On $p(x)$-Laplacian equations in $\mathbb{R}^{N}$ with nonlinearity sublinear at zero

TL;DR

This paper studies $p(x)$-Laplacian equations with variable exponents in $ R^N$, establishing existence of solutions and sequences of solutions tending to zero and infinity using variational methods and compact embeddings.

Contribution

It introduces a suitable subspace for variational analysis of $p(x)$-Laplacian problems and proves compact embedding results, leading to existence and multiplicity of solutions.

Findings

Existence of nontrivial solutions established.

Two sequences of solutions tending to zero and infinity identified.

Compact embedding of the subspace into $L^{q(x)}$ proved.

Abstract

Let $p,q$ be functions on $\mathbb{R}^{N}$ satisfying $1\ll q\ll p\ll N$, we consider $p(x)$-Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert ^{q(x)-2}u+g(x,u)\text{,}\\ u\in W^{1,p(x)}(\mathbb{R}^{N})\text{.}% \end{array} \right. \] To apply variational methods, we introduce a subspace $X$ of $W^{1,p(x)}(\mathbb{R}^N)$ as our working space. Compact embedding from $X$ into $L^{q(x)}(\mathbb{R}^N)$ is proved, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to $\infty$ and $0$ respectively, when $g(x,\cdot)$ is odd.