Existence of nontrivial weak solutions for a quasilinear Choquard equation

Jongrak Lee,Jung-Hyun Bae,Jae-Myoung Kim,Kisoeb Park

doi:10.1186/s13660-018-1632-z

Abstract

We are concerned with the following quasilinear Choquard equation: −Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫0tf(s)ds,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta_{p} u+V(x)|u|^{p-2}u=\\lambda\\bigl(I_{\\alpha} \\ast F(u)\\bigr)f(u) \\quad \\text{in } \\mathbb {R}^{N}, \\qquad F(t)= \\int_{0}^{t}f(s) \\,ds, $$\\end{document} where 1< p<infty, Delta_{p} u=nablacdot(|nabla u|^{p-2}nabla u) is the p-Laplacian operator, the potential function V:mathbb {R}^{N}to(0,infty) is continuous and F in C^{1}(mathbb {R}, mathbb {R}). Here, I_{alpha}: {mathbb {R}}^{N}rightarrow {mathbb {R}} is the Riesz potential of order alphain(0,p). We study the existence of weak solutions for the problem above via the mountain pass theorem and the fountain theorem. Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f.

Highlights

We are concerned with the following quasilinear Choquard equation:– pu + V (x)|u|p–2u = λ Iα ∗ F(u) f (u) in RN, (P)where 1 < p < N, pu = ∇ · (|∇u|p–2∇u) is the p-Laplacian operator, the potential functionV : RN → (0, ∞) is continuous and F ∈ C1(R, R) with F(t) = t 0 f (s) ds
We prove the existence of weak solutions for the quasilinear Choquard equation (P) under the Cerami condition, as a weak version of the Palais–Smale condition
Using Lemma 3.2, we prove the existence of a nontrivial weak solution for our problem under the assumptions

Summary

Introduction

We are concerned with the following quasilinear Choquard equation:. where 1 < p < N , pu = ∇ · (|∇u|p–2∇u) is the p-Laplacian operator, the potential function. Under the assumption (V), it is obvious that the functional is well defined on X, ∈ C1(X, R) and its Fréchet derivative is given by (u), v = |∇u|p–2∇u · ∇v dx + V (x)|u|p–2uv dx. This plays a key role in obtaining the existence of a nontrivial weak solution for the given problem. 3.1 Existence of weak solutions: approach to the mountain pass theorem We give the following result to show that the energy functional Iλ satisfies the geometric conditions of the mountain pass theorem based on the idea of Lemma 3.2 in [30]. Using Lemma 3.2, we prove the existence of a nontrivial weak solution for our problem under the assumptions.

Existence of a sequence of weak solutions: approach to the fountain theorem

Existence of a sequence of weak solutions converging to zero

Conclusion