On a system involving a critically growing nonlinearity

TL;DR

This paper investigates a nonlinear elliptic system involving a critical growth term, establishing conditions for the existence and nonexistence of solutions based on the parameter lambda.

Contribution

It provides new existence and nonexistence results for a nonlinear PDE system with critical growth, depending on the parameter lambda.

Findings

Existence of solutions for certain lambda values.

Nonexistence results for other lambda ranges.

Analysis of critical nonlinearities in elliptic systems.

Abstract

This paper deals with the system \[\{{array}{ll} -\Delta u = \lambda u + q |u|^3 u \phi & \hbox{in} B_R, -\Delta \phi=q |u|^5 & \hbox{in} B_R, u=\phi=0 & \hbox{on} \partial B_R. {array}.\] We prove existence and nonexistence results depending on the value of $\lambda$.