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)).     

含下有界非线性项的一类弹性梁方程解的存在性与多解性

利用锥拉伸和锥压缩型的Krasnosel‘skii不动点定理，对于一类含下有界非线性项的四阶两点边值问题考察了解和正解的存在性和多解性，这类问题通常描述两端固定的弹性梁的形变。     

四阶边值问题正解的存在性与多解性

本文讨论了非线性四阶边值问题u^(4)(t)=φ（t）f(u(t),u“(t),t∈(0,1),u(0)=u(1)=u“(0) =u“(1)=0正确的存在性，其中φ（t)∈C([0,1],[0,∞)),f(u,v)∈C（[0,∞],[0,∞))。利用锥压缩与锥拉伸不动点定理，给出了该问题正解存在与多个正解存在的充分条件。     

MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION

We study a quasilinear elliptic equation with polynomial growth coefficients. The existence of infinitely many solutions is obtained by a dual method and a nonsmooth critical point theory.     

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.     

STABILITY AND BOUNDEDNESS OF SOLUTIONS FOR A CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATION

In this paper, with the help of Lyapunov functional approach, sufficient conditions for the asymptotic stability of zero solution for a certain fourthorder non-linear delay differential equation are given.     

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-△u+(λa(x)+1)u=(1/|x|α*F(u))f(u) in R~N,where N≥3,0 αmin{N,4},λ is a positive parameter and the nonnegative potential function a(x) is continuous.Using variational methods,we prove that if the potential well int(a~(-1)(0)) consists of k disjoint components,then there exist at least 2~k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed as λ→+∞.     

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.     

含有Sobolev-Hardy临界指数的拟线性椭圆方程解的存在性和多重性

本文研究了一类含有Sobolev-Hardy临界指数的拟线性奇异椭圆方程解的存在性和多重性.利用Ekeland变分原理和Clark临界点定理证明了该问题非平凡解和无穷多解的存在性,推广了已有结果.     

一类半线性抛物方程组的爆破临界指标

本文考察一类半线性抛物方程组的Cauchy问题，计算出了该问题的爆破临界指标。     

LOCATION OF THE BLOW UP POINT FOR POSITIVE SOLUTIONS OF A BIHARMONIC EQUATION INVOLVING NEARLY CRITICAL EXPONENT

In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin‘s function corresponding to the Green‘s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.     

具快速增长非线性的定态Cahn-Hilliard方程解的遍历有限性

本文考虑具快速增长非线性的定态Ｃａｈｎ－Ｈｉｌｌｉａｒｄ方程的Ｎｅｕｍｍａｎ边值问题：－△（－△ｕ＋ｆ（ｕ））＝ｇ,１／｜Ω｜ｎｕｄｘ＝ａ证明了解的存在性和遍历有限性。     

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 MULTIPLICITY OF NONTRIVIAL SOLUTIONS TO p-KIRCHHOFF TYPE EQUATION

In this paper, by applying Fountain theorem, we study the existence of nontrivial solutions to the nonlinear kirchhof nonlocal equation under Ambrosetti-Rabinowitztype growth conditions.     

ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF A CERTAIN FOURTH-ORDER DIFFERENTIAL EQUATION

The present paper establishes sufficient conditions for the uniformly bounded of any solution of (1.1) and which tend to zero as t → ∞.     

MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A FOURTH ORDER NONLINEAR ELLIPTIC EQUATION

0Introducti0nWeinvestigatetheexistellce0fs0lutiolls0fthef0urthordernonlillearellipticb0undaryvalueproblemwhereu =max{u,0},sisreal,andcisn0taneigenvalue0f-A11llderDirichletb0undarycondition.Hereweassumethatflisab0unded0pensetinR"withsmootl1b0undaryOn.The0perat0rA2denotesthebihalm0nic0perat0r.Weassumethatbisllotancigenvalue0fAa cAunderDirichletb0undaryc0llditi0n.Theno11lillcarequati0uwithjumpillgn0l1lillearityhavebeenextcllsivclystudi(}dbymany.uth.rs[3'4'6'7'8].Theyst11dicdtheexistence()fs0l…     

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: △2 u = K(x)u p , u > 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p + 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.     

MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHR?DINGER EQUATIONS WITH SATURABLE NONLINEARITY

In this paper, we construct sign-changing radial solutions for a class of Schr?dinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer k, by using a minimization argument, we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda’s theorem, that the minimizer is the desired solution.     

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.