ball centered in zero of radius R and f is a subcritical continuous function. The Kirchhoff term is allowed to vanish at the origin covering the degenerate case. The main technical approach is based on variational and topological methods.

Abstract: Publication date: Available online 20 November 2018Source: Nonlinear AnalysisAuthor(s): Iwona Chlebicka We study a general nonlinear elliptic equation in the Orlicz setting with data not belonging to the dual of the energy space. We provide several Lorentz-type and Morrey-type estimates for the gradients of solutions under various conditions on the data.

s
-fractional+Laplacian+

Abstract: Publication date: Available online 17 November 2018Source: Nonlinear AnalysisAuthor(s): Begoña Barrios, Luigi Montoro, Ireneo Peral, Fernando Soria In this paper we study a variational Neumann problem for the higher order fractional Laplacian (−Δ)s, s>1. In the process we introduce some nonlocal Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.

Abstract: Publication date: Available online 13 November 2018Source: Nonlinear AnalysisAuthor(s): Jijiang Sun, Lin Li, Matija Cencelj, Boštjan Gabrovšek In this paper, we consider the following nonlinear Kirchhoff type problem: −a+b∫R3 ∇u 2Δu+V(x)u=f(u),inR3,u∈H1(R3),where a,b>0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik–Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity u p−2u with p∈(2,4].

Abstract: Publication date: Available online 29 October 2018Source: Nonlinear AnalysisAuthor(s): The Anh Bui, The Quan Bui In this paper we prove the regularity estimates on the weighted Lorentz spaces for the higher-order elliptic system of equations with non-smooth coefficients on a Lipschitz domain. As a by product, we obtain the regularity estimate on the Lorentz–Morrey spaces. Our results extend previous results to the larger class of Muckenhoupt weights.

Abstract: Publication date: Available online 24 October 2018Source: Nonlinear AnalysisAuthor(s): Shuang Liang, Shenzhou Zheng We prove a global Calderón–Zygmund type estimate in the framework of Lorentz spaces for the variable power of the gradients to the solution pair (u,P) of the conormal derivative problems of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one of the spatial variables and have sufficiently small bounded mean oscillation seminorm in the other variables, the boundary of domain belongs to the Reifenberg flatness, and the variable exponents p(x) satisfy the log-Hölder continuity.

Abstract: Publication date: Available online 12 October 2018Source: Nonlinear AnalysisAuthor(s): Alessio Fiscella, Pawan Kumar Mishra In the present paper, we study the following singular Kirchhoff problem M∬R2N u(x)−u(y) 2 x−y N+2sdxdy(−Δ)su=λf(x)u−γ+g(x)u2s∗−1inΩ,u>0inΩ,u=0inRN∖Ω,where Ω⊂RN is an open bounded domain, dimension N>2s with s∈(0,1), 2s∗=2N∕(N−2s) is the fractional critical Sobolev exponent, parameter λ>0, exponent γ∈(0,1), M models a Kirchhoff coefficient, f∈L2s∗2s∗+γ−1(Ω) is a positive weight, while g∈L∞(Ω) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ. This approach allows us to avoid any restriction on the boundary of

Abstract: Publication date: Available online 13 September 2018Source: Nonlinear AnalysisAuthor(s): Luís H. de Miranda, Adilson E. Presoto This paper addresses the gain of global fractional regularity in Nikolskii spaces for solutions of a class of quasilinear degenerate equations with (p,q)-growth. Indeed, we investigate the effects of the datum on the derivatives of order greater than one of the solutions of the (p,q)-Laplacian operator, under Dirichlet’s boundary conditions. As it turns out, even in the absence of the so-called Lavrentiev phenomenon and without variations on the order of ellipticity of the equations, the fractional regularity of these solutions ramifies depending on the interplay between the growth parameters p, q and the data. Indeed, we are going to exploit the absence of this phenomenon in order to prove the validity up to the boundary of some regularity results, which are known to hold locally, and as well provide new fractional regularity for the associated solutions. In turn, there are obtained certain global regularity results by means of the combination between new a priori estimates and approximations of the differential operators, whereas the nonstandard boundary terms are handled by means of a careful choice for the local frame.

Abstract: Publication date: Available online 5 September 2018Source: Nonlinear AnalysisAuthor(s): Sun-Sig Byun, Seungjin Ryu We study a nonlinear elliptic double obstacle problem with irregular data and establish an optimal Calderón–Zygmund theory. The partial differential operator is of the p-Laplacian type and includes merely measurable coefficients in one variable. We prove that the gradient of a weak solution is as integrable as both the gradient of assigned two obstacles and the nonhomogeneous divergence term under a small BMO semi-norm assumption on the coefficients in the other variables.

Abstract: Publication date: Available online 9 August 2018Source: Nonlinear AnalysisAuthor(s): Mouhamed Moustapha Fall In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.

Abstract: Publication date: Available online 2 August 2018Source: Nonlinear AnalysisAuthor(s): Minhyun Kim, Ki-Ahm Lee We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in Caffarelli and Silvestre (2009). Since the order of differentiability of the kernel is not characterized by a single number, we use the constant Cφ=∫Rn1−cosy1 y nφ( y )dy−1instead of 2−σ, where φ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Hölder estimates of viscosity solutions to the nonlinear integro-differential equations.

Abstract: Publication date: Available online 23 July 2018Source: Nonlinear AnalysisAuthor(s): Fengping Yao, Chao Zhang, Shulin Zhou In this paper we use the Hardy–Littlewoodmaximal functions to obtain the following global BMO estimates f∈BMO(Rn)⇒∇u∈BMO(Rn)for the weak solutions of a class of quasilinear elliptic equations diva∇u∇u=divfinRn,where B(t)=∫0tτa(τ)dτ for t≥0. Meanwhile, we use the iteration-covering procedure to prove that Bf∈Lq(Rn)⇒B∇u∈Lq(Rn)for anyq>1for the weak solutions of diva∇u∇u=divaffinRn.Moreover, we remark that a(t)=tp−2(p-Laplace equation)anda(t)=tp−2log(1+t)satisfy the given conditions in this work.