Critical double phase problems involving sandwich-type nonlinearities

Abstract

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= λw(x)|u|^{s-2}u+θB\left(x,u\right) \quad \text{in } Ω,\quad u= 0 \quad\text{on } \partial Ω, \end{align*} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary $\partial \hat{I}\text{c◯}$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\le a(\cdot )\in C^{0,1}(\overline{\hat{I}}\text{c◯})$, $\hat{I}»$, $\hat{I}¸$ are real parameters, $w$ is a suitable weight and $B:\overline{\hat{I}}\text{c◯}\times \mathbb{R}\to \mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^{\ast }:=Nr/(N-r)$ for $r\in \{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot ,\cdot )$ and a sandwich-type perturbation with exponent $s\in (p,q)$. Under different values of the parameters $\hat{I}»$ and $\hat{I}¸$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot )$, $b_0(\cdot )$ and $b(\cdot )$.