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Singly periodic solutions of a semilinear equation.

Abstract : 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.
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Contributor : Anne Beaulieu Connect in order to contact the contributor
Submitted on : Tuesday, September 4, 2012 - 1:20:41 PM
Last modification on : Saturday, January 15, 2022 - 4:09:13 AM

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Geneviève Allain, Anne Beaulieu. Singly periodic solutions of a semilinear equation.. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2009, 26 (4), pp.1277-1297. ⟨10.1016/j.anihpc.2008.10.001⟩. ⟨hal-00727776⟩



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