Ground state solutions for nonlinearly coupled systems of Choquard type with lower critical exponent

Abstract

In this paper, we study the existence of ground state solutions for the following nonlinearly coupled systems of Choquard type with lower critical exponent by variational methods \begin{equation*} \begin{cases} \displaystyle-\Delta u+V(x)u=(I_\alpha\ast|u|^{\frac{\alpha}{N}+1})|u|^{\frac{\alpha}{N}-1}u+p|u|^{p-2}u|\upsilon|^q,&\mbox{in }\mathbb{R}^N,\\ \displaystyle-\Delta\upsilon+V(x)\upsilon=(I_\alpha\ast|\upsilon|^{\frac{\alpha}{N}+1})|\upsilon|^{\frac{\alpha}{N}-1}\upsilon+q|\upsilon|^{q-2}\upsilon|u|^p,&\mbox{in } \mathbb{R}^N. \end{cases} \end{equation*} Where $N\geq3$, $\alpha\in(0,N)$, $I_\alpha$ is the Riesz potential, $p,q\in\big(1,\sqrt{\frac{N}{N-2}}\big)$ and $Np+(N+2)q<2N+4$, $\frac{N+\alpha}{N} $ is the lower critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $V\in C(\mathbb{R}^N,(0,\infty))$ is a bounded potential function. As far as we have known, little research has been done on this type of coupled systems up to now. Our research is a promotion and supplement to previous research.

Keywords

• nonlinearly coupled systems
• lower critical exponent
• choquard type equation
• ground state solutions
• variational methods