Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities

Abstract

We consider the existence and asymptotic behavior of solutions to a doubly nonlocal elliptic equation

$$\begin{aligned} -\Big (1+\lambda \displaystyle \int \limits _{\mathbb {R}^{N}}|\nabla u|^{2}\,\textrm{d} x\Big )\Delta u+\omega u=\displaystyle \big ( |x|^{-\alpha }*G(u)\big )g(u) \ \ \ \text{ in } \ \mathbb {R}^{N}, \end{aligned}$$

where $N\ge 3$, $\lambda >0$ is a parameter, $\omega >0$ is a constant, $\alpha \in (0,N)$, $G\in {\mathcal {C}}^{1}(\mathbb {R},\mathbb {R})$, and $g=G'$. Under almost necessary assumptions on G, by using the variational arguments and employing a Pohozaev-type constraint technique, we prove that there exists a ground state solution $u_{\lambda }$ for the above equation. Moreover, the asymptotic behavior of the ground state solution is also explored as $\lambda \rightarrow 0^{+}$.