Localized concentration of semiclassical solutions for double phase problems with nonlocal reaction

Source abstract

Abstract This paper focuses on the study of multiplicity and localized concentration properties of positive solutions for the following singularly perturbed double phase problem with nonlocal Choquard reaction $$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{p}\Delta _{p} u-\epsilon ^{q}\Delta _{q} u +V(x)(|u|^{p-2}u+|u|^{q-2}u)\\ \quad =\epsilon ^{\mu -N}\left( \frac{1}{|x|^{\mu }}*G(u)\right) g(u),& \hbox {in}~\mathbb {R}^{N},\\ u\in W^{1,p}(\mathbb {R}^{N})\cap W^{1,q}(\mathbb {R}^{N}),u>0, & \hbox {in}~\mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$ - ϵ p Δ p u - ϵ q Δ q u + V ( x ) ( | u | p - 2 u + | u | q - 2 u ) = ϵ μ - N 1 | x | μ ∗ G ( u ) g ( u ) , in R N , u ∈ W 1 , p ( R N ) ∩ W 1 , q ( R N ) , u &gt; 0 , in R N , where $$1< p<q<N$$ 1 &lt; p &lt; q &lt; N , $$0<\mu <p$$ 0 &lt; μ &lt; p , $$\epsilon$$ ϵ is a small positive parameter and V is the absorption potential. We assume that the potential V satisfies only a local condition introduced by del Pino and Felmer. Applying suitable variational and topological methods combined with penalization technique, we obtain multiple semiclassical positive solutions for $$\epsilon >0$$ ϵ &gt; 0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum. Moreover, we also investigate the decay property of semiclassical positive solutions. The main results included in this paper complement several recent contributions to the study of concentration phenomena.