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In this paper we study the existence and multiplicity of two distinctnontrivial weak solutions of the following equation in Nehari manifold. We havealso 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+\lambdaf(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 certainconditions. The domain $\Omega\subset\mathbb{R}^N (N\geq 2)$ is smooth andbounded. 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, $\alpha(x)\leqp(x,x)\;\forall\;x\in\overline{\Omega}$ and $s(x,y)p(x,y) < N\;\forall\;(x,y)\in\overline{\Omega}\times\overline{\Omega}$.

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