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Abstract:In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-\epsilon})u^{2_{s}^{\sharp}-2-\epsilon} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}\Omega,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}\Omega,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminus\Omega, \end{aligned} \right.\end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $\epsilon>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the fractional Laplace operator $(-\Delta)^{s}$ in $\Omega$ with outside zero Dirichlet boundary condition. The above problem is reduced to the subcritical fractional system $$ A_{s}u=u^{2_{s}^{\sharp}-2-\epsilon}v,\hspace{2mm}A_{s}v=u^{2_{s}^{\sharp}-1-\epsilon},\hspace{2mm}u,v>0\hspace{2mm}\mbox{in}\hspace{2mm}\Omega\hspace{2mm}\mbox{and}\hspace{2mm}u=(-\Delta)^sv=0\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminus\Omega.$$ For a general domain $\Omega$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution this http URL these results, we study the asymptotic behavior of solutions as $\epsilon\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are this http URL,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

Submission history

From: Shunneng Zhao [view email]
[v1] Fri, 22 May 2026 16:14:59 UTC (36 KB)