Journal of Elliptic and Parabolic Equations
Number of Followers: 0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 22969020  ISSN (Online) 22969039 Published by SpringerVerlag [2468 journals] 
 Decay estimates for quasilinear elliptic equations and a
Brezis–Nirenberg result in $$D^{1,p}({\mathbb R}^N)$$
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Abstract: Abstract We prove decay estimates for solutions of quasilinear elliptic equations in the whole \({\mathbb R}^N\) of the form $$\begin{aligned} u\in X: \text{ div }\,A(x,\nabla u)=a(x) f(x,u), \end{aligned}$$ where \(X=D^{1,p}({\mathbb R}^N)\) is the BeppoLevi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional \(\Phi : X\rightarrow {\mathbb R}\) related to the pLaplacian equation in \({\mathbb R}^N\) in the form $$\begin{aligned} u\in X: \Delta _p u=a(x) g(u), \end{aligned}$$ saying that for the subspace V of bounded continuous functions with weight \(1+ x ^{\frac{Np}{p}},\) a local minimizer of \(\Phi \) in the finer V topology is also a local minimizer in the Xtopology. Global \(L^\infty \) estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.
PubDate: 20240221

 Positive solutions, positive radial solutions and uniqueness results for
some nonlocal elliptic problems
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Abstract: Abstract In this paper, we prove existence of a positive solution to a onedimensional nonlocal elliptic problem and existence of a positive radial solution to a multidimensional nonlocal elliptic problem under weak conditions on the reaction terms and the diffusion coefficients. We use Krasnoselskii’s fixed point theorem. Uniqueness results are also given.
PubDate: 20240129

 Entropy solutions for some elliptic anisotropic problems involving
variable exponent with Fourier boundary conditions and measure data
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Abstract: Abstract This paper is devoted to the study of some nonlinear elliptic anisotropic Fourier boundaryvalue problems, whose prototype is given by $$\begin{aligned} {\left\{ \begin{array}{ll}  Au + g(x,u,\nabla u) +\delta \vert u\vert ^{p_0(x)2}u = \mu \text{ div }\phi (u)\ \hspace{0.2cm} \ \text{ in } \ \ \Omega ,\\ \ \displaystyle \ Bu+\lambda u=h\ \hspace{3.1cm} \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where the right hand side \(\mu \) belongs to \(L^1(\Omega ) + W^{1,\vec {p}\,'(x)}(\overline{\Omega })\) , the operator Au is a LerayLions anisotropic operator and \(\phi \in {\mathcal {C}}^0({\mathbb {R}}, {\mathbb {R}}^{N})\) , the nonlinear term \(g: \Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N}\longrightarrow {\mathbb {R}}\) satisfying some growth condition but no sign condition. We provide an existence result of entropy solutions for this class of anisotropic problems.
PubDate: 20240109

 A variational approach for some singular elliptic problems with Hardy
potential
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Abstract: Abstract In this paper, we are interested in applying a three critical points theorem by B. Ricceri, to establish the existence of three weak solutions for some singular elliptic problems involving a pLaplace operator, subject to Dirichlet boundary conditions in a smooth bounded domain in \(\mathbb {R}^N.\)
PubDate: 20240105

 Existence and multiplicity of nontrivial solutions for fractional
Schrödinger–Poisson systems with a combined nonlinearity
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Abstract: Abstract In this paper, we are concerned with the following fractional Schrödinger–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle (\Delta )^s u + V(x)u +\lambda \phi u=K(x){ u }^ {q2}u+ f(x,u), &{}\qquad \ x \in \mathbb {R}^3 \\ \displaystyle (\Delta )^t\phi = u^2,&{} \qquad \ x \in \mathbb {R}^3\\ \end{array}\right. \end{aligned}$$ where \(\lambda >0\) is a constant, \(s,t \in (0,1]\) , \(2t+4s>3\) , \(1<q<2\) and f(x, u) is linearly bounded in u at infinity. With some assumptions on K, V and f we get the existence and multiplicity of nontrivial solutions with the help of the variational methods.
PubDate: 20240103

 Three solutions to a Neumann boundary value problem driven by
p(x)biharmonic operator
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Abstract: Abstract In this article, we establish the existence of at least three distinct weak solutions for a specific class of quasilinear elliptic equations. These equations incorporate the p(x)biharmonic operator and are constrained by Neumann boundary conditions. Our technical approach is primarily founded on Ricceri’s three critical points theorem (Nonlinear Anal 70:3084–3089, 2009). In addition, we give an example to show our key findings.
PubDate: 20240102

 Entropy solutions for some noncoercive quasilinear p(x)parabolic
equations with L1data
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Abstract: Abstract In this paper, we consider the following quasilinear noncoercive \(p(\cdot )\) parabolic problem $$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t} \text{ div }\, a(x,t,u,\nabla u) + u ^{p(x)2}u = f &{}\quad \text{ in } \ Q_T, \\ u = 0&{}\quad \text{ on } \ \Sigma _T,\\ u(x,0)= u_0 &{}\quad \text{ in } \ \Omega , \end{array}\right. \end{aligned}$$ with \( f\in L^{1}(Q_T) \) and \( u_0 \in L^1(\Omega ). \) We show the existence of solutions in the parabolic space with variable exponent V by using approximate method and some a priori estimation technics. Moreover, we show that these solutions are also renormalized solutions for our quasilinear p(x)parabolic problem. The uniqueness of solution will be concluded by using some additional assumption on the Carathéodory function \(a(x,t,s,\xi ).\)
PubDate: 20231227

 Chaotic behavior and generation theorems of conformable time and space
partial differential equations in specific Lebesgue spaces
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Abstract: Abstract This work aims to investigate the chaotic behavior of conformable partial differential equations. The main focus is on a specific conformable partial differential equation involving both time and space derivatives. To achieve this objective, the study begins by developing the theory of conformable Sobolev spaces, which provides a suitable framework for analyzing the operator associated with the proposed conformable partial differential equation. The investigation further utilizes the concept of conformable semigroups to establish a generation theorem for the solutions of the conformable partial differential equation. Drawing inspiration from previous research on chaos in semigroups, the work introduces a characterization of chaos specific to conformable equations. This characterization allows for a thorough analysis of the chaotic behavior associated with the semigroup generated by the proposed model.
PubDate: 20231222

 Pseudo asymptotically Bloch periodic functions: applications for some
models with piecewise constant argument
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Abstract: Abstract In this paper, we show the existence and uniqueness of a pseudo asymptotically Bloch periodic solution for two equation models with piecewise constant argument, one contains a generator of a semi group and the second a generator of an evolutionary process. The concluding part of the work is crowned with two examples to confirm the reliability and feasibility.
PubDate: 20231218

 Local integrability of $$G(\cdot )$$ superharmonic functions in Lebesgue
and Musielak–Orlicz spaces
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Abstract: Abstract In this article, we study local integrability properties of superharmonic functions related to partial differential equations with MusielakOrlicz growth conditions in Lebesgue and MusielakOrlicz spaces.
PubDate: 20231210

 Breathers and rogue waves for semilinear curlcurl wave equations

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Abstract: Abstract We consider localized solutions of variants of the semilinear curlcurl wave equation \(s(x) \partial _t^2 U +\nabla \times \nabla \times U + q(x) U \pm V(x) \vert U \vert ^{p1} U = 0\) for \((x,t)\in {\mathbb {R}}^3\times {\mathbb {R}}\) and arbitrary \(p>1\) . Depending on the coefficients s, q, V we can prove the existence of three types of localized solutions: timeperiodic solutions decaying to 0 at spatial infinity, timeperiodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to 0 both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curloperator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution U(x, t) already generates a full continuum of phaseshifted solutions \(U(x,t+b(x))\) where the continuous function \(b:{\mathbb {R}}^3\rightarrow {\mathbb {R}}\) belongs to a suitable admissible family.
PubDate: 20231201
DOI: 10.1007/s4180802300215x

 Multiple positive solutions for a fractional $$ p \& q$$ Laplacian
system with concave and critical nonlinearities
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Abstract: Abstract In this paper, we study the following nonlinear fractional p &qLaplacian system with critical exponent $$\begin{aligned} {\left\{ \begin{array}{ll}(\Delta )_p^{s_1} u+(\Delta )_q^{s_2} u=\lambda \vert u\vert ^{r2} u+\frac{2 \alpha }{\alpha +\beta }\vert u\vert ^{\alpha 2} u\vert v\vert ^{\beta }, &{} \text{ in } \Omega , \\ (\Delta )_p^{s_1} v+(\Delta )_q^{s_2} v=\mu \vert v\vert ^{r2} v+\frac{2 \beta }{\alpha +\beta }\vert u\vert ^{\alpha }\vert v\vert ^{\beta 2} v, &{} \text{ in } \Omega , \\ u=v=0, &{} \text{ in } {\mathbb {R}}^{N} \backslash \Omega ,\end{array}\right. } \end{aligned}$$ where \(\Omega \) is a smooth bounded set in \({\mathbb {R}}^{N}, \lambda , \mu >0\) are two parameters, \(0<s_{2}<s_{1}<1\) , \( 1<r<q<\frac{N(ps_1)}{Ns_1}<p<p_{s_1}^{*}\) , \(N>p s_{1}\) , \(\alpha ,\beta >1\) satisfy \(\alpha +\beta =p_{s_{1}}^{*}\) with \(p_{s_{1}}^{*}=\frac{n p}{np s_{1}}\) is the fractional Sobolev critical exponent and \((\Delta )_{t}^{s}\) is the fractional tLaplacian operator. With the help of Nehari manifold and LjusternikSchnirelmann category, we show that the above system has at least \(cat(\Omega )+1\) distinct positive solutions, where \(cat(\Omega )\) denotes the LusternikSchnirelman category of \(\Omega \) in itself.
PubDate: 20231201
DOI: 10.1007/s4180802300222y

 Solvability of the acidmediated tumor growth model with nonlinear acid
production term
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Abstract: Abstract The main goal of the paper is to establish the solvability of the 3D acidmediated tumor growth model combined with the Dirichlet and Neuman boundary conditions in the fractionalorder Sobolev spaces using the Schauder fixed point theorem. Further, the uniqueness of solutions is also proved using the duality approach and Gronwall’s lemma.
PubDate: 20231201
DOI: 10.1007/s41808023002277

 Existence of weak solution for pKirchoff type problem via topological
degree
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Abstract: Abstract In the present paper, we use the topological degree methods of Berkovits to prove the existence of weak solutions of the following pKirchhoff type problems with Dirichlet boundary condition: $$\begin{aligned}{} & {}  M\left( \int _{\Omega }\left( A(x,\nabla u)+ \frac{1}{p} \vert \nabla u\vert ^{p}\right) \,dx \right) \,\left[ \text { div}\,\left( a(x,\nabla u)\vert \nabla u \vert ^{p2}\nabla u\right) \right] \\{} & {} \quad \quad =\lambda H(x,u,\nabla u)\;\textrm{in}\quad \Omega , \end{aligned}$$ where \(\Omega \) is a smooth bounded domain of \({\mathbb {R}}^N\) ,M is a positive function and H, a are the Carathéodory’s functions that satisfy some assumptions.
PubDate: 20231201
DOI: 10.1007/s41808023002200

 A hybrid technique based on Lucas polynomials for solving fractional
diffusion partial differential equation
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Abstract: Abstract This paper presents a new numerical technique to approximate solutions of diffusion partial differential equations with Caputo fractional derivatives. We use a spectral collocation method based on Lucas polynomials for time fractional derivatives and a finite difference scheme in space. Stability and error analyses of the proposed technique are established. To demonstrate the reliability and efficiency of our new technique, we applied the method to a number of examples. The new technique is simply applicable, and the results show high efficiency in calculation and approximation precision.
PubDate: 20230918
DOI: 10.1007/s41808023002464

 Solutions of a Schrödinger–Kirchhoff–Poisson system with
concave–convex nonlinearities
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Abstract: Abstract We consider the following nonlinear Schrödinger–Kirchhoff–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle (a+b\int _{\mathbb {R}^3} \nabla u ^2)\Delta u + \phi u=\mu g(x,u)+\lambda f(x,u), \qquad &{} x \in \mathbb {R}^3,\\ \displaystyle \Delta \phi =u^2, \qquad \lim _{\vert x \vert \rightarrow \infty } \phi (x)=0, &{}x \in \mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(a,b> 0\) . Under certain assumptions, we prove the existence of infinitely many solutions with high energy by using Fountain theorem.
PubDate: 20230827
DOI: 10.1007/s41808023002437

 Multiplicity of solutions for discrete 2nth order periodic boundary value
problem
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Abstract: Abstract In this article, we investigate the existence of multiple solutions to the discrete 2nth order periodic boundary value issue by employing critical point theory.
PubDate: 20230811
DOI: 10.1007/s41808023002419

 Multiplicity of solutions for fractional $$q(\cdot )$$ laplacian
equations
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Abstract: Abstract In this paper, we deal with the following elliptictype problem $$\begin{aligned} {\left\{ \begin{array}{ll} (\Delta )_{q(\cdot )}^{s(\cdot )}u + \lambda Vu = \alpha \left u\right ^{p(\cdot )2}u+\beta \left u\right ^{k(\cdot )2}u &{} \text { in }\Omega , \\ u =0 &{} \text { in }\mathbb {R}^{n}\backslash \Omega , \end{array}\right. } \end{aligned}$$ where \(q(\cdot ):{\overline{\Omega }}\times {\overline{\Omega }}\rightarrow \mathbb {R}\) is a measurable function and \(s(\cdot ):\mathbb {R}^n\times \mathbb {R}^n\rightarrow (0,1)\) is a continuous function, \(n>q(x,y)s(x,y)\) for all \((x,y)\in \Omega \times \Omega \) , \((\Delta )_{q(\cdot )}^{s(\cdot )}\) is the variableorder fractional Laplace operator, and V is a positive continuous potential. Using the mountain pass category theorem and Ekeland’s variational principle, we obtain the existence of at least two different solutions for all \(\lambda >0\) . Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as \(\lambda \rightarrow +\infty \) .
PubDate: 20230729
DOI: 10.1007/s41808023002393

 On the local everywhere Hölder continuity for weak solutions of a class
of not convex vectorial problems of the Calculus of Variations
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Abstract: Abstract In this paper we study the regularity of the local minima of the following integral functional 0.1 $$\begin{aligned} J\left( u,\Omega \right) =\int \limits _{\Omega }\sum \limits _{\alpha =1}^{n}\left \nabla u^{\alpha }\left( x\right) \right ^{p}+G\left( x,u\left( x\right) ,\nabla u\left( x\right) \right) \,dx \end{aligned}$$ where \(\Omega \) is a open subset of \( {\mathbb {R}} ^{n}\) and \(u\in W^{1,p}\left( \Omega , {\mathbb {R}} ^{m}\right) \) with \(n\ge 2\) , \(m\ge 1\) and \(1<p<n\) . In particular, not convexity (quasiconvexity, policonvexity or rank one convexity) hypothesis will be made on the density G, neither structure hypothesis nor radial nor diagonal.
PubDate: 20230630
DOI: 10.1007/s4180802300232w

 Binonlocal sixth order p(x)problem with indefinite weight

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Abstract: Abstract This paper is concerned with the existence and the multiplicity of a nontrivial weak solutions for a class of sixth order binonlocal p(x)Kirchhoff type problem with indefinite weight under Navier boundary condition, by means of a variational arguments based on Mountain Pass Theorem and Ekland’s variational principle.
PubDate: 20230628
DOI: 10.1007/s41808023002348
