Sequences of high and low energy solutions for weighted (<i>p</i>, <i>q</i>)-equations

Abstract

<p style='text-indent:20px;'>We consider a Dirichlet elliptic equation driven by a weighted <inline-formula><tex-math id="M2">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula>-Laplace differential operator. The weights are in general different. When the reaction is "superlinear", using the fountain theorem, we show the existence of a sequence of distinct smooth solutions with energies diverging to <inline-formula><tex-math id="M3">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. When the reaction is "sublinear" (possibly resonant), we establish the existence of a sequence of nodal solutions converging to zero in <inline-formula><tex-math id="M4">\begin{document}$ C^1_0(\bar{\Omega}) $\end{document}</tex-math></inline-formula> (in particular, the energies converge to zero).</p>