Concentrating nonnegative solutions for double phase Choquard problems

Abstract

This paper deals with the multiplicity and concentration phenomenon of nonnegative solutions for the following double phase Choquard equation $$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_{\epsilon }{(\mathrm{x})(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$where $\epsilon$ is a positive parameter, $N\ge 2,$$1<p<q<N,$$q<2p,$$q<p^{\ast }$ with $p^{\ast }=\frac{\mathit{Np}}{N-p},$$0<\mu <p,$ the function $\mathcal{U}:{\mathbb{R}}^N\to \mathbb{R}$ is continuous, ${\mathcal{U}}_{\epsilon }(x)=\mathcal{U}(\epsilon x),$$V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential and satisfies a local minimum condition, $V_{\epsilon }(x)=V(\epsilon x),$$f:\mathbb{R}\to \mathbb{R}$ is a continuous subcritical nonlinearity in the sense of Hardy–Littlewood–Sobolev inequality and F is the primitive of f. Based on the variational methods and topological arguments, the connection between the multiplicity of solutions and the topological structure of the potential at the local minimum points is established.