Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Multiple solutions for semilinear elliptic equations in unbounded cylinder domains

Tsing-San Hsua1

a1 Department of Center of General Education, Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan (


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) xs2208 RN with y xs2208 Rm, z xs2208 Rn, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω xs2286 Rm a bounded C1,1 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.

(Received July 10 2003)

(Accepted March 01 2004)