Multiple solutions for critical Choquard-Kirchhoff type equations

Abstract

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

Ground state solutions for a generalized quasilinear Choquard equation

“…There are also existence of infinitely many solutions 5 and multibump solutions 6 . For more research on related nonlinear partial differential equations, please refer to previous studies 7‐12 and the references therein.…”

On the nonlocal Schrödinger‐poisson type system in the Heisenberg group

“…Next, we note that models of Kirchhoff‐type arise in various models of physical and biological systems, and the study for this problem by variational methods has received more and more attention in recent years, we refer the reader to previous studies 14–22 . Other important references can be found in other works 23–25 and the references therein.…”

On critical Kirchhoff problems driven by the fractional Laplacian

“…and a similar inequality ensures that the associated Euler functional is weakly lower semicontinuous. For some very recent results on Kirchhoff-type problems, see [15,18] as well as [27] for related topics.…”