On existence and multiplicity of solutions for generalized (p, q)-Laplacian equations on unbounded domains

Abstract

This paper deals with the existence and multiplicity of solutions for the generalized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x, u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x, u)|\nabla u|^{q-2}\nabla u) \\ &\quad\qquad+\frac1q B_t(x, u)|\nabla u|^q + V(x)|u|^{p-2} u+ W(x)|u|^{q-2} u= g(x, u)\quad\qquad\mbox{ in } \mathbb{R}^N, \end{align*} where $1<q\le p< N$, $A, B:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ are suitable $C^1$ Carath\'eodory functions with $A_t(x, u)=\frac{\partial A}{\partial t}(x, u), B_t(x, u)=\frac{\partial B}{\partial t}(x, u)$, $V, W:\mathbb{R}^N\to\mathbb{R}$ are proper ``weight functions" and $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a Carath\'eodory map. Notwithstanding the occurrence of some coefficients which rely upon the solution itself makes the use of variational techniques more challenging, under suitable assumptions on the involved functions, we are able to exploit the variational nature of our problem. In particular, the existence of a nontrivial solution is derived via a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, based on a weaker version of the classical Cerami-Palais-Smale condition. Finally, the multiplicity result, which is thoroughly new also even in the simpler case $q=p$, is gained under symmetry assumptions and a sharp decomposition of the ambient space.