Abstract
In the present paper, we consider the following singularly perturbed problem: {−ε2Δu+V(x)u−ε2Δ(u2)u=ε−α(Iα∗G(u))g(u),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}$$ \\textstyle\\begin{cases} -\\varepsilon ^{2}\\Delta u+V(x)u-\\varepsilon ^{2}\\Delta (u^{2})u= \\varepsilon ^{-\\alpha }(I_{\\alpha }*G(u))g(u), \\quad x\\in \\mathbb{R}^{N}; \\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where varepsilon >0 is a parameter, Nge 3, alpha in (0, N), G(t)=int _{0}^{t}g(s),mathrm{d}s, I_{alpha }: mathbb{R}^{N}rightarrow mathbb{R} is the Riesz potential, and Vin mathcal{C}(mathbb{R}^{N}, mathbb{R}) with 0<min_{xin mathbb{R}^{N}}V(x)< lim_{|y|to infty }V(y). Under the general Berestycki–Lions assumptions on g, we prove that there exists a constant varepsilon _{0}>0 determined by V and g such that for varepsilon in (0,varepsilon _{0}] the above problem admits a semiclassical ground state solution hat{u}_{varepsilon } with exponential decay at infinity. We also study the asymptotic behavior of {hat{u}_{varepsilon }} as varepsilon to 0.
Highlights
In this paper, we consider the following singularly perturbed quasilinear Choquard equation: ⎧⎨–ε2 u + V (x)u – ε2 (u2)u = ε–α(Iα ∗ F(u))f (u), x ∈ RN ; ⎩u ∈ H1(RN ), (1.1)where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N), and Iα : RN → R is the Riesz potential defined by Iα(x) = ( N–α ) ( α )2α π
Berestycki–Lions assumptions on g, we prove that there exists a constant ε0 > 0 determined by V and g such that for ε ∈
We introduce the following conditions: (V2) V ∈ C1(RN, R) and t → NV(tx)+∇tαV(tx)·(tx) is nonincreasing on (0, ∞) for all x ∈ RN \ {0}; (G4) g(t) = o(t) as t → 0
Summary
Lemma 2.1 The function f (t) and its derivative satisfy the following properties: (f1) f is uniquely defined, C∞ and invertible, and 0 < f (t) ≤ 1 for all t ∈ R; (f2) |f (t)| ≤ |t| and |f (t)| ≤ 21/4|t|1√/2 for all t ∈ R; (f3) f (t)/t → 1 as t → 0 and f (t)/ t → 21/4 as t → +∞; (f4) f (t)/2 ≤ tf (t) ≤ f (t) for all t > 0 and f (t) ≤ tf (t) ≤ f (t)/2 for all t ≤ 0; (f5) There exists a positive constant θ0 such that Arguing as in the proof of [25, Lemma 2.8], we get the following result.
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