Existence and Symmetry of Positive Solutions for a Modified Schrödinger System Under the Keller–Osserman Type Conditions

Abstract

Our purpose is to investigate the existence of entire solutions $\left( u_{1},u_{2}\right)$ which are of class $C^{1}$ and satisfy the system

$$\begin{aligned} \left\{ \begin{array}{l} \Delta u_{1}+\Delta (|u_{1}|^{2\gamma _{1}})\left| u_{1}\right| ^{2\gamma _{1}-2}u_{1}=p_{1}(\left| x\right| )\Psi _{1}\left( u_{1}\right) F_{1}(u_{2})\quad \text { in }\mathbb {R}^{N}(N\ge 3)\text {,} \\ \Delta u_{2}+\Delta (|u_{2}|^{2\gamma _{2}})\left| u_{2}\right| ^{2\gamma _{2}-2}u_{2}=p_{2}(\left| x\right| )\Psi _{2}\left( u_{2}\right) F_{2}(u_{1})\quad \text { in }\mathbb {R}^{N}(N\ge 3).\text { } \end{array} \right. \end{aligned}$$

Here $p_{1}$, $F_{1}$, $\Psi _{1}$, $p_{2}$, $\Psi _{2}$ and $F_{2}$ are continuous functions satisfying new conditions. The original motivation of our work comes from the Schrödinger equation.