Existence of ground‐state solutions for p‐Choquard equations with singular potential and doubly critical exponents

Senli Liu and Haibo Chen

Mathematische Nachrichten, 2023, vol. 296, issue 6, 2467-2502

Abstract: We are concerned with the following Choquard equation: −Δpu+A|x|θ|u|p−2u=Iα∗F(u)f(u),x∈RN,$$\begin{equation*} \hspace*{5pc}-\Delta _{p}u + \frac{A}{|x|^{\theta }}|u|^{p-2}u = {\left(I_{\alpha }*F(u)\right)}f(u), \, x\in \mathbb {R}^{N}, \end{equation*}$$where p∈(1,N)$p\in (1,N)$, α∈(0,N)$\alpha \in (0,N)$, θ∈[0,p)∪p,(N−1)pp−1$\theta \in [0,p)\cup \left(p,\frac{(N-1)p}{p-1}\right)$, A>0$A>0$, Δp$\Delta _{p}$ is the p‐Laplacian, Iα$I_{\alpha }$ is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.

Date: 2023
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