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Existence of weak solutions to some stationary Schr{ö}dinger equations with singular nonlinearity

Pascal Bégout, Jesús Ildefonso Díaz

Abstract

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.

Existence of weak solutions to some stationary Schr{ö}dinger equations with singular nonlinearity

Abstract

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 -space.

Paper Structure

This paper contains 7 sections, 19 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations
  4. A priori estimates
  5. Proofs of the main results
  6. On the existence of solutions of the Dirichlet problem for data beyond $\boldsymbol{L^2(\Omega)}$
  7. Conclusions

Key Result

Theorem 2.1

Let $\Omega$ an open subset of $\mathbb{R}^N$ be such that $|\Omega|<\infty$ and assume $0<m<1,$$(a,b)\in\mathbb{C}^2$ and $F\in L^2(\Omega).$ If $\mathrm{Re}(b)<0$ then assume further that $\mathrm{Im}(b)\neq0$ or $-\frac{1}{C_\mathrm{P}^2}<\mathrm{Re}(b),$ where $C_\mathrm{P}$ is the Poincaré's co

Theorems & Definitions (30)

  • Theorem 2.1: Existence
  • Theorem 2.3: A priori bound
  • Theorem 2.4: Existence
  • Remark 2.5
  • Theorem 2.6: A priori bound
  • Theorem 2.8: Existence
  • Theorem 2.9: A priori bound
  • Theorem 2.10: Uniqueness
  • Remark 2.11
  • Theorem 2.12: Regularity
  • ...and 20 more