In this paper we prove an existence result for the nonlinear elliptic problem：-△u = Ku5,u 〉 0 in Ω,u = 0 on Ω,where Ω is a smooth bounded domain of R3 and K is a positive function in Ω.Our method relies on stud...In this paper we prove an existence result for the nonlinear elliptic problem：-△u = Ku5,u 〉 0 in Ω,u = 0 on Ω,where Ω is a smooth bounded domain of R3 and K is a positive function in Ω.Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.展开更多
Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(...Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(n))is defined by||u||F=(∫_(R_(n)(F^(n)(↓△u)+|u|^(n)dx)^(1/n)In this paper,the following sharp anisotropic Moser-Trudinger inequality involving L^(n)norm u∈W^(1,n)^(SUP)(R^(n),||u||F≤1∫_(R^(n))Ф(λ_(n)|u|n/n-1(1+a||u||^(n)_(n)1/n-1)dx<+∞in the entire space R^(n)for any 0<a<1 is estabished,whereФ(t)=e^(t)-∑^(n-2)_(j=0)tj/j!,λ_(n)=n^(n/n-1)k_(n)1/n-1 and kn is the volume of the unit Wulf ball in Rn.It is also shown that the above supremum is infinity for all α≥1.Moreover,we prove the supremum is attained,that is,there exists a maximizer for the above supremum whenα>O is sufficiently small.展开更多
We study a prescribing involving Poisson kernel on the unit of PDE. As in Nirenberg problem, solutions. We prove existence in the functions problem of a conformally invariant integral equation ball. This integral equa...We study a prescribing involving Poisson kernel on the unit of PDE. As in Nirenberg problem, solutions. We prove existence in the functions problem of a conformally invariant integral equation ball. This integral equation is not the dual of any standard type there exists a Kazdan-Warner type obstruction to existence of antipodal symmetry functions class.展开更多
In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=｜x｜2αk V k eukinΩ,（0.1）whereΩR2,V k→V in C1,｜Vk｜≤A,0
文摘In this paper we prove an existence result for the nonlinear elliptic problem：-△u = Ku5,u 〉 0 in Ω,u = 0 on Ω,where Ω is a smooth bounded domain of R3 and K is a positive function in Ω.Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.
基金Supported by Natural Science Foundation of China(Grant Nos.11526212,11721101,11971026)Natural Science Foundation of Anhui Province(Grant No.1608085QA12)+1 种基金Natural Science Foundation of Education Committee of Anhui Province(Grant Nos.KJ2016A506,KJ2017A454)Excellent Young Talents Foundation of Anhui Province(Grant No.GXYQ2020049)。
文摘Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(n))is defined by||u||F=(∫_(R_(n)(F^(n)(↓△u)+|u|^(n)dx)^(1/n)In this paper,the following sharp anisotropic Moser-Trudinger inequality involving L^(n)norm u∈W^(1,n)^(SUP)(R^(n),||u||F≤1∫_(R^(n))Ф(λ_(n)|u|n/n-1(1+a||u||^(n)_(n)1/n-1)dx<+∞in the entire space R^(n)for any 0<a<1 is estabished,whereФ(t)=e^(t)-∑^(n-2)_(j=0)tj/j!,λ_(n)=n^(n/n-1)k_(n)1/n-1 and kn is the volume of the unit Wulf ball in Rn.It is also shown that the above supremum is infinity for all α≥1.Moreover,we prove the supremum is attained,that is,there exists a maximizer for the above supremum whenα>O is sufficiently small.
基金Supported in part by NSFC(Grant Nos.11501034 and 11571019)a key project of NSFC(Grant No.11631002)
文摘We study a prescribing involving Poisson kernel on the unit of PDE. As in Nirenberg problem, solutions. We prove existence in the functions problem of a conformally invariant integral equation ball. This integral equation is not the dual of any standard type there exists a Kazdan-Warner type obstruction to existence of antipodal symmetry functions class.
基金Supported by the German Research Foundation Collaborative Research Center Collaborative Research/Tran-sregio 71
文摘In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=｜x｜2αk V k eukinΩ,（0.1）whereΩR2,V k→V in C1,｜Vk｜≤A,0
