Singly periodic solutions of a semilinear equation

Abstract

We consider the solutions of the equation −ɛ2Δu+u−|u|p−1u=0 in S1×R, where ɛ and p are positive real numbers, p>1. We prove that the set of the positive bounded solutions even in x1 and x2, decreasing for x1∊]−π,0[ and tending to 0 as x2 tends to +∞ is the first branch of solutions constructed by bifurcation from the ground-state solution (ɛ,w0(x2ɛ)). We prove that there exists a positive real number ɛ⋆ such that for every ɛ∊]0,ɛ⋆] there exists a finite number of solutions verifying the above properties and none such solution for ɛ>ɛ⋆. The proves make use of compactness results and of the Leray–Schauder degree theory.