Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity

Pascal Bégout (1) , Jesús Ildefonso Díaz (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Departamento de Matematica Aplicada [Madrid]

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\"{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors), the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual $L^2$-space.

Mots clés: uniqueness; existence; smoothness; data in weighted spaces; non local terms; unbounded domains; different boundary conditions; nonlinear Schrödinger equation

Domaines: Equations aux dérivées partielles [math.AP]

Pascal Bégout, Jesús Ildefonso Díaz. Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity. RACSAM. Real Academia de Ciencias. Serie A, Matemáticas - Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 2015, 109 (1), pp.43-63. ⟨hal-00812195v4⟩