Existence of solutions for a class of quasilinear elliptic equations with a parameter

Abstract

<p style='text-indent:20px;'>This paper is concerned with the existence of nontrivial solutions for the quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \notag -\Delta u-\Delta(|u|^{2\alpha})|u|^{2\alpha-2}u+V(x)u = |u|^{q-2}u,\quad x\in \mathbb{R}^{N}, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 3/4\leq\alpha\leq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> is a positive continuous function. Using Mountain-Pass lemma we establish the existence of nontrivial solutions. When <inline-formula><tex-math id="M3">\begin{document}$ q\geq2\alpha 2^{\ast} $\end{document}</tex-math></inline-formula>, by using a Pohozaev type variational identity we prove that the equation has no nontrivial solutions.</p>