论文标题
具有二维吸收非线性的Choquard类型方程
A Choquard type equation with a singular absorption nonlinearity in two dimension
论文作者
论文摘要
在本文中，我们展示了在$ \ mb r^2 $中提出的单数问题$（\ mc p_ \ la）$的非负解决方案（见下文）。我们通过近似函数$ u^{ - β} \ log（u）$通过函数$ l_ \ e（u）$来实现这一目标来实现这一目标，从而将pointWisely收敛到$ -U^β\ log（u）$ as $ \ e \ e \ ra 0 $。使用变异技术，扰动的方程$ - \ de u+l_ \ e（u）= \ ds \ la \ left（\ i {\ i {\ om} \ frac {f（y（y））} {| x-y |^μ} h_0^{1}（\ om）$当参数$ \ la> 0 $很小时。让$ \ e \ ra 0 $并证明了一个渐变估算值，我们表明解决方案$ u_ \ e $收敛到原始问题$（\ mc p_ \ la）$的非平凡的非负解决方案。
In this article, we show the existence of a nonnegative solution to the singular problem $(\mc P_\la)$ posed in a bounded domain $Ω$ in $\mb R^2$ (see below). We achieve this by approximating the singular function $u^{-β}\log(u)$ by a function $l_\e(u)$ which pointwisely converges to $-u^β\log(u)$ as $\e \ra 0$. Using variational techniques, the perturbed equation $-\De u+l_\e(u)=\ds\la \left(\I{\Om}\frac{F(u(y))}{|x-y|^μ}dy\right)f(u(x))$ is shown to have a solution $u_\e \in H_0^{1}(\Om)$ when the parameter $\la >0$ is small enough. Letting $\e \ra 0$ and proving a pointwise gradient estimate, we show that the solution $u_\e$ converges to a nontrivial nonnegative solution of the original problem $(\mc P_\la)$.
