Existence and multiplicity of solutions to a nonlocal elliptic PDE with variable exponent in a Nehari manifold using the Banach fixed point theorem (1907.09009v1)

Abstract: In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\infty}(\Omega)$. \begin{align*} \begin{split} -\Delta_{p(x,y)}^{s(x,y)}u &= \beta|u|^{\alpha(x)-2}u+\lambda f(x,u)\,\,\mbox{in}\,\,\Omega,\ u &= 0\,\, \mbox{in}\,\, \mathbb{R}^{N}\setminus\Omega \end{split} \end{align*} Here, $\lambda, \beta > 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $\Omega\subset\mathbb{R}^N (N\geq 2)$ is smooth and bounded. The relation between the exponents are assumed in the order $2 < \alpha^{-}\leq\alpha(x)\leq\alpha^{+} < p^{-}\leq p(x,y)\leq p^{+} < q^{+} < r^{+} < r^{+2} < p_{s}^{*}(x)$. Also, $\alpha(x)\leq p(x,x)\;\forall\;x\in\overline{\Omega}$ and $s(x,y)p(x,y) < N \;\forall\;(x,y)\in\overline{\Omega}\times\overline{\Omega}$.