Abstract
The aim of this paper is establishing the existence of a nontrivial solution for the following quasilinear Schrödinger–Poisson system: {−Δu+V(x)u−uΔ(u2)+K(x)ϕ(x)u=g(x,u),x∈R3,−Δϕ=K(x)u2,x∈R3,u∈H1(R3),u>0,\\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}$$ \\left \\{ \\textstyle\\begin{array}{l} -\\Delta u+V(x)u-u\\Delta(u^{2})+K(x)\\phi(x)u=g(x, u),\\quad x\\in\\mathbb {R}^{3}, \\\\ -\\Delta\\phi=K(x)u^{2}, \\quad x\\in\\mathbb{R}^{3},\\\\ u\\in H^{1}(\\mathbb{R}^{3}),\\qquad u>0, \\end{array}\\displaystyle \\right . $$\\end{document} where V, K, g are continuous functions. To overcome the technical difficulties caused by the quasilinear term, we change the variable to guarantee the feasibility of applying the mountain pass theorem to solve the above problems. We use the mountain pass theorem and the concentration–compactness principle as basic tools to gain a nontrivial solution the system possesses under an asymptotic periodicity condition at infinity.
Highlights
Introduction and main resultsIn this paper, we consider the following quasilinear asymptotically periodic Schrödinger– Poisson system:⎧ ⎪⎨ – u + V (x)u – u (u2) + K(x)φ(x)u = g(x, u), x ∈ R3, ⎪⎩– u φ = K (x)u2, ∈ H1(R3), x ∈ R3, u > 0, (1.1)where V, K : R3 → R and g : R3 × R → R are continuous functions
The so-called quasilinear Schrödinger–Poisson system was introduced in [5, 25] and is a quantum mechanical model of extremely small devices in semiconductor nanostructures taking into account the quantum structure and longitudinal field oscillations during the beam propagation
– u + V (x)u + K(x)φ(x)u = g(x, u), x ∈ R3, – φ = K (x)u2, x ∈ R3, which has been widely studied, and many meaningful results were achieved for the subcritical growth [7, 27, 44, 45, 47] or the critical exponent [46] under various assumptions on the potentials and nonlinearities
Summary
By using Ekeland’s variational principle and the mountain pass theorem the authors in [33] obtained the existence of the ground state solution for a generalized quasilinear Schrödinger–Poisson system in R3. The asymptotic periodicity of g at infinity is given by the following condition: (G4) there exist a constant 4 ≤ q3 < 12 and functions h2 ∈ F , and g0 ∈ C(R3 × R, R+), 1-periodic in xi, 1 ≤ i ≤ 3, such that (i) To show the existence of a solution for the periodic problem, condition (G4) is not necessary.
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