A remark on the radial solutions of a modified Schr\"{o}dinger system by dual approach

Abstract

By using some reorganized ideas combined with successive approximationtechnique we establish conditions for the existence of positive entireradially symmetric solutions for a modified Schr\"{o}dinger system%\begin{equation*}\left\{ \begin{array}{l}\Delta u_{1}+\Delta (|u_{1}|^{2\gamma _{1}})\left\vert u_{1}\right\vert^{2\gamma _{1}-2}u_{1}=a_{1}(\left\vert x\right\vert )\Psi _{1}\left(u_{1}\right) F_{1}(u_{2})\text{ in }\mathbb{R}^{N}\text{,} \\ \Delta u_{2}+\Delta (|u_{2}|^{2\gamma _{2}})\left\vert u_{2}\right\vert^{2\gamma _{2}-2}u_{2}=a_{2}(\left\vert x\right\vert )\Psi _{2}\left(u_{2}\right) F_{2}(u_{1})\text{ in }\mathbb{R}^{N}\text{,}%\end{array}%\right. \end{equation*}%where $\gamma _{1},\gamma _{2}\in \left( \frac{1}{2},\infty \right) $, $%N\geq 3$ and the functions $a_{1}$, $a_{2}$, $\Psi _{1}\left( u_{1}\right) $%, $\Psi _{2}\left( u_{2}\right) $, $F_{1}$, $F_{2}$ are suitably chosen. Ourobtained results improve and extend some previous works and haveapplications in several areas of mathematics and various applied sciencesincluding the study of nonreactive scattering of atoms and molecules.