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

Abstract

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