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

The present paper is devoted to the study of the following double-phase equation \(\begin{aligned} -\text {div}(|\nabla u|^{p-2}\nabla u+\mu _{\varepsilon }(x)|\nabla u|^{q-2}\nabla u)+V_{\varepsilon }(x)(|u|^{p-2}u+\mu _{\varepsilon }(x)|u|^{q-2}u)=f(u)\quad \text{ in }\quad \mathbb {R}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -{\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u+{\mu }_{\epsilon }{(x)|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u)+V_{\epsilon }{(x)(|u|}^{p-2}u+{\mu }_{\epsilon }{(x)|u|}^{q-2}u)=f(u)\text{in}{\mathbb{R}}^N,\end{array}$$ where \(N\ge 2\) $N\ge 2$ , \(1<p<q<N\) $1<p<q<N$ , \(q<p^{*}\) $q<p_{}^{\ast }$ with \(p^{*}=\frac{Np}{N-p}\) $p_{}^{\ast }=\frac{\mathit{Np}}{N-p}$ , \(\mu :\mathbb {R}^{N}\rightarrow \mathbb {R}\) $\mu :{\mathbb{R}}^N\to \mathbb{R}$ is a continuous non-negative function, \(\mu _{\varepsilon }(x)=\mu (\varepsilon x)\) ${\mu }_{\epsilon }(x)=\mu (\epsilon x)$ , \(V:\mathbb {R}^{N}\rightarrow \mathbb {R}\) $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive potential satisfying a local minimum condition, \(V_{{{\,\mathrm{\varepsilon }\,}}}(x)=V({{\,\mathrm{\varepsilon }\,}}x)\) $V_{\epsilon }(x)=V(\epsilon x)$ , and the nonlinearity \(f:\mathbb {R}\rightarrow \mathbb {R}\) $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with subcritical growth. Under natural assumptions on \(\mu \) $\mu$ , 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.