Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent

Abstract

<abstract><p>In this paper, we are concerned with the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations with critical growth</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -{\rm{div}}(g^{p}(u)|\nabla u|^{p-2}\nabla u)+ g^{p-1}(u)g'(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u = h(x, u), \; \; x\in \mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ 1 < p < N $. Under some suitable conditions, we prove that the above equation has a nontrivial positive solution by variational methods. To some extent, our results improve and supplement some existing relevant results.</p></abstract>