Study on a class of Schrödinger elliptic system involving a nonlinear operator

Guotao Wang,Zedong Yang,Lihong Zhang,Ravi P Agarwal

doi:10.15388/namc.2020.25.18385

Abstract

This paper considers a class of Schrödinger elliptic system involving a nonlinear operator. Firstly, under the simple condition on and ', we prove the existence of the entire positive bounded radial solutions. Secondly, by using the iterative technique and the method of contradiction, we prove the existence and nonexistence of the entire positive blow-up radial solutions. Our results extend the previous existence and nonexistence results for both the single equation and systems. In the end, we give two examples to illustrate our results.

Highlights

Introduction and preliminaryIn this paper, our main objective is to show the positive radial solutions of the following nonlinear Schrödinger elliptic system involving a nonlinear operator: div(G |∇y|p−2 ∇y) = b |x| ψ(z), x ∈ Rn, div(G |∇z|p−2 ∇z) = h |x| φ(y), x ∈ Rn, (1)where n 3, b, h, ψ, φ ∈ C([0, +∞), [0, +∞)), and G is a nonlinear operator on Θ = {G ∈ C2([0, +∞), (0, +∞)) | ∃p = const > 2: G(ls) lp−2G(s), 0 < l < 1}.c 2020 Authors
Many rich results on the Schrödinger elliptic system have been obtained by using nonlinear functional analysis methods such as the variational method [4,5,6,7,15,18,33,35,38], the fixed point theorem [1, 12, 14, 16, 20, 21], the upper and lower solution method [13, 39] and the method of moving planes [27, 31]
To the best of our knowledge, many papers have been found on the Schrödinger system, there is no work on the existence and nonexistence of blow-up radial solutions for the Schrödinger system (1) involving a nonlinear operator by using the monotone iterative method

Summary

Introduction and preliminary

In a recent paper [35], by using the iterative method and the dual method, Zhang and Liu studied the existence and nonexistence of entire blow-up radial solutions for the following quasilinear p-Laplacian Schrödinger elliptic equation with a nonsquare diffusion term:. In 2018, by using the iterative technique and introducing a growth condition, Zhang and Wu [37] focused on the existence and nonexistence of the entire blow-up radial solutions for the following nonlinear Schrödinger elliptic equation: div G |∇z| ∇z = b |x| ψ(z), x ∈ Rn,. To the best of our knowledge, many papers have been found on the Schrödinger system, there is no work on the existence and nonexistence of blow-up radial solutions for the Schrödinger system (1) involving a nonlinear operator by using the monotone iterative method.

Existence of the positive bounded radial solutions

Existence and nonexistence of the positive blow-up radical solutions

Examples