Fractional (p, q)-Schrödinger Equations with Critical and Supercritical Growth

Abstract

In this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: (-Δ)psu+(-Δ)qsu+V(εx)(up-1+uq-1)=f(u)+uqs∗-1inRN,u∈Ws,p(RN)∩Ws,q(RN),u>0inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} (-\\Delta )^{s}_{p}u+(-\\Delta )^{s}_{q}u + V(\\varepsilon x) (u^{p-1} + u^{q-1})= f(u)+u^{q^{*}_{s}-1} \\, \ ext{ in } \\mathbb {R}^{N}, \\\\ u\\in W^{s, p}(\\mathbb {R}^{N})\\cap W^{s,q}(\\mathbb {R}^{N}), \\, u>0 \ ext{ in } \\mathbb {R}^{N}, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon >0 is a small parameter, sin (0, 1), 1<p<q<frac{N}{s}, q^{*}_{s}=frac{Nq}{N-sq} is the fractional critical Sobolev exponent, (-Delta )^{s}_{r}, with rin {p, q}, is the fractional r-Laplacian operator, V:mathbb {R}^{N}rightarrow mathbb {R} is a positive continuous potential such that inf _{partial Lambda }V>inf _{Lambda } V for some bounded open set Lambda subset mathbb {R}^{N}, and f:mathbb {R}rightarrow mathbb {R} is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f(t)=t^{gamma -1}+mu t^{tau -1}, with 1< p<q<gamma<q^{*}_{s}<tau and mu >0 sufficiently small, by combining a truncation argument with a Moser-type iteration.