Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

Abstract

<p style='text-indent:20px;'>This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ q $\end{document}</tex-math></inline-formula> type and singular nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {} = \lambda \frac{f(u)}{u^\gamma}, \ u>0 && \quad\mbox{ in } \, \Omega, \\ u & {} = 0 && \quad\mbox{ on } \partial\Omega, \end{alignedat} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> boundary, <inline-formula><tex-math id="M6">\begin{document}$ N \geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \lambda >0 $\end{document}</tex-math></inline-formula> is a real parameter,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \mathcal{L}_{p,q} u : = {\rm{div}}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><inline-formula><tex-math id="M8">\begin{document}$ 1<p<q< \infty $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \gamma \in (0,1) $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M10">\begin{document}$ f $\end{document}</tex-math></inline-formula> is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [<xref ref-type="bibr" rid="b1">1</xref>], we prove existence of three positive solutions in the positive cone of <inline-formula><tex-math id="M11">\begin{document}$ C_\delta(\overline{\Omega}) $\end{document}</tex-math></inline-formula> and in a certain range of <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>.</p>