Nonradial solutions of nonlinear Neumann problems in radially symmetric domains

Abstract

where Ω ⊂ R is a bounded domain with smooth boundary, λ > 0 and p > 1 are constants and ν is the unit outer normal to ∂Ω. Concerning the existence, multiplicity, and qualitative properties of solutions of (1.1)λ many interesting results have appeared; especially after Ni and Takagi ([NT1]) first discovered the spike-layer structure on the shape of least energy solutions for the subcritical problems, a lot of work has been devoted to the study of qualitative properties of solutions of (1.1)λ. For more references, we refer to [NT2] and [Wz5], in which both the subcritical exponent case (i.e. 1 < p < N+2 N−2 ) and the critical exponent case (i.e. p = N+2 N−2 ) are surveyed. In this paper, we shall focus on the case where Ω is a spherically symmetric domain, especially on the case where Ω is a ball domain. We are mainly interested in the existence and the shape of nonradial solutions of (1.1)λ. When we replace the Neumann boundary condition by the Dirichlet boundary condition the well known Gidas-Ni-Nirenberg result ([GNN]) asserts that any positive solutions must be radially symmetric. However, we shall see that contrary to its Dirichlet counterpart, (1.1)λ possesses many nonradial solutions when Ω is a ball domain. In [Wz6], we have presented an approach to this problem to construct multi-peaked solutions for (1.1)λ with the critical Sobolev exponent when Ω is a symmetric domain. We