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Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity

Abdelaaziz Sbai, Youssef El hadfi, Mounim El Ouardy

Abstract

In this paper we are going to prove existence for positive solutions of the following Schrödinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(Ω):-\operatorname{div}\left(a(x) \nabla u\right)+ψ|u|^{r-2} u=\frac{f(x)}{u^θ}, ψ\in W_{0}^{1,2}(Ω):-\operatorname{div}(M(x) \nabla ψ)=|u|^{r} \end{array}\right. \end{equation} where $Ω$ is a bounded open set of $\mathbb{R}^{N}, N>2,$ $r>,1,$ $u>0,$ $ψ>0,$ $0 < θ<1$ and $f$ belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions.

Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity

Abstract

In this paper we are going to prove existence for positive solutions of the following Schrödinger-Maxwell system of singular elliptic equations: begin{equation} \left\{\begin{array}{l} u \in W_{0}^{1,2}(Ω):-\operatorname{div}\left(a(x) \nabla u\right)+ψ|u|^{r-2} u=\frac{f(x)}{u^θ}, ψ\in W_{0}^{1,2}(Ω):-\operatorname{div}(M(x) \nabla ψ)=|u|^{r} \end{array}\right. \end{equation} where is a bounded open set of and belongs to a suitable Lebesgue space. In particular, we take advantage of the coupling between the two equations of the system by demonstrating how the structure of the system gives rise to a regularizing effect on the summability of the solutions.

Paper Structure

This paper contains 7 sections, 7 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. A priori estimates and main results
  3. The Approximated Problem
  4. A Priori Estimates
  5. Proof of Theorem \ref{['th21']}
  6. Saddle points
  7. Appendix: Basic Results and Existence for Bounded Data

Key Result

Theorem 2.1

Let $0< \theta <1$ and let a and $M$ be such that 134 and 135 hold. Let $r>1$ and let $f$ in $L^{m}(\Omega) .$ We have the following: (i) if $r \geq \frac{2 N}{\theta(N-2)+N+2}$, and if $m \geq(\frac{r+1}{1-\theta})^{\prime}$, there exist $u$ and $\psi$ in $W_{0}^{1,2}(\Omega)$, solutions of 133; fu

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.1
  • Theorem 4.1