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Bound state solutions for a class of nonlinear Schrödinger equations

par Bonheure, Denis ;Van Schaftingen, Jean
Référence Revista matemática iberoamericana, 24, 1, page (297-351)
Publication Publié, 2008

Article révisé par les pairs

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Résumé :

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form -ε2Δu+V(x)u = K(x)up, x ∈ ℝN, where V, K are positive continuous functions and p > 1 is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential V is allowed to vanish at infinity and the competing function K does not have to be bounded. In the semi-classical limit, i.e. for ε ∼ 0, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function A - VθK - 2/p-1, where θ - (p+1)/(p - 1) - N/2. A special attention is devoted to the qualitative properties of these solutions as ε goes to zero.