Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition

<p style='text-indent:20px;'>In this paper, we study the existence and multiplicity of weak solutions for a general class elliptic equations <inline-formula><tex-math id="M1">\begin{document} $( \mathscr{P}_\lambda)$\end{document}</tex-math></inline-formula> in smooth bounded domain, driven by nonlocal integrodifferential operator id="M2">\begin{document}$ \mathscr{L}_{\mathcal{A}K} $\end{document}</tex-math></inline-formula> with Dirichlet boundary conditions involving variable exponents without Ambrosetti Rabinowitz type growth conditions. Using different versions Mountain Pass Theorem, as well as, Fountain Theorem Dual Cerami condition, obtain problem id="M3">\begin{document} show that treated has at least one nontrivial solution any parameter id="M4">\begin{document}$ \lambda >0 small enough blows up, fractional Sobolev norm, id="M5">\begin{document}$ \to 0 $\end{document}</tex-math></inline-formula>. Moreover, sublinear case, imposing some additional hypotheses on nonlinearity id="M6">\begin{document}$ f(x,\cdot) $\end{document}</tex-math></inline-formula>, using new version symmetric due to Kajikiya [<xref ref-type="bibr" rid="b18">18</xref>], infinitely many which tend zero, id="M7">\begin{document}$ As far know, results paper are literature.</p>