Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti-Rabinowitz condition in $\mathbb{R}^{N}$

TL;DR

This paper establishes the existence of multiple solutions for a class of variable exponent elliptic problems in rica without relying on the traditional Ambrosetti-Rabinowitz growth condition, using new growth assumptions and critical point theory.

Contribution

It introduces new growth conditions compatible with variable exponents and verifies the Cerami compactness condition to prove multiple solutions without the Ambrosetti-Rabinowitz condition.

Findings

Proved existence of multiple solutions under new growth conditions.

Characterized asymptotic behaviors of solutions.

Verified Cerami condition for the problem.

Abstract

We investigate the existence and multiplicity of solutions to the following $p(x)$-Laplacian problem in $\mathbb{R}^{N}$ via critical point theory \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u),\text{ in } \mathbb{R}^{N}, \\ u\in W^{1,p(\cdot )}(\mathbb{R}^{N}). \end{array} \right. \end{equation*} We propose a new set of growth conditions which matches the variable exponent nature of the problem. Under this new set of assumptions, we manage to verify the Cerami compactness condition. Therefore, we succeed in proving the existence of multiple solutions to the above problem without the well-known Ambrosetti--Rabinowitz type growth condition. Meanwhile, we could also characterize the pointwise asymptotic behaviors of these solutions. In our main argument, the idea of localization, decomposition of the domain, regularity of weak solutions and comparison principle are crucial ingredients among others.