Authors:Ivan Naumkin; Pierre Raphaël Abstract: We consider the mass critical fractional (NLS) $$\begin{aligned} i\partial _{t}u-\left D\right ^{s}u+u\left u\right ^{2s}=0,\text { }x\in \mathbb {R},\text { }1<s<2. \end{aligned}$$ We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in Gérard et al (A two soliton with transient turbulent regime for the one dimensional cubic half wave, 2018) for the critical case \(s=1\) , but with a completely different asymptotic profile when the mass vanishes. PubDate: 2018-05-16 DOI: 10.1007/s00526-018-1355-5 Issue No:Vol. 57, No. 3 (2018)

Authors:Blair Davey; Jiuyi Zhu Abstract: In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new \(L^p - L^q\) Carleman estimate for the Laplacian in \(\mathbb {R}^2\) . PubDate: 2018-05-11 DOI: 10.1007/s00526-018-1345-7 Issue No:Vol. 57, No. 3 (2018)

Authors:Yangqin Fang; Sławomir Kolasiński Abstract: We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed m dimensional subsets of \({\mathbf {R}}^n\) which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an \(({\mathscr {H}}^m,m)\) rectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimension m and the co-dimension \(n-m\) other than \(1 \le m < n\) . An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren’s (Ann Math (2) 87:321–391, 1968) paper. In the end we show that classes of sets spanning some closed set B in homological and cohomological sense satisfy our axioms. PubDate: 2018-05-11 DOI: 10.1007/s00526-018-1348-4 Issue No:Vol. 57, No. 3 (2018)

Authors:Wei Zhang; Jiguang Bao; Bo Wang Abstract: We extend a theorem of Jörgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampère equation to parabolic Monge–Ampère equation, and obtain delicate asymptotic behavior of solutions at infinity. For the dimension \(n\ge 3\) , the work of Gutiérrez and Huang in Indiana Univ. Math. J. 47, 1459–1480 (1998) is an easy consequence of our result. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations. PubDate: 2018-05-09 DOI: 10.1007/s00526-018-1363-5 Issue No:Vol. 57, No. 3 (2018)

Authors:Colette De Coster; Antonio J. Fernández Abstract: We consider the boundary value problem where \(\Omega \subset \mathbb {R}^{N}\) , \(N \ge 2\) , is a bounded domain with smooth boundary. We assume \(c,\, h \in L^q(\Omega )\) for some \(q > \max \{N/p,1\}\) with \(c \gneqq 0\) and \(\mu \in L^{\infty }(\Omega )\) . We prove existence and uniqueness results in the coercive case \( \lambda \le 0\) and existence and multiplicity results in the non-coercive case \( \lambda >0\) . Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case. PubDate: 2018-05-05 DOI: 10.1007/s00526-018-1346-6 Issue No:Vol. 57, No. 3 (2018)

Authors:P. Bousquet; L. Brasco; C. Leone; A. Verde Abstract: We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every \(p\ge 2\) and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces. PubDate: 2018-05-05 DOI: 10.1007/s00526-018-1349-3 Issue No:Vol. 57, No. 3 (2018)

Authors:Suting Wei; Bin Xu; Jun Yang Abstract: We consider the problem $$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$ where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, the exponent p is greater than 1, \(\epsilon >0\) is a small parameter, V is a uniformly positive, smooth potential on \(\bar{\Omega }\) , and \(\nu \) denotes the outward unit normal of \(\partial \Omega \) . Let \(\Gamma \) be a curve intersecting orthogonally \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Moreover, \(\Gamma \) satisfies stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma }V^{\sigma }\) , where \(\sigma =\frac{p+1}{p-1}-\frac{1}{2}\) . We prove the existence of a solution \(u_\epsilon \) concentrating along the whole of \(\Gamma \) , exponentially small in \(\epsilon \) at any fixed distance from it, provided that \(\epsilon \) is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004). PubDate: 2018-05-04 DOI: 10.1007/s00526-018-1347-5 Issue No:Vol. 57, No. 3 (2018)

Authors:S. Aiyappan; A. K. Nandakumaran; Ravi Prakash Abstract: Unfolding operators have been introduced and used to study homogenization problems. Initially, they were introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. In this article, we develop new unfolding operators, where the oscillations can be smooth and hence they have wider applications. We have demonstrated by developing unfolding operators for circular domains with rapid oscillations with high amplitude of O(1) to study the homogenization of an elliptic problem. PubDate: 2018-05-04 DOI: 10.1007/s00526-018-1354-6 Issue No:Vol. 57, No. 3 (2018)

Authors:Jürgen Jost; Ruijun Wu; Miaomiao Zhu Abstract: We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler–Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By assuming some higher integrability of the vector spinor, we can show a partial regularity result for stationary solutions, provided the gravitino is critical, which means that the corresponding supercurrent vanishes. Moreover, in dimension \(<6\) , partial regularity holds for stationary solutions with respect to general gravitino fields. PubDate: 2018-05-03 DOI: 10.1007/s00526-018-1366-2 Issue No:Vol. 57, No. 3 (2018)

Authors:Jungang Li; Guozhen Lu; Maochun Zhu Abstract: Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\) . We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\) . Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\) . Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\) : $$\begin{aligned} -\mathrm {div}\left( \left \nabla _{\mathbb {H}}u\right ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left u\right ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$ with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \) . All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument. PubDate: 2018-05-02 DOI: 10.1007/s00526-018-1352-8 Issue No:Vol. 57, No. 3 (2018)

Authors:João Vítor da Silva; Ariel M. Salort Abstract: In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type ( \(1< p< \infty \) ) with strong absorption condition: $$\begin{aligned} -\mathrm {div}(\Phi (x, u, \nabla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ where \(\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N\) is a vector field with an appropriate p-structure, \(\lambda _0\) is a non-negative and bounded function and \(0\le q<p-1\) . Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved \(C^{\gamma }\) regularity estimates along free boundary points, namely \(\mathfrak {F}_0(u, \Omega ) = \partial \{u>0\} \cap \Omega \) , where the regularity exponent is given explicitly by \(\gamma = \frac{p}{p-1-q} \gg 1\) . Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of \((N-1)\) -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator \(-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0\) for any \(\lambda _0>0\) . PubDate: 2018-04-30 DOI: 10.1007/s00526-018-1344-8 Issue No:Vol. 57, No. 3 (2018)

Authors:Alessio Brancolini; Benedikt Wirth Abstract: A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form. PubDate: 2018-04-30 DOI: 10.1007/s00526-018-1364-4 Issue No:Vol. 57, No. 3 (2018)

Authors:Kentaro Nagahara; Eiji Yanagida Abstract: This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population. PubDate: 2018-04-28 DOI: 10.1007/s00526-018-1353-7 Issue No:Vol. 57, No. 3 (2018)

Authors:Siao-Hao Guo Abstract: Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\) , and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\) , we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\) , where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \) ) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity. PubDate: 2018-04-28 DOI: 10.1007/s00526-018-1356-4 Issue No:Vol. 57, No. 3 (2018)

Authors:Inwon Kim; Alpár Richárd Mészáros Abstract: We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, we find a stable initial configuration which allows the densities to be segregated. This leads to the evolution of a stable interface between the two densities, and to a stronger convergence result to the continuum limit. In particular derivation of a standard weak solution to the system is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow. PubDate: 2018-04-28 DOI: 10.1007/s00526-018-1351-9 Issue No:Vol. 57, No. 3 (2018)

Authors:Woocheol Choi; Younghun Hong; Jinmyoung Seok Abstract: We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann (Anal PDE 2:1–27, 2009) using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work (Choi et al. 2017. arXiv:1705.09068) to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit. PubDate: 2018-04-27 DOI: 10.1007/s00526-018-1362-6 Issue No:Vol. 57, No. 3 (2018)

Authors:Shane Cooper Abstract: The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta. PubDate: 2018-04-27 DOI: 10.1007/s00526-018-1365-3 Issue No:Vol. 57, No. 3 (2018)

Authors:Hiroyoshi Mitake; Kohei Soga Abstract: Weak KAM theory for discounted Hamilton–Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of \(\alpha \) -limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of \(\alpha \) -limit points is effectively exploited with properties of the corresponding dynamical systems. PubDate: 2018-04-27 DOI: 10.1007/s00526-018-1359-1 Issue No:Vol. 57, No. 3 (2018)

Authors:Roberto Monti; Alessandro Pigati; Davide Vittone Abstract: We show that length minimizing curves in Carnot–Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. PubDate: 2018-04-27 DOI: 10.1007/s00526-018-1361-7 Issue No:Vol. 57, No. 3 (2018)

Authors:Karthik Adimurthi; Cong Phuc Nguyen Abstract: We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = \nabla u ^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\) . Here \(\Delta _p\) , \(p>1\) , is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, ( \nabla u ^{p-2}\nabla u)\) , and the datum \(\sigma \) is a signed distribution in \(\Omega \) . The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \( \nabla u \in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\) , a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that $$\begin{aligned} \int _{\Omega } f ^{p} \varphi ^p dx \le C \int _{\Omega } ( \nabla \varphi ^p + \varphi ^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$ for some \(C>0\) . This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \) . We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \( F ^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\) . As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\) , we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\) , \(v\ge 0\) in \(\Omega \) , and \(v=1\) on \(\partial \Omega \) , which is interesting in its own right. PubDate: 2018-04-27 DOI: 10.1007/s00526-018-1357-3 Issue No:Vol. 57, No. 3 (2018)