Multiplicity of nontrivial solutions of a class of fractional \(p\)-Laplacian problem
Summary: In this paper, we deal with existence of nontrivial solutions to the fractional \(p\)-Laplacian problem of the type \[ \begin{cases} (-\Delta)^\alpha_pu=\frac{1}{r}\frac{\partial F(x,u)}{\partial u}+\lambda a(x)|u|^{q-2}u\quad & \text{ in }\Omega, \\ u=0\quad & \text{ in }\mathbb R^n\setminus\Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial \Omega\), \(a\in C(\Omega)\), \(p\geq 2\), \(\alpha\in (0,1)\) such that \(p\alpha<n\), \(1<q<p<r<\frac{np}{n-\alpha p}\), and \(F\in C^1(\overline{\Omega}\times\mathbb R,\mathbb R)\). Using the decomposition of the Nehari manifold, we prove that the non-local elliptic problem has at least two nontrivial solutions.
scientific article; zbMATH DE number 6484444