The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L K p . Specifically, we get multiple solutions to the following fractional p -Laplacian equations with the help of fibering maps and Nehari manifold. − Δ p s u x = λ u q + u r , u > 0   in   Ω , u = 0 , in   ℝ N \ Ω . . Our results extend the previous results in some respects.

In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent ∆ 2 p(x) u = λV (x)|u| q(x)−2 u, in a smooth bounded domain,under Neumann boundary conditions, where λ is a positive real number, p, q : Ω → R, are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.

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<p style='text-indent:20px;'>In this paper, we study the existence of positive solutions of the following equation</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE10000">\begin{document}$\begin{equation} (P_{\lambda}) \left\{ \begin{array}{rclll} - \Delta_{p(x)} u+V(x)\vert u\vert^{p(x)-2}u &amp; = &amp; \lambda k(x) \vert u\vert^{\alpha(x)-2}u\\ &amp;+&amp; h(x) \vert u\vert^{\beta(x)-2}u&amp;\mbox{ in }&amp;\Omega\\ u&amp; = &amp;0 &amp;\mbox{ on }&amp; \partial \Omega. \end{array} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document}</tex-math></disp-formula></p> <p style='text-indent:20px;'>The study of the problem <inline-formula><tex-math id="M2">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> needs generalized Lebesgue and Sobolev spaces. In this work, under suitable assumptions, we prove that some variational methods still work. We use them to prove the existence of positive solutions to the problem <inline-formula><tex-math id="M3">\begin{document}$ (P_{\lambda}) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ W_{0}^{1,p(x)}(\Omega) $\end{document}</tex-math></inline-formula>.</p>

For an anisotropic discrete nonlinear problem with variable exponent, we demonstrate both the existence and multiplicity of nontrivial solutions in this study. The variational principle and critical point theory are the key techniques employed here.