Multiple solutions for a quasilinear Schr\xf6dinger\u2013Poisson system

Jing Zhang

doi:10.1186/s13661-021-01553-2

Abstract

In this article, we consider the following quasilinear Schrödinger–Poisson system \t\t\t0.1{−Δu+V(x)u−uΔ(u2)+K(x)ϕ(x)u=g(x,u),x∈R3,−Δϕ=K(x)u2,x∈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}$$ \\textstyle\\begin{cases} -\\Delta u+V(x)u-u\\Delta (u^{2})+K(x)\\phi (x)u=g(x,u), \\quad x\\in \\mathbb{R}^{3}, \\\\ -\\Delta \\phi =K(x)u^{2}, \\quad x\\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where V,K:mathbb{R}^{3}rightarrow mathbb{R} and g:mathbb{R}^{3}times mathbb{R}rightarrow mathbb{R} are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

Highlights

In this article, we consider the following quasilinear Schrödinger–Poisson system: ⎧⎨– u + V (x)u – u (u2) + K(x)φ(x)u = g(x, u), x ∈ R3,⎩– φ = K (x)u2, x ∈ R3, (1.1)where V, K : R3 → R and g : R3 × R → R are continuous functions
The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth
The quasilinear Schrödinger–Poisson system had been introduced in [4, 20], which is a quantum mechanical model of extremely small devices in semiconductor nanostructures taking into account the quantum structure and the longitudinal field oscillations during the beam propagation

Summary

Introduction

We consider the following quasilinear Schrödinger–Poisson system:. where V , K : R3 → R and g : R3 × R → R are continuous functions. The authors in [24] considered a system with radial potentials and discontinuous nonlinearity, and obtained the multiplicity results of radial solutions by nonsmooth critical point theory. Lemma 2.2 The function f satisfies the following properties: (1) f is uniquely defined, C∞, and invertible; (2) |f (t)| ≤ 1 for all t ∈ R; (3) |f (t)| ≤ |t| for all t ∈ R; (4) f (t)/t√→ 1 as t → 0; (5) f (t)/ t → 21/4 as t → +∞; (6) f (t)/2 ≤ tf (t) ≤ f (t) for all t ≥ 0; (7) |f (t)| ≤ 21/4|t|1/2 for all t ∈ R; (8) f 2(t)/2 ≤ f (t)f (t)t ≤ f 2(t) for all t ∈ R;.

R3 p R3

Methods