Existence and multiplicity of solutions for singular fractional elliptic system via the Nehari manifold approach

Kamel Akrout  ; Nor Elhouda Soltani ; Sounia Zediri

Novi Sad Journal of Mathematics, Tome 51 (2021) no. 1.

Résumé

In this paper, we study the existence and multiplicity of positive nontrivial solutions of the following singular fractional elliptic system. \begin{equation*} \left\{ \begin{array}{l} \left( -\Delta \right) _{p}^{s}u=a(x)u^{-\gamma }+\lambda f(x,u,v)\text{ in }% \Omega , \\ \\ \left( -\Delta \right) _{p}^{s}v=b(x)v^{-\gamma }+\lambda g(x,u,v)\text{ in }% \Omega , \\ \\ u=v=0\text{\ on\ }\mathbb{R}^{N}\backslash \Omega ,% \end{array}% \right. \end{equation*}% where $\Omega $ is a smooth bounded domain in $\mathbb{R}^{N},$ $N>ps$ with $% s\in (0,1),$ $\lambda $ is a positive parameter and $0\gamma 1$ $a,b\in L^{\infty }(\Omega ,\mathbb{R}_{\ast }^{+}),f,g\in C(\Omega \times \mathbb{R}\times \mathbb{R},\mathbb{R}^{+})$ are positively homogeneous functions of degree $% \left( r-1\right) $. The results are obtained by using the fibering method, Nehari Manifold technique and applying Ekeland's variational principle.

Détail

Publié le : 2020-07-08

DOI : 10.30755/NSJOM.11180

Classification : 35J35, 35J50, 35J60
Keywords: fractional operator, Nehari manifold, Ekeland's principle