Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian

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}\partial\Omega,\end{aligned} \right.\end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $0 0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the spectral fractional Laplacian. 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 term.Among 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 studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.