FAST AND SLOW DECAY SOLUTIONS FOR SUPERCRITICAL FRACTIONAL ELLIPTIC PROBLEMS IN EXTERIOR DOMAINS

<正>1 Introduction and Main ResultsWe construct classic solutions of the following supercritical nonlinear fractional exteriorproblem■where ■ and B₁ is the unit ball in■ As usual,the operator (■)^s is the fractional Laplacian,defined at any point ■ as     

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT

This paper is concerned with the following nonlinear Dirichlet problem:{-Δpu=|u|^p*-2 u λf(x,u) x∈Ω；u=0 x∈эΩ} whereΔp^u = div(|∧u|^p-2∧u) is the p-Laplacian of u,Ω is a bounded in R^n(n≥3），1＜p＜n, p=pn/n-p is the critical exponent for the Sobolev imbedding,λ＞0 and f(x,u)satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p=2 or f(x,u) = |u|^q-2 u, where 1＜q＜p, are generalized.     

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.     

MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved.Furthermore,by the improved inequalities and the linking theorem,in a new k-order Sobolev-Hardy space,we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△(k)u:=-△u-(((N-2)2)/4)U/︱X︱2-1/4 sum from i=1 to(k-1) u/(︱x︱2(In(i)R/︱x︱2))=f(x,u),x ∈Ω,u=0,x ∈Ω,where 0 ∈ΩBa(0)RN,N≥3,ln(i)=i éj=1 ln(j),and R=ae(k-1),where e(0)=1,e(j) = ee(j-1) for j≥1,ln(1)=ln,ln(j)=ln ln(j-1) for j≥2.Besides,positive andnegative solutions are obtained by a variant mountain pass theorem.     

SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions to the problem.     

POSITIVE SOLUTIONS FOR NONHOMOGENEOUS ELLIPTIC EQUATIONSWITH CRITICAL GROWTH ON R~(2)

1IntroductionIntillspaper,weare(follccrlledwitlltileexistellccofPositly(tsollltiollsoftilefollowillgnonhonlogelleousellipticProblclll:whereg(x)EL'(R'),g(:v)Z0alldg(x)t0,f(x,t)=h(x,t).hi=withb>0,h(x,t)EC(R=xR,R)alldtilefollowing(CI)-(C3)11old:(CI)sliphillM0.linljfl- x,h(T,t)(t--if-=011llif'orllllyforxeR2.hill}t:l-:,t)(axle(~ltJ')= lx,11lliforllllyl'Ora:6RZ.ltl~la(C3)ThereexistM>0,aE(0,1]sucllthatFOllowing[1,5],wesaythatf(x,t)=h(x,f…     

一类含临界增长的多重调和方程组非平凡解的存在性

本文研究了带临界指标的多重调和半线性椭圆方程组.利用变分法,得到了此类方程组非平凡解的存在性和非存在性的条件.     

QUASILINEAR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT AND HARDY TERM

This paper is concerned with a p-Laplacian elliptic problem with critical Sobolev-Hardy exponent and Hardy term. By variational methods and genus theory, we guarantee that this problem has at least one positive solution and admits many solutions with negative energy under sufficient conditions.     

MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS

In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions to this problem is obtained.     

具有临界指数的半线性椭圆组弱解的正则性

本文研究具有临界指数的方程组。用ＤｅＧｉｏｒｇｅ估计方法，获得方程组的解的有界性。     

带对流项的非线性椭圆型问题爆炸解的存在性与渐近行为

设Ω是RN（N≥3）中的C2有界区域,对带负对流项的情形,对更广泛的非线性项,构造一种新型的非线性变换将爆炸解问题,转化成等价的带奇异项的Dirichlet问题,应用极大值原理得到了爆炸解问题解的最小爆炸速度.应用三种摄动方法,结合上下解方法、二阶椭圆型偏微分方程的估计理论得到了爆炸解的存在性.特别允许非线性项的系数不仅在Ω的内部子区域恒为零而且在Ω上可适当无界.随后再应用摄动方法,将所得结果推广到RN,得到了整体爆炸解的存在性以及在无穷远附近的最小爆炸速度.而对带正对流项的情形,对更广泛的非线性项,构造爆炸上下解u和u在Ω上满足u≤u,得到了爆炸解u的存在性且在Ω上满足u≤u≤u.     

带对流项的非线性椭圆型问题爆炸解的存在性与渐近行为

设Ω是RN(N≥3)中的C2有界区域,对带负对流项的情形,对更广泛的非线性项,构造一种新型的非线性变换将爆炸解问题,转化成等价的带奇异项的Dirichlet问题,应用极大值原理得到了爆炸解问题解的最小爆炸速度.应用三种摄动方法,结合上下解方法、二阶椭圆型偏微分方程的估计理论得到了爆炸解的存在性.特别允许非线性项的系数不仅在Ω的内部子区域恒为零而且在Ω上可适当无界.随后再应用摄动方法,将所得结果推广到RN,得到了整体爆炸解的存在性以及在无穷远附近的最小爆炸速度.而对带正对流项的情形,对更广泛的非线性项,构造爆炸上下解-u和u-在Ω上满足u-≤-u,得到了爆炸解u的存在性且在Ω上满足u-≤u≤-u.     

SINGULAR LIMIT SOLUTIONS FOR TWO-DIMENSIONAL ELLIPTIC PROBLEMS WITH EXPONENTIALLY DOMINATED NONLINEARITY

51. IntroductionThe pmpose of this paper is to consider the eristence of solutions u: n C C - R for thefollowing Din~ Problem:where 7 6 (0, 1). Our motiVation is to study the etotence of non-rnuumal solutions withsingular ~ as the parameter p tends to 0 and eXtend the results of 13, 2] to more generalfunctions which are juSt exponentially dominated. The additional term e7u wilds the possibility Of better steady state models for physical phenomena having exponential nonlinewhies(see for ex…     

EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities.It is shown,by means of variational methods,that under certain conditions,the system has at least two positive solutions.     

SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN R^N

For the following elliptic problem where 2-(s)=2(N-s)/N-2 is the critical Sobolev-Hardy exponent, h(x)∈(D1,2(RN))*, the dual space of (D1,2(RN)), 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     

INFINITELY MANY SOLUTIONS FOR AN ELLIPTIC PROBLEM INVOLVING CRITICAL NONLINEARITY

We study the following elliptic problem:{-div(a(x)Du)=Q(x)|u|2-2u+λu x ∈Ω ,u=0 on ΩUnder certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.     

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES

In this article, we study the following critical problem involving the fractional Laplacian:■where ? ? R~N(N α) is a bounded smooth domain containing the origin, α∈(0, 2),0 ≤ s, t α, 1 ≤ q 2, λ 0, 2*_α(t) =2(N-t)/(N-α) is the fractional critical Sobolev-Hardy exponent, 0 ≤γ γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.     

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

This paper is concerned with the harmonic equation(P_(?ε)) : ?u = 0, u 0 in B~n and ?u/?ν+((n-2)/2)u =((n-2)/2) Ku~(n/(n-2)?ε) on S~(n-1)where B~n is the unit ball in R~n, n ≥ 4 with Euclidean metric g_0, ?B~n= S~(n-1)is its boundary, K is a function on S~(n-1)and ε is a small positive parameter. We construct solutions of the subcritical equation(P_(-ε)) which blow up at one critical point of K. We give also a sufficient condition on the function K to ensure the nonexistence of solutions for(P_(-ε)) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation(P_(+ε)).