Strauss's radial compactness and nonlinear elliptic equation involving a variable critical exponent
type: Item
instance of: scholarly article
title: Strauss's radial compactness and nonlinear elliptic equation involving a variable critical exponent
zbMATH Open document ID: 1405.35065
DOI: 10.1155/2018/5497172
author: Masato Hashizume
author: Megumi Sano
published in: Journal of Function Spaces
publication date: 2018-11-15T00:00:00Z
review text: Summary: We study existence of a nontrivial solution of \(-\Delta_p u(x) + u(x)^{p - 1} = u(x)^{q(x) - 1}\), \(u(x) \geq 0\), \(x \in \mathbb{R}^N\), \(u \in W_{\operatorname{rad}}^{1, p}(\mathbb{R}^N)\), under some conditions on \(q(x)\), especially, \(\liminf_{| x | \rightarrow \infty} q(x) = p\). Concerning this problem, we firstly consider compactness and noncompactness for the embedding from \(W_{\operatorname{rad}}^{1, p}(\mathbb{R}^N)\) to \(L^{q(x)}(\mathbb{R}^N)\). We point out that the decaying speed of \(q(x)\) at infinity plays an essential role on the compactness. Secondly, by applying the compactness result, we show the existence of a nontrivial solution of the elliptic equation.
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 6979876
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: existence of solutions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/5497172
OpenAlex ID: W2891813378
cites work: Multiple solutions for a class of quasilinear problems involving variable exponents
cites work: A multiplicity result for \(p(x)\)-Laplacian problem in \(\mathbb R^N\)
cites work: Existence of solutions for a class of \(p(x)\)-Laplacian equations involving a concave-convex nonlinearity with critical growth in \({\mathbb R}^N\)
cites work: Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb R^N\)
cites work: \(p(x)\)-Laplacian equations in \(\mathbb R^N\) with periodic data and nonperiodic perturbations
cites work: A minimization problem with variable growth on Nehari manifold
cites work: Existence of Solutions for a Class of Problems in IR N Involving the p(x)-Laplacian
cites work: Local existence conditions for an equations involving the \(p(x)\)-Laplacian with critical exponent in \(\mathbb R^N\)
cites work: Q4021390
cites work: On the Sobolev-type inequality for Lebesgue spaces with a variable exponent
cites work: Sobolev-type inequality for spaces L^{p(x)}(\mathbb{R}^{N})
cites work: Sobolev embeddings with variable exponent
cites work: Q4797974
cites work: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spacesLp(·) andWk,p(·)
cites work: The fractional maximal operator and fractional integrals on variable \(L^p\) spaces
cites work: Sobolev inequalities with variable exponent attaining the values 1 and \(n\)
cites work: Compact embedding from \(W_0^{1,2}(\Omega)\) to \(L^{q(x)}(\Omega )\) and its application to nonlinear elliptic boundary value problem with variable critical exponent
cites work: Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\)
cites work: Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-Laplacian and its critical exponent
cites work: Variable Lebesgue spaces. Foundations and harmonic analysis
cites work: On the Sobolev embedding theorem for variable exponent spaces in the critical range
cites work: On the Sobolev trace theorem for variable exponent spaces in the critical range
cites work: Existence of solitary waves in higher dimensions
cites work: Symétrie et compacité dans les espaces de Sobolev
cites work: On compact embeddings of radial sobolev spaces and their applications
cites work: On the (non)compactness of the radial Sobolev spaces
cites work: Compact imbedding theorems with symmetry of Strauss-Lions type for the space \(W^{1,p(x)}(\Omega)\)
cites work: C1 + α local regularity of weak solutions of degenerate elliptic equations
cites work: The maximum principle
cites work: On the Ambrosetti-Rabinowitz Superlinear Condition
cites work: Q3876715
cites work: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity
cites work: Functional analysis, Sobolev spaces and partial differential equations
cites work: Q3749777
cites work: Variational Methods
cites work: Dual variational methods in critical point theory and applications
description: scientific article; zbMATH DE number 6979876