arXiv subjects

Jinkai Gao

Publications and source records attributed to Jinkai Gao.

6 recordsLinked to original sources

arXiv2026

Existence and asymptotics for the upper critical Choquard equation in dimension three

In this paper, we are interested in the existence and asymptotic behavior of least energy solutions to the upper critical Choquard equation \begin{equation*} \begin{cases} -Δu+au=\displaystyle\left(\int_Ω\frac{u^{6-α}(y)}{|x-y|^α}dy\right)u^{5-α}&\mbox{in}\ Ω, u>0 \ \ &\mbox{in}\ Ω, u=0 \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^{3}$ is a bounded domain with a $C^{2}$ boundary, $α\in (0,3)$, $a \in C(\overlineΩ) \cap C^{1}(Ω)$, and the operator $-Δ+ a$ is coercive. We first establish that the following three properties are equivalent: the existence of least energy solutions, the validity of a strict inequality in the associated minimization problem, and the positivity of the Robin function somewhere in the domain. This leads naturally to the definition of a critical function $a$. Under the perturbation $a \mapsto a + \varepsilon V$ with $a$ critical and $V \in L^{\infty}(Ω)$, we prove that least energy solutions exist. Furthermore, we establish a refined energy estimate and describe their asymptotic profile.

math.AP↗

arXiv2026

Nonexistence of single-bubble solutions for a slightly supercritical Choquard equation

In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{2^*_α+\varepsilon}(y)}{|x-y|^α}dy\Big)u^{2^*_α-1+\varepsilon},\quad u>0\ \ &\mbox{in}\ Ω, \quad \ \ u=0 \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $N\geq 3$, $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $α\in (0,N)$, $2^*_α:=\frac{2N-α}{N-2}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\varepsilon>0$ is a small parameter. In contrast with the slightly subcritical Choquard equation studied by Chen and Wang (Calculus of Variations and Partial Differential Equations, 63:235, 2024), we find that there is no chance to construct a family of single-bubble solutions as $\varepsilon\to 0^{+}$.

math.AP↗

arXiv2025

Asymptotic behavior of the least energy solutions to the Choquard equation in dimension two

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω, \quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $Ω$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $α\to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation.

math.AP↗

arXiv2025

Nondegeneracy of bubble solutions to the Choquard equation in two dimension

In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -Δu=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^α}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $α\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic decomposition. The main result of this paper can be viewed as an extension of the nondegeneracy of solutions for both the planar Liouville equation and the higher-dimensional upper critical Choquard equation.

math.AP↗

arXiv2025

Asymptotic behavior for the Brezis-Nirenberg problem. The subcritical perturbation case

In this paper, we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{2^*-1}+\varepsilon u^{q-1},\quad u>0, &{\text{in}~Ω},\\ \quad \ \ u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^N$ with $N\ge 3$ is a bounded domain, $q\in(2,2^*)$ and $2^*=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. It is well-known (H. Brézis and L. Nirenberg, \newblock {\em Comm. Pure Appl. Math.}, 36(4):437--477, 1983) that the above problem admits a positive least energy solution for all $\varepsilon >0$ and $q>\max\{2,\frac{4}{N-2}\}$. In the present paper, we first analyze the asymptotic behavior of the positive least energy solution as $\varepsilon\to 0$ and establish a sharp asymptotic characterisation of the profile and blow-up rate of the least energy solution. Then, we prove the uniqueness and nondegeneracy of the least energy solution under some mild assumptions on domain $Ω$. The main results in this paper can be viewed as a generalization of the results for $q=2$ previously established in the literature. But the situation is quite different from the case $q=2$, and the blow-up rate not only heavily depends on the space dimension $N$ and the geometry of the domain $Ω$, but also depends on the exponent $q\in(\max\{2,\frac{4}{N-2}\}, 2^*)$ in a non-trivial way.

math.AP↗

arXiv2025

Asymptotic behavior of multi-peak solutions to the Brezis-Nirenberg problem. The sub-critical perturbation case

In this paper, we consider the following well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{2^*-1}+\varepsilon u^{q-1}, \quad u>0, &{\text{in}~Ω},\\ \quad \ \ u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\geq 3$, $Ω$ is a smooth and bounded domain in $\R^{N}$, $\varepsilon>0$ is a small parameter, $q\in (2,2^*)$ and $2^*:=\frac{2N}{N-2}$ denotes the critical Sobolev exponent. The existence of solutions to the above problem has been obtained by many authors in the literature. However, as far as the authors know, the asymptotic behavior of solutions to the above problem is still open. Here we first describe the asymptotic profile of solutions to the above problem as $\varepsilon\to 0$. Then, we derive the exact blow-up rate and characterize the concentration speed and the location of concentration points in the general case of multi-peak solutions. Finally, we prove the uniqueness, nondegeneracy and count the exact number of blow-up solutions. The main results in this paper give a complete picture of multi-peak blow-up phenomena in the framework of Brezis-Peletier conjecture in the case of sub-critical perturbation. On the other hand, compared with the special case $q=2$ previously studied in the literature, we observe that the exponent $q$ has a significant impact on the asymptotic behavior, uniqueness and nondegeneracy of solutions in addition to the geometry of domain $Ω$ and space dimension $N$ which is already known in the literature.

math.AP↗