一类带奇异型Trudinger-Moser项的非线性\椭圆方程非负解存在性

Let $Omega$ be a bounded domain in $R^2$containing the origin, we study the singular Trudinger-Moserembedding [sup_{smallsubstack{uin W_{0}^{1,2}(Omega) \ {| ablau|}_{L^2}leq 1}} int_{Omega} { rac{e^{ lpha{|u|}^{2}}-1}{|x|^{ eta}}}dx=int_{Omega} { rac{e^{ lpha{|v|}^{2}}-1}{|x|^{ eta}}}dx<infty,]where $u=u(x),{| abla v|}_{L^2}leq 1,v=v(x)inW_0^{1,2}(Omega), lpha>0, etain[0,2), rac{ lpha}{4pi}+ rac{ eta}{2}leq1.$And the conclusions are as follows:[limsup_{n ightarrowinfty}int_Omega rac{f(u_n)}{|x|^{ eta}} ightarrow int_Omega rac{f(u)}{|x|^{ eta}}=C(b, eta),]where $f(u)=g(u)e^{bu^2},g(0)=0,g(t)=-g(-t)>0,t>0$, $ orall b>0, orall eta in [0,2)$,$ rac{b}{4pi}+ rac{ eta}{2}leq 1$. Theexistence of non negative solutions of nonlinear equation withsingular Trudinger-Moser terms $- riangleu= rac{f(u)}{{|x|}^{ eta}}$ is proved by using the mountain passlemma.