Ground state sign-changing solutions for critical Choquard equations with steep well potential

Abstract

In this paper, we study sign-changing solution of the Choquard type equation \begin{align*} -\Delta u+\left(\lambda V(x)+1\right)u =\big(I_\alpha\ast|u|^{2_\alpha^*}\big)|u|^{2_\alpha^*-2}u +\mu|u|^{p-2}u\quad \mbox{in}\ \mathbb{R}^N, \end{align*} where $N\geq3$, $\alpha\in((N-4)^+,N)$, $I_\alpha$ is a Riesz potential, $p\in\big[2_\alpha^*,\frac{2N}{N-2}\big)$, $2_\alpha^*:=\frac{N+\alpha}{N-2}$ is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, $\mu>0$, $\lambda>0$, $V\in C(\mathbb{R}^N,\mathbb{R})$ is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for $\lambda,\mu$ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as $\lambda\rightarrow+\infty$ and $\mu\rightarrow+\infty$, respectively.

Keywords

• choquard equation
• upper critical exponent
• steep well potential
• ground state sign-changing solution
• asymptotic behaviour