Multiple solutions for asymptotically $q$-linear $(p,q)$-Laplacian problems

TL;DR

This paper studies the existence of multiple solutions for a class of nonlinear elliptic equations involving the $(p,q)$-Laplacian with asymptotic $q$-linear growth, using variational methods and critical point theory.

Contribution

It introduces new conditions for multiple solutions of $(p,q)$-Laplacian problems with asymptotically $q$-linear nonlinearities, including the resonant case.

Findings

Multiple solutions are obtained via minimax critical points.

The approach covers resonant and non-resonant cases.

Variational methods effectively handle asymptotic $q$-linear growth.

Abstract

We investigate the existence and the multiplicity of solutions of the problem $$ \begin{cases} -\Delta_p u-\Delta_q u = g(x, u)\quad & \mbox{in } \Omega,\\ \displaystyle{u=0} & \mbox{on } \partial\Omega, \end{cases} $$ where $\Omega$ is a smooth, bounded domain of $\mathbb R^N$, $1<p<q<\infty$, and the nonlinearity $g$ behaves as $u^{q-1}$ at infinity. We use variational methods and find multiple solutions as minimax critical points of the associated energy functional. Under suitable assumptions on the nonlinearity, we cover also the resonant case.