Radial and nonradial solutions for a semilinear elliptic system of Schr\"odinger type

TL;DR

This paper investigates the existence and properties of positive solutions, including radial and nonradial types, for a class of semilinear elliptic systems of Schrödinger type on ^N, using approximation and lower-upper solution methods.

Contribution

It establishes conditions for bounded positive solutions, characterizes when radial solutions are large, and constructs nonradial solutions using lower and upper solution techniques.

Findings

Existence of bounded positive entire solutions under certain conditions.

Necessary and sufficient conditions for radial solutions to be large.

Construction of nonradial solutions via lower and upper solution methods.

Abstract

In this article we consider the system of equations {\Delta}u_{i}=p_{i}(x)f_{i}(u_{1},...,u_{d}) for i=1,...,d on R^{N}, N\geq3 and d\in{1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some conditions on p_{i} and f_{i}. Also, we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Furthermore, a non-radially symmetric solution is obtained by using a lower and upper solution method.