Multiplicity of symmetric solutions for a nonlinear eigenvalue problem in ℝn

In this paper, we study the nonlinear eigenvalue field equation -Δu + V(|x|)u + ε(-Δpu + W′(u)) = μu where u is a function from ℝn to ℝn+1 with n ≥ 3, ε is a positive parameter and p > n. We find a multiplicity of solutions, symmetric with respect to an action of the orthogonal group O(n): For any q ∈ ℤ we prove the existence of finitely many pairs (u, μ) solutions for ε sufficiently small, where u is symmetric and has topological charge q. The multiplicity of our solutions can be as large as desired, provided that the singular point of W and ε are chosen accordingly.