Positive nonradial solutions of an elliptic equation with critical advection term

Abstract

Consider the problem(1)$-\Delta u-\frac{\beta x\cdot \nabla u}{|x{|}^2}+u=u^{p-1}\text{ in }{R}^N,$ where $p>2$, $\beta >0$ and $N=2n\ge 4$ an even integer. In this work we are interested in finding positive classical nonradial solutions of (1). We also consider the problem on the unit ball. We demonstrate the existence of a positive bounded solution for the range $2<p<p_{\beta }:=\frac{N+2\beta }{N-2+\beta }$, and show that nonradial solutions exist for a specific range of p.