ON POSITIVE G-SYMMETRIC SOLUTIONS OF A WEIGHTED QUASILINEAR ELLIPTIC EQUATION WITH CRITICAL HARDY-SOBOLEV EXPONENT

In this paper, we are concerned with a weighted quasilinear elliptic equation involving critical Hardy–Sobolev exponent in a bounded G-symmetric domain. By using the symmetric criticality principle of Palais and variational method, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on the potential and the nonlinearity.     

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.     

PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT

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.     

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 OF AN ELLIPTIC EIGENVALUE PROBLEM INVOLVING SUBCRITICAL OR CRITICAL EXPONENTS

In this paper, we prove the existence of a positive solution, a negative solution and a sign-changing solution of a semilinear elliptic eigenvalue problem with constraint involving subcritical and critical Sobolev exponents. The solutions are obtained in the ω-limit sets of some descending flow curves whose starting points are specifically chosen.     

EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS AND HARDY TERMS

This paper deals with the existence of solutions to the elliptic equation -△uμu/|x|2=λu |u|2*-2u f(x, u) in Ω, u = 0 on ( a)Ω, where Ω is a bounded domain in RN(N≥3),0∈Ω,2*=2N/N-2,λ＞0,λ(a)σμ, σμ is the spectrum of the operator -△- μI/|x|2with zero Dirichlet boundary condition, 0 ＜μ＜-μ,-μ=(N-2)2/4,f(x,u) is an asymmetric lower order perturbation of |u|2*-1 at infinity. Using the dual variational methods, the existence of nontrivial solutions is proved.     

EXISTENCE RESULTS FOR DEGENERATE ELLIPTIC EQUATIONS WITH CRITICAL CONE SOBOLEV EXPONENTS

In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.     

对角形拟线性椭圆方程组的一般特征问题

本文对拟线性椭圆方程组的一般特征问题得到极小解在L∞中的界,并利用变分方法证明了它的极小解的存在性.     

INFINITELY MANY SOLUTIONS FOR A SINGULAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS IN RN

In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△u -μ u |x|2 = α|u|2*|x(s|s)*2u+ βa(x)|u|r-2u, x ∈RN.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.     

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents ■where Ω is a smooth bounded domain in R^N with 0 ∈ ?Ω and all the principle curvatures of ?Ω at 0 are negative,a ∈ C¹(Ω,R^*+),μ> 0,0 2(q+1)/(q-1).By2^*:=2N/(N-2) and 2^*(s):(2(N-s))/(N-2) we denote the critical Sobolev exponent and Hardy-...     

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS INVOLVING m CRITICAL HARDY-SOBOLEV EXPONENTS AND m SIGN-CHANGING WEIGHT FUNCTION

In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.     

一类含凹凸非线性项拟线性椭圆方程的多重解

本文研究了一类拟线性椭圆方程,其中非线性项f在无穷远处(p-1)-次线性增长,非线性项g在无穷远处超线性增长.利用三临界点定理,获得了该类方程多重解的存在性,结果推广了Kristaly等人最近的相关结果.     

EXISTENCE OF POSITIVE SOLUTIONS TO QUASI-LINEAR EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT

This paper is concerned with the quasi-linear equation with critical SobolevHardy exponent where Ω 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, 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.     

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.     

REGULARITY RESULTS FOR SOME QUASI-LINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL EXPONENTS

The authors show the regularity of weak solutions for some typical quasi-linear elliptic systems governed by two p-Laplacian operators. The weak solutions of the following problem with lack of compactness are proved to be regular when α(x) and α,β,p, q satisfy some conditions: where Ω(?) RN (N≥3) is a smooth bounded domain.     

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, we study the multiplicity results of positive solutions for a class of quasi-linear elliptic equations involving critical Sobolev exponent. With the help of Nehari manifold and a mini-max principle, we prove that problem admits at least two or three positive solutions under different conditions.     

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.     

ON NEUMANN PROBLEM FOR SOMESEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

1IntroductionLetnbeaboundeddomaininRewithCIboundarytn23.Weconsidertheprobleminwhichp=ac,q=M,7denotestheunitoutwardnormaltoon,A>0,p20.Inthisp8Per,wesuppose(Al)a(z)eL"(n),a(z)50.(afl)op(2)EC(on),op(x)20,M==op(:).(op2)op(zo)=M,icEOf.InasmallneighborhoodUofic,thedoesnQliesononesideOfthetangentplaneOfonatic,andthe(n--l)principlecurvaturesofxo(withrespecttotheinnernormalOfon)arepositive.(gi)g(z,u)ismeasurablein2,continuousinu,sup{g(2,u)::En,0Su5an}< coforeveryfixedan>0andg(x,0)=0.(g2)!5op=0…     

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.