Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation
The paper deals with the existence and multiplicity of solutions to the following nonlocal equation having two nonlocal terms: \[ -M\left(\int_\Omega |\nabla u|^pdx\right)\Delta_pu = \lambda |u|^{q-2}u + \mu g(x)|u|^{\gamma-2}u\left[\frac{1}{\gamma}\int_\Omega g(x) |u|^{\gamma}dx\right]^{2r},\;\;x\in \Omega, \] with Dirichlet boundary conditions. Here, \(\Omega\subset {\mathbb R}^N\) is a bounded smooth domain, \(1<p<N,\; 1<\gamma<p^*\), \(r>0\), \(p(\alpha+1) <q\leq p^*\) or \(p-1<q<p(\alpha+1)\), and \(M\) and \(g\) are some continuous functions. This problem is considered in a variational framework, solutions for this problem are searched as critical points of some functional. In this paper, the authors obtain existence and multiplicity results of nontrivial solutions for the cases: \(\lambda =0\) and \(\mu =1\) or \(\lambda =1\) and \(\mu=1\). The analysis is based on Krasnolselskii genus theory, the mountain-pass theorem and the concentration-compactness principle.
scientific article; zbMATH DE number 6166942