Concentration of solutions for non-autonomous double-phase problems with lack of compactness (Q6597373)

This article is concerned with the study of the problem \(-\operatorname{div}(|\nabla u|^{p-2}\nabla u+\mu_\varepsilon(x)|\nabla u|^{q-2}\nabla u)+V_\varepsilon (x)(|u|^{p-2}u+\mu_\varepsilon(x)|u|^{q-2} u)=f(u)\) in \(\mathbb{R}^N\), where \(N\geq 2\), \(1<p<q<N\), \(q<\frac{Np}{N-p}\), \(\mu:\mathbb{R}^N\to \mathbb{R}\) is continus, \(\mu_\varepsilon(x)=\mu(\varepsilon x)\). \N\NThe purpose of the article is two-fold. First, by using the Lusternik-Schnirelmann theory, the authors establish the existence of multiple non-negative solutions. The second aim of the article is to derive concentration properties of solutions. The approach is variational and combines penalization methods with concentration phenomena and properties of Musielak-Orlicz-Sobolev spaces.