Abstract
In this paper, we focus on the existence of solutions for the Choquard equation {−Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+λ|u|p−2u,x∈RN;u∈H1(RN),\\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} $$\\begin{aligned} \\textstyle\\begin{cases} {-}\\Delta {u}+V(x)u=(I_{\\alpha }* \\vert u \\vert ^{\\frac{\\alpha }{N}+1}) \\vert u \\vert ^{ \\frac{\\alpha }{N}-1}u+\\lambda \\vert u \\vert ^{p-2}u,\\quad x\\in \\mathbb{R}^{N}; \\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where lambda >0 is a parameter, alpha in (0,N), Nge 3, I_{alpha }: mathbb{R}^{N}to mathbb{R} is the Riesz potential. As usual, alpha /N+1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if lambda >lambda _{*} for some given number lambda _{*} in three cases: (i) 2< p<frac{4}{N}+2, (ii) p=frac{4}{N}+2, and (iii) frac{4}{N}+2< p<2^{*}. Our result improves the previous related ones in the literature.
Highlights
In this paper, we mainly study the Choquard equation with a variable potential and a local nonlinearity: ⎧ ⎨– V (x)u λ|u|p–2u,⎩u ∈ H1(RN ), x ∈ RN ; (1.1)where λ > 0, α ∈ (0, N), N ≥ 3, 2 < p < 2∗, and Iα : RN → R is the Riesz potential defined by Iα(x) = ( N–α )
In this paper, we focus on the existence of solutions for the Choquard equation
By using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if λ > λ∗ for some given number λ∗
Summary
Under (V1), (V2), (1.2), and the Sobolev embedding theorem, the weak solutions of (1.1) correspond to the critical points of the energy functional I : H1(RN ) → R defined by Wei, and Chen [25] proved that (1.10) has ground state solutions in the following assumptions: (i) 2 < p < N4 + 2 and λ > 0; (ii) p = 4 + 2 and λ > Motivated by the above works, especially [25, 26], in this paper, we establish the existence result of ground state solutions for (1.1).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have