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Asymptotics and symmetries of ground-state and least energy nodal solutions for boundary value problems with slowly growing superlinearities

par Bonheure, Denis ;Bouchez, Vincent;Grumiau, Christopher
Référence Differential and integral equations, 22, 9-10, page (1047-1074)
Publication Publié, 2009

Article révisé par les pairs

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Résumé :

We study the problems - Δu = fθ(u) in Ω, u = 0on ∂Ω, - Δu + u = fθ(u) in Ω, ∂vu = 0 on ∂Ω, where fθ is a slowly superlinearly growing nonlinearity, and Ω is a bounded domain. Namely, we are interested in generalizing the results obtained in [4], where the model nonlinearity fθ(u) = \u\ θ-2u was considered in the case of Dirichlet boundary conditions. We derive the asymptotic behaviour of ground state and least energy nodal solutions when θ→2, leading to symmetry results for θ small. Our assumptions permit us to study some typical nonlinearities such as a superlinear perturbation of a small pure power or the sum of small powers and slowly exponentialy growing nonlinearities in dimension 2.