Elliptic systems involving Schrödinger operators with vanishing potentials

Abstract

<p style='text-indent:20px;'>We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{cll} -\Delta u+V_{1}(x)u = \lambda\rho_{1}(x)(u+1)^{r}(v+1)^{p}&amp;\mbox{ in }&amp;\mathbb{R}^{N}\\ -\Delta v+V_{2}(x)v = \mu\rho_{2}(x)(u+1)^{q}(v+1)^{s}&amp;\mbox{ in }&amp;\mathbb{R}^{N},\\ u(x),v(x)\to 0&amp; \mbox{ as}&amp;|x|\to\infty \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p,q,r,s\geq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ V_{i} $\end{document}</tex-math></inline-formula> is a nonnegative vanishing potential, and <inline-formula><tex-math id="M3">\begin{document}$ \rho_{i} $\end{document}</tex-math></inline-formula> has the property <inline-formula><tex-math id="M4">\begin{document}$ (\mathrm{H}) $\end{document}</tex-math></inline-formula> introduced by Brezis and Kamin [<xref ref-type="bibr" rid="b4">4</xref>].As in that celebrated work we will prove that for every <inline-formula><tex-math id="M5">\begin{document}$ R&gt; 0 $\end{document}</tex-math></inline-formula> there is a solution <inline-formula><tex-math id="M6">\begin{document}$ (u_R, v_R) $\end{document}</tex-math></inline-formula> defined on the ball of radius <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula> centered at the origin. Then, we will show that this sequence of solutions tends to a bounded solution of the previous system when <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula> tends to infinity. Furthermore, by imposing some restrictions on the powers <inline-formula><tex-math id="M9">\begin{document}$ p,q,r,s $\end{document}</tex-math></inline-formula> without additional hypotheses on the weights <inline-formula><tex-math id="M10">\begin{document}$ \rho_{i} $\end{document}</tex-math></inline-formula>, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.</p>