Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well

• Published in: Calculus of variations and partial differential equations Vol. 53; no. 1-2; pp. 413 - 439
• Main Authors: Cingolani, Silvia, Jeanjean, Louis, Tanaka, Kazunaga
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2015; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Analysis of PDEs; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Manifolds; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; Partial differential equations; Schrodinger equation; Schroedinger equation; Searching; Systems Theory; Texts; Theoretical; 35Q55; 35J20; 58E05
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-014-0754-5

We consider singularly perturbed nonlinear Schrödinger equations 0.1 - ε 2 Δ u + V ( x ) u = f ( u ) , u > 0 , v ∈ H 1 ( R N ) where V ∈ C ( R N , R ) and f is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain Ω ⊂ R N such that m 0 ≡ inf x ∈ Ω V ( x ) < inf x ∈ ∂ Ω V ( x ) and we set K = { x ∈ Ω | V ( x ) = m 0 } . For ε > 0 small we prove the existence of at least cupl ( K ) + 1 solutions to (0.1) concentrating, as ε → 0 around K . We remark that, under our assumptions of f , the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.