In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequal...In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.展开更多
For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy expon...For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.展开更多
In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^...In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β.展开更多
In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).
Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves ...Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problem △^u-μ u/{x}^4=｜μ｜^2＊-2u＋f（x） with Dirichlet boundary condition on 偏dΩ under some assumptions on f（x） and μ.展开更多
This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponent whereΩ(?)RN(N(?)3)is a smooth bounded domain,0∈Ω,0(?)s<p,1<p<N,p(s):=p(N-s)/N-p is the critical Sobolev-Hardy exp...This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponent whereΩ(?)RN(N(?)3)is a smooth bounded domain,0∈Ω,0(?)s<p,1<p<N,p(s):=p(N-s)/N-p is the critical Sobolev-Hardy exponent,λ>0,p(?)r<p,p:=Np/N-p is the critical Sobolev exponent,μ>,0(?)t<p,p(?)q<p(t)=p(N-t)/N-p.The existence of a positive solution is proved by Sobolev-Hardy inequality and variational method.展开更多
In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation me...In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.展开更多
In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）...In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）,（12-n）/（n-4）].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=（Ωa）a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.展开更多
This paper concerns the existence and multiplicity of solutions for some semilinear elliptic equations with critical Sobolev exponent, Hardy term and the sublinear nonlinearity at origin. By using Ekeland,s variationa...This paper concerns the existence and multiplicity of solutions for some semilinear elliptic equations with critical Sobolev exponent, Hardy term and the sublinear nonlinearity at origin. By using Ekeland,s variational principle, we conclude the existence of nontrivial solution for this problem, the Clark＇s critical point theorem is used to prove the existence of infinitely many solutions for this problem with odd nonlinearity.展开更多
基金Supported by NSFC(10471047)NSF Guangdong Province(05300159).
文摘In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.
文摘For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.
文摘In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ｜x｜^2^-μ=α｜x｜^s^-｜μ｜^2＊（s）-2u＋βα（x）｜u｜^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β.
文摘In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).
文摘Abstract： Let Ω belong to R^N be a smooth bounded domain such that 0 ∈ Ω, N ≥ 5, 2^＊ ：2N/N-4 is the critical Sobolev exponent, and f（x） is a given function. By using the variational methods, the paper proves the existence of solutions for the singular critical in the homogeneous problem △^u-μ u/{x}^4=｜μ｜^2＊-2u＋f（x） with Dirichlet boundary condition on 偏dΩ under some assumptions on f（x） and μ.
基金This research is supported by the National Natural Science Foundation of China(l0171036) and the Natural Science Foundation of South-Central University For Nationalities(YZZ03001).
文摘This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponent whereΩ(?)RN(N(?)3)is a smooth bounded domain,0∈Ω,0(?)s<p,1<p<N,p(s):=p(N-s)/N-p is the critical Sobolev-Hardy exponent,λ>0,p(?)r<p,p:=Np/N-p is the critical Sobolev exponent,μ>,0(?)t<p,p(?)q<p(t)=p(N-t)/N-p.The existence of a positive solution is proved by Sobolev-Hardy inequality and variational method.
基金Supported by National Natural Science Foundation of China(11071198)
文摘In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory.
文摘In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）,（12-n）/（n-4）].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=（Ωa）a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.
基金Supported by National Natural Science Foundation of China（No. 10471113） and Research Award Program for 0utstanding Young Teachers in Higher Education Institutions of M0E, P.R.C. and by Doctor Foundation of Southwest Normal University（No. SWNUB2005021）.
文摘This paper concerns the existence and multiplicity of solutions for some semilinear elliptic equations with critical Sobolev exponent, Hardy term and the sublinear nonlinearity at origin. By using Ekeland,s variational principle, we conclude the existence of nontrivial solution for this problem, the Clark＇s critical point theorem is used to prove the existence of infinitely many solutions for this problem with odd nonlinearity.
