Nonradial solutions of a Neumann Hénon equation on a ball

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 , {equation*} {cases} -Δ u +u = |x|^α uᵖ⁻¹ & in B_1, \\ u>0 & in B_1, \\ ∂ _ν u= 0 & on ∂ B_1, {cases} {equation*} where p > 1 p>1 , α > 0 α >0 and B 1 B_1 is the unit ball in R N {R}^N where N ≥ 4 N ≥ 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,α and N N there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.