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- W3210195123 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 = lambdarho_{1}(x)(u+1)^{r}(v+1)^{p}&mbox{ in }&mathbb{R}^{N} -Delta v+V_{2}(x)v = murho_{2}(x)(u+1)^{q}(v+1)^{s}&mbox{ in }&mathbb{R}^{N}, u(x),v(x)to 0& mbox{ as}&|x|toinfty 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,sgeq0 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=M2>begin{document}$ 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> 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>" @default.
- W3210195123 created "2021-11-08" @default.
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- W3210195123 date "2022-01-01" @default.
- W3210195123 modified "2023-10-18" @default.
- W3210195123 title "Elliptic systems involving Schrödinger operators with vanishing potentials" @default.
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- W3210195123 doi "https://doi.org/10.3934/dcds.2021156" @default.
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