Ground state solutions for nonlinear Choquard equations with doubly critical exponents

doi:10.1016/j.aml.2021.107715

Applied Mathematics Letters

Volume 125, March 2022, 107715

https://doi.org/10.1016/j.aml.2021.107715 Get rights and content

Abstract

This paper is concerned with nonlinear Choquard equations with doubly critical growth. We apply the Pohozăev-type identity to overcome loss of compactness caused by the doubly critical nonlinearities and obtain the existence of a ground state solution.

Section snippets

Introduction and main result

In this paper, we study the existence of ground state solutions for the following Choquard equation: $- Δ u + u = ({| x |}^{α - N} * {| u |}^{p}) {| u |}^{p - 2} u + ({| x |}^{α - N} * {| u |}^{q}) {| u |}^{q - 2} u, in R^{N},$ where $N \geq 3$ , $α \in (0, N)$ , $q = \frac{N + α}{N}$ and $p = \frac{N + α}{N - 2}$ are called as the lower and the upper critical exponents with respect to the Hardy–Littlewood–Sobolev inequality respectively. It is known that the equation $- Δ u + u = ({| x |}^{α - N} * {| u |}^{r}) {| u |}^{r - 2} u, in R^{N}$ arises in the description of the quantum theory of a polaron at rest by Pekar in 1954 and the modeling of an electron

Proof of Theorem 1.1

In this section, we shall prove that Eq. (1.1) admits a ground state solution on $M$ . We firstly give the following useful observation.

Lemma 2.1

The functional $J$ is unbounded from below.

Proof

Let $u \in X$ , and $u_{t} = t u (t^{b} x)$ , $b = \frac{2 α}{N (2 + α)}$ , $t > 0$ . The standard scaling implies that $\begin{matrix} J (u_{t}) & = & \frac{t^{\frac{4 q}{2 + α}}}{2} \int_{R^{N}} {| \nabla u |}^{2} d x + \frac{t^{\frac{4}{2 + α}}}{2} \int_{R^{N}} {| u |}^{2} d x - \frac{t^{\frac{4 q}{2 + α}}}{2 q} L_{1} (u) - \frac{t^{\frac{4 p q}{2 + α}}}{2 p} L_{2} (u) . \end{matrix}$ It is easy to follow from our assumptions that $J (u_{t}) \to - \infty$ as $t \to + \infty$ . As desired. □

Lemma 2.2

Let $p > q$ and $a_{1}, a_{2}, a_{3}, a_{4}$ be positive constants, and $f (t) = a_{1} t^{\frac{4 q}{2 + α}} + a_{2} t^{\frac{4}{2 + α}} - a_{3} t^{\frac{4 q}{2 + α}} - a_{4} t^{\frac{4 p q}{2 + α}}$ for $t \geq 0$ .

Acknowledgments

B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199), the Shandong Provincial Natural Science Foundation, PR China (No. ZR2020MA006), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province .

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Ground state solutions for nonlinear Choquard equations with doubly critical exponents

Abstract

Section snippets

Introduction and main result

Proof of Theorem 1.1

Acknowledgments

J. Differential Equations

Appl. Math. Lett.

J. Funct. Anal.

J. Differential Equations

Appl. Math. Lett.

J. Math. Anal. Appl.

Untersuchung über Die Elektronentheorie Der Kristalle

Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation

Stud. Appl. Math.

Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth

Adv. Nonlinear Anal.