Nonradial solutions of a Neumann Hénon equation on a ball

Abstract:

In this work we examine the existence of positive classical solutions of \begin{equation*} \begin {cases} -\Delta u +u = |x|^\alpha u^{p-1} & \text { in } B_1, \\ u>0 & \text { in } B_1, \\ \partial _\nu u= 0 & \text { on } \partial B_1, \end{cases} \end{equation*} where $p>1$, $\alpha >0$ and $B_1$ is the unit ball in ${\mathbb {R}}^N$ where $N \ge 4$ and is even. Of particular interest is the existence of nonradial position classical solutions. We show that under suitable conditions on $p,\alpha$ and $N$ there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.