Positive radial solutions for coupled Schr\"{o}dinger system with critical exponent in $\R^N\,(N\geq5)$

TL;DR

This paper establishes the existence of positive radial solutions for a coupled Schrödinger system with critical nonlinearities in ^N, employing variational methods and the Mountain Pass Theorem.

Contribution

It introduces new existence results for coupled Schrödinger systems with critical exponents, covering both positive and negative coupling parameters.

Findings

Positive radial solutions exist for ^N with positive  and some negative .

Uses variational methods including Mountain Pass Theorem and Nehari manifold.

Addresses critical nonlinearities in coupled Schrödinger equations.

Abstract

We study the following coupled Schr\"odinger system \ds -\Delta u+u=u^{2^*-1}+\be u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}+\la_1u^{\al-1}, &x\in \R^N, \ds -\Delta v+v=v^{2^*-1}+\be u^{\frac{2^*}{2}}v^{\frac{2^*}{2}-1}+\la_2v^{r-1}, &x\in \R^N, u,v > 0, &x\in \R^N, where $N\geq 5, \la_1,\la_2>0,\be\neq 0, 2<\al,r<2^*,2^*\triangleq \frac{2N}{N-2}.$ Note that the nonlinearity and the coupling terms are both critical. Using the Mountain Pass Theorem, Ekeland's variational principle and Nehari mainfold, we show that this critical system has a positive radial solution for positive $\be$ and some negative $\be$ respectively.