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

Abstract

We investigate the existence and the multiplicity of solutions of the problem $$ \begin{cases} -Δ_p u-Δ_q u = g(x, u)\quad & \mbox{in } Ω,\\ \displaystyle{u=0} & \mbox{on } \partialΩ, \end{cases} $$ 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.

arXiv comment19 pages, 0 figures. In version v1 a mistake is corrected, leading to a different statement of Theorem 1.1-(H_+)