Singly periodic solutions of a semilinear equation.
Résumé
We consider the solutions of the equation −ε2Δu+u−|u|p−1u=0 in S1×R, where ε 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. We prove that there exists a positive real number ε⋆ such that for every ε∈]0,ε⋆] there exists a finite number of solutions verifying the above properties and none such solution for ε>ε⋆. The proves make use of compactness results and of the Leray-Schauder degree theory.