Radial quasilinear elliptic problems with singular or vanishing potentials

Abstract

<p style='text-indent:20px;'>In this paper we continue the work that we began in [<xref ref-type="bibr" rid="b6">6</xref>]. Given <inline-formula><tex-math id="M1">\begin{document}$ 1<p<N $\end{document}</tex-math></inline-formula>, two measurable functions <inline-formula><tex-math id="M2">\begin{document}$ V\left(r \right)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ K\left(r\right)> 0 $\end{document}</tex-math></inline-formula>, and a continuous function <inline-formula><tex-math id="M4">\begin{document}$ A(r) >0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$ r>0 $\end{document}</tex-math></inline-formula>), we consider the quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where all the potentials <inline-formula><tex-math id="M6">\begin{document}$ A,V,K $\end{document}</tex-math></inline-formula> may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space <inline-formula><tex-math id="M7">\begin{document}$ X $\end{document}</tex-math></inline-formula> into the sum of Lebesgue spaces <inline-formula><tex-math id="M8">\begin{document}$ L_{K}^{q_{1}}+L_{K}^{q_{2}} $\end{document}</tex-math></inline-formula>. The nonlinearity has a double-power super <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>-linear behavior, as <inline-formula><tex-math id="M10">\begin{document}$ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ q_1,q_2>p $\end{document}</tex-math></inline-formula> (recovering the power case if <inline-formula><tex-math id="M12">\begin{document}$ q_1 = q_2 $\end{document}</tex-math></inline-formula>). With respect to [<xref ref-type="bibr" rid="b6">6</xref>], in the present paper we assume some more hypotheses on <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula>, and we are able to enlarge the set of values <inline-formula><tex-math id="M14">\begin{document}$ q_1 , q_2 $\end{document}</tex-math></inline-formula> for which we get existence results.</p>