Nonradial solutions of a Neumann Hénon equation on a ball

Abstract

In this work we examine the existence of positive classical solutions of { − Δ u + u = | x | α u p − 1 a m p ; in B 1 , u > 0 a m p ; in B 1 , ∂ ν u = 0 a m p ; on ∂ B 1 , \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 p>1 , α > 0 \alpha >0 and B 1 B_1 is the unit ball in R N {\mathbb {R}}^N where N ≥ 4 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 , α p,\alpha and N N there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.