Concentration of solutions for non-autonomous double-phase problems with lack of compactness

Abstract

The present paper is devoted to the study of the following double-phase equation

$$\begin{aligned} -\text {div}(|\nabla u|^{p-2}\nabla u+\mu _{\varepsilon }(x)|\nabla u|^{q-2}\nabla u)+V_{\varepsilon }(x)(|u|^{p-2}u+\mu _{\varepsilon }(x)|u|^{q-2}u)=f(u)\quad \text{ in }\quad \mathbb {R}^{N}, \end{aligned}$$

where \(N\ge 2\), \(1<p<q<N\), \(q<p^{*}\) with \(p^{*}=\frac{Np}{N-p}\), \(\mu :\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a continuous non-negative function, \(\mu _{\varepsilon }(x)=\mu (\varepsilon x)\), \(V:\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a positive potential satisfying a local minimum condition, \(V_{{{\,\mathrm{\varepsilon }\,}}}(x)=V({{\,\mathrm{\varepsilon }\,}}x)\), and the nonlinearity \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous function with subcritical growth. Under natural assumptions on \(\mu \), V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.