Abstract
In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term: \t\t\t{−Δu+V(x)u+(Iα∗|u|q)|u|q−2u=f(u),x∈R3,u∈H1(R3),\\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}$$ \\textstyle\\begin{cases} -\\Delta u+V(x)u+(I_{\\alpha }\\ast \\vert u \\vert ^{q}) \\vert u \\vert ^{q-2}u=f(u), \\quad x\\in {\\mathbb{R}}^{3}, \\\\ u\\in H^{1}(\\mathbb{R}^{3}), \\quad \\end{cases} $$\\end{document} where ain (0,3), qin [1+frac{alpha }{3},3+alpha ), I_{alpha }:mathbb{R}^{3}rightarrow mathbb{R} is the Riesz potential, Vin mathcal{C}(mathbb{R}^{3},[0,infty )), fin mathcal{C}(mathbb{R},mathbb{R}) and F(t)=int _{0}^{t}f(s),ds satisfies lim_{|t|to infty }F(t)/|t|^{sigma }=infty with sigma =min {2,frac{2beta +2}{beta }} where beta =frac{ alpha +2}{2(q-1)}. By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f.
Highlights
In this paper we are concerned with the existence of ground state solutions for the nonlinear problem: ⎧⎨– u + V (x)u + (Iα ∗ |u|q)|u|q–2u = f (u), x ∈ R3, ⎩u ∈ H1(R3), (1.1)where α ∈ (0, 3), q ∈ [1 + α3, 3 + α), V ∈ C(R3, [0, ∞)), f ∈ C(R, R) and Iα : R3 → R is the Riesz potential of order α ∈ (0, 3), given byAα Iα(x) = |x|N–α Γ ( N–α ) with Aα Γ
By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f
1 Introduction In this paper we are concerned with the existence of ground state solutions for the nonlinear problem:
Summary
1 Introduction In this paper we are concerned with the existence of ground state solutions for the nonlinear problem: Let β = 2(αq+–21) , and in addition to (V1), (F1) and (F2), we need to introduce the following assumptions: (V2) V ∈ C1(R3), the set {x ∈ R3 : |∇V (x) · x| ≥ } has finite Lebesgue measure for every > 0, and the function t → t2[(2β –3)V (tx)–∇V (tx)·(tx)] is increasing on (0, +∞) for every x ∈ R3; (V3) V ∈ C1(R3), ∇V (x) · x ∈ L∞(R3) and there exists > 0 such that
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