On quasilinear elliptic problems with finite or infinite potential wells

TL;DR

This paper studies quasilinear elliptic equations with variable potential wells, establishing existence and multiplicity of solutions using variational methods, especially when the potential well is finite or infinite, and analyzing the effect of a steep potential well parameter.

Contribution

It introduces new existence results for solutions to quasilinear elliptic problems with variable potential wells, including cases with infinite potential and steep well parameters.

Findings

Infinitely many solutions when potential tends to infinity.

Existence of solutions for large steep potential well parameter.

Compact embedding results for unbounded domains.

Abstract

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,\Phi }(\mathbb{R}^{N})$ into $L^{\Phi}(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda a(x) + 1$ exhibits a steep potential well controlled by a positive parameter $\lambda$, we get nontrivial solutions for large $\lambda$.