Double phase problems with competing potentials: concentration and multiplication of ground states

Abstract

In this paper, we establish concentration and multiplicity properties of ground state solutions to the following perturbed double phase problem with competing potentials:

$$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{p}\Delta _{p} u-\epsilon ^{q}\Delta _{q} u +V(x)(|u|^{p-2}u+|u|^{q-2}u)=K(x)f(u),&{} \quad \hbox {in}~\mathbb {R}^{N},\\ u\in W^{1,p}(\mathbb {R}^{N})\cap W^{1,q}(\mathbb {R}^{N}),\ u>0, &{} \quad \hbox {in}~\mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$

where $1<p<q<N$, $\Delta _{s}u=\hbox {div}(|\nabla u|^{s-2}\nabla u)$, with $s\in \{p,q\}$, is the s-Laplacian operator, and $\epsilon$ is a small positive parameter. We assume that the potentials V, K and the nonlinearity f are continuous but are not necessarily of class $C^{1}$. Under some natural hypotheses, using topological and variational tools from Nehari manifold analysis and Ljusternik–Schnirelmann category theory, we study the existence of positive ground state solutions and the relation between the number of positive solutions and the topology of the set where V attains its global minimum and K attains its global maximum. Moreover, we determine two concrete sets related to the potentials V and K as the concentration positions and we describe the concentration of ground state solutions as $\epsilon \rightarrow 0$. The asymptotic convergence and the exponential decay of positive solutions are also explored. Finally, we establish a sufficient condition for the non-existence of ground state solutions.