Multiple solutions for semilinear elliptic equations in unbounded cylinder domains

Abstract

In this paper, we show that if b(x) ≥ b^∞ > 0 in Ω̄ and there exist positive constants C, δ, R₀ such that where x = (y, z) ∈ R^N with y ∈ R^m, z ∈ Rⁿ, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω ⊆ R^m a bounded C^1,1 domain and Ω = ω × Rⁿ, then the Dirichlet problem −Δu + u = b(x)|u|^p−1u in Ω has a solution that changes sign in Ω, in addition to a positive solution.