IR^N上拟线性椭圆型方程两解的存在性

本文证明了拟线性椭圆型方程至少有两个弱解，其中Q并且适当小．

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

类渐近线性p&q-Laplace方程弱解的全局衰减性

该文研究椭圆型方程■弱解在全空间R^N上的衰减性,其中m,n≥0,N≥3,1<q<p<N,g(x,u)关于u满足类渐近线性.证明了该方程的弱解在无穷远处关于|x|呈指数衰减性.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ H 1 (R N ) × H 1 (R N ) to the following semilinear elliptic system{-u + u = f(x, v), x ∈RN ,-v + v = g(x,u), x ∈ R N ,(0.1) by using a linking theorem and the concentration-compactness principle. The main con-ditions we imposed on the nonnegative functions f, g ∈ C 0 (R N × R 1 ) are that, f (x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem{-u + u = f(x, u), x ∈Ω,u ∈H10(Ω)where ΩRN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 & 6.pp.925–954, 2004] concerning (0.1) when f and g are asymptotically linear.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

In the present paper,we consider the nonlocal Kirchhoff problem-(ε^2a+εb∫|■u|^2)Δu+u=Q(x)u^p,u>0 in R^3,,where a,b>0,1<p<5 andε>0 is a parameter.Under some assumptions on Q(x),we show the existence and local uniqueness of positive multi-peak solutions by LyapunovSchmidt reduction method and the local Pohozaev identity method,respectly.

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

In this paper,we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:-△pu-△qu = |u|p*-2u + μ|u|r-2u in Ω,u|Ω = 0,whereΩRN is a bounded domain,N > p,p* = Np/N-p is the critical Sobolev exponent and μ > 0. We prove that if 1 < r < q < p < N,then there is a μ0 > 0,such that for any μ∈ (0,μ0),the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].

THE EXISTENCE OF INFINITELY MANY SO UTIONS OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS

In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R^1.

THE EXISTENCE OF A WEAK SOLUTION OF QUASILINEAR ELLIPTIC EQUATION WITH CRITICAL SOBOLV EXPONENT ON UNBOUNDED DOMAIN

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THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

This paper considers the following quasilinear elliptic problem where Ω is a bounded regular domain in RN(N≥3), N ＞ p＞ 1. Wheng(u) satisfies suitable conditions and g(u)u -βfou g(s)ds is unbounded, a(x) is a H?ldcr continuous function which changes sign onΩamid fΩ -a(x)dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N＞p≥1, in particular, the existence of a positive solution to the problem for N＞p≥2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p 2. They prove also that if g(u)=u(q-2)u with p ＜q＜p and Ω+ = xΩa(x) ＞ 0 is a nonempty open set, then the above problem possesses infinitely many solutions.

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN R^N WITH ZERO MASS

In this paper,we prove the existence of at least one positive solution pair(u,v) ∈ D1,2(RN) × D1,2(RN) to the following semilinear elliptic system △u = K(x)f(v),x ∈ RN,△v = K(x)g(u),x ∈ RN(0.1) by using a linking theorem,where K(x) is a positive function in Ls(RN) for some s > 1 and the nonnegative functions f,g ∈ C(R,R) are of quasicritical growth,superlinear at infinity.We do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.Our main result can be viewed as a partial extension of a recent result of Alves,Souto and Montenegro in [1] concerning the existence of a positive solution to the following semilinear elliptic problem △u = K(x)f(u),x ∈ RN,and a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in RN.

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrdinger-Kirchhoff type -εpMεp_N∫RN|▽u|p△pu+V(x)|u|p-2u=f(u) in R^N, where △_p is the p-Laplacian operator, 1 < p < N, M :R^+→R^+ and V :R^N→R^+are continuous functions,ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and LyusternikSchnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

L~∞ESTIMATE FOR SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS IN R^N

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EIGENVALUE PROBLEM FOR FIELD EQUATION INVOLVING LIMITING NONLINEARITY

In this paper, we study the following eigenvalue problem involving limiting nonlinearity by a new way of using the concentration-compactness principle. Even in the nonlimiting situation, we improve the known result in the positive mass case in the sense of seeking a nontrivial solution of the above eigenvalue problem.

EXISTENCE OF NONTRIVIAL SOLUTION OF QUASILINEAR ELLIPTIC EIGENVALUE PROBLEM ON R^n WITH NATURAL GROWTH CONDITIONS

In this paper, we get the existence result of the nontrivial weak solution (λ, u) of the following eigenvalue problem with natural growth conditions.

C^(1,α)PARTIAL REGULARITY FOR SOLUTIONS OF NONLINEAR ELLIPTIC SYSTEMS

We use the blow-up method to get the C1,α partial regularity results for the following elliptic systems satisfying natural growth conditions:

带非齐次扰动项和Hardy-Sobolev临界指数项的双调和方程的两个弱解的存在性

本文用变分方法研究如下RN中包含0的有界光滑区域Ω上带非齐次扰动项和Hardy奇异项及Sobolev临界指数项的非线性双调和问题:的非平凡解的存在性,其中n是∂Ω的单位外法向量,λ∈R,0≤s≤4,N≥5,且2**=2N/(N-4)是H02(Ω)嵌入到Lp(Ω)的Sobolev临界指数,∆2是重调和算子,f∈H0-2(Ω).本文在f的范数适当小且相关参数满足适当的条件时证明(*)至少有两个非平凡解.本文的主要结果将Tarantello(1992)关于调和方程的结果推广到了双调和方程,同时也将Deng和Wang(1999)的结果推广到了含Hardy奇异项的情形,更重要的是本文考虑了2≤s≤4的情形.

R^N上临界增长p-Kirchhoff型方程的非平凡解的存在性

本文主要研究以下具临界增长的非线性p-Kirchhoff型方程的非平凡解的存在性:{-(a+b∫_(R^N)|▽u|~p)?_pu=|u|^(p*-2)u+μf (x)|u|^(q-2)u, x∈R^N,(0.1) u∈D^(1,p)(R^N),其中a≥0,b>0,1<p<N,1<q<p,p*=N_p/(N-p),μ≥0,?_pu=div(|▽u|^(p-2)▽u)表示p-Laplace算子对函数u的作用, f∈L(p*/(p*-q))(R^N)\{0}且f是非负的.本文利用Ekeland变分原理和山路定理证明方程(0.1)在适当条件下至少存在两个非平凡解.

拟线性椭圆组弱解的部分正则性

本文考虑下面椭圆组弱解的部分正则性:■其中Ω是R^n的一个有界区域,n≥3,N≥1。这里,重复的英文(希腊)字母表示从1到N(n)求和。

PARTIAL REGULARITY FOR WEAK SOLUTIONS OF QUASILINEAR ELLIPTIC SYSTEMS

In this note, we study the partial regularity for the weak solutions of the elliptic systems:Dα(Aαβij（x,u）Dβuj)=fi（x,u,Du）, x∈Ω,i=1,2,…,N, （1）where Ω is a bounded domain in Rn, n≥3 and N≥1. Here, the repeated Latin letters andrepeated Greek letters are summed from 1 to N and 1 to n respectively. We assume thefollowing conditions: