Nehari manifold and multiplicity result for elliptic equation involving p-laplacian problems

• Main Authors: Khaled Ben Ali, Abdeljabbar Ghanmi
• Format: Article
• Language: English
• Published: Sociedade Brasileira de Matemática 2018-10-01
• Series: Boletim da Sociedade Paranaense de Matemática
• Subjects: Multiple positive solutions; sign-changing weight function; Nehari manifold
• Online Access: http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/34078

This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem $$\displaystyle -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad \mbox{for } x\in\partial\Omega$$ where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.