Existence of positive solutions for a class of singular and quasilinear elliptic problems with critical exponential growth

Suellen Cristina Q. Arruda¹ ; Giovany M. Figueiredo² ; Rubia G. Nascimento³

¹ Universidade Federal do Pará - UFPA, Faculdade de Ciências Exatas e Tecnologia
² Universidade de Brasília - UNB, Departamento de Matemática
³ Universidade Federal do Pará - UFPA, Instituto de Ciências Exatas e Naturais

Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 395-420.

Résumé

In this paper we use Galerkin method to investigate the existence of positive solution for a class of singular and quasilinear elliptic problems given by \[\begin{cases}-\operatorname{div}(a_0(|\nabla u|^{p_0})|\nabla u|^{p_{0}-2}\nabla u)= \displaystyle\frac {\lambda_0}{u^{\beta_0}} + f_0(u),\ u>0 &in\ \Omega,\\ u=0 &on\ \partial\Omega,\end{cases}\] and its version for systems given by \[\begin{cases}-\operatorname{div}(a_1(\vert\nabla u\vert^{p_1})\ \vert \nabla u\vert ^{p_1-2}\ \nabla u)=\dfrac{\lambda_1}{u^{\beta_1}}+f_1(v) &in\ \Omega,\\ -\operatorname{div}(a_2(\vert\nabla v\vert^{p_2})\ \vert \nabla v\vert ^{p_2-2}\ \nabla v)=\dfrac{\lambda_2}{v^{\beta_2}}+f_2(u) &in\ \Omega,\\ u,v>0 &in\ \Omega,\\ u=v=0 &on\ \partial\Omega,\end{cases}\] where $\Omega\subset\mathbf{R}^{N}$ is bounded smooth domain with $N\geq 3$ and for $i=0,1,2$ we have $2 \leq p_i < N$, $0<\beta_i \leq 1$, $\lambda_i>0$ and $f_i$ are continuous functions. The hypotheses on the $C^1$-functions $a_i\colon \mathbf{R}^+\rightarrow \mathbf{R}^+$ allow to consider a large class of quasilinear operators.  

Détail

Keywords: Galerkin method, exponential growth, Trudinger-Moser inequality, Hardy-Sobolev inequality