EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|~(N-2) u + f(x, u), x ∈ ?,u ∈ W_0~(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R~N(N 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ λ_1, λ = λ_?(? = 2, 3, ···), and λ_? is the eigenvalues of the operator(-?_N, W_0~(1,N)(?)),which is defined by the Z_2-cohomological index.     

Degenerate Nonlinear Elliptic Equations Lacking in Compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|~(p-2)u) in R~N, where ▽_pu =|▽u|~(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.     

THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω→ R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|p* |u|p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|p+|u|p* + a(x)), (2) where L≥1, 1pN,p* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.     

The existence and multiplicity of solutions of a fractional Schrdinger-Poisson system with critical growth

In this paper, we study the existence and multiplicity of solutions for the following fractional Schr¨odinger-Poisson system:ε~(2s)(-?)~su + V(x)u + ?u = |u|~2_s~*-2 u + f(u) in R~3,ε~(2s)(-?)~s? = u~2 in R~3,(0.1)where 3/4 s 1, 2_s~*:=6/(3-2s)is the fractional critical exponent for 3-dimension, the potential V : R~3→ R is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System(0.1) via variational methods.     

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R~N with Exponential Critical Growth

We study a quasilinear Schr?dinger equation{-ε~NΔNu+V(x)|u|~(N-2)u=Q(x)f(u) in R~N,0u∈W~(1,N)(R~N),u(x)|x|→∞→0,where V,Q are two continuous real functions on R~N and ε0 is a real parameter.Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger–Moser inequality,we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL

In this article, we study the following nonhomogeneous Schr¨odinger-Poisson equations -?u + λV(x)u + K(x)φu = f(x, u) + g(x), x ∈ R~3,?-?φ = K(x)u~2, x ∈ R~3,where λ 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.     

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.     

Multiplicity of positive solutions for a class of concave-convex elliptic equations with critical growth

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g(x)|u|2*-2u+λf(x)|u|q-2u,x∈Ω,u=0,x∈■Ω where Ω■R~N(N≥3) is an open bounded domain with smooth boundary, 1 q 2, λ 0.2*=2 N/(N-2)is the critical Sobolev exponent,f∈L2~*/(2~*-q)(Ω)is nonzero and nonnegative,and g ∈ C(■) is a positive function with k local maximum points. By the Nehari method and variational method,k+1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].     

一类不确定薛定谔基尔霍夫方程解的存在性和集中性（英文）

This paper is concerned with the nonlinear Schrodinger-Kirchhoff system -(a+b∫_(R~3)|▽u|~2 dx)△u+λV(x)u=f(x,u) in R~3,where constants a 0,b≥ 0 and λ 0 is a parameter.We require that V(x) ∈C(R~3)and has a potential well V~(-1)(0).Combining this with other suitable assumptions on K and f,the existence of nontrivial solutions is obtained via variational methods.Furthermore,the concentration behavior of the nontrivial solution is also explored on the set V~(-1)(0) as λ→+∞ as well.It is worth noting that the(PS)-condition can not be directly got as done in the literature,which makes the problem more complicated.To overcome this difficulty,we adopt different method.     

INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

In this article, we give a new proof on the existence of infinitely many signchanging solutions for the following Brézis-Nirenberg problem with critical exponent and a Hardy potential -?u- μu/(|x|~2)= λu + |u|~2~(*-2)u in ?, u = 0 on ??,where ? is a smooth open bounded domain of R~N which contains the origin, 2~*=(2N)/(N-2) is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7, μ∈ [0, -4),2and =(N-2)~2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ 0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.     

The Ground State Solutions for Kirchhoff-Schr¨odinger Type Equations with Singular Exponential Nonlinearities in RN*

In this paper, the authors consider the following singular Kirchhoff-Schr¨odinger problem M Z RN |u|N + V (x)|u|N dx (N u + V (x)|u|N?2u) = f(x, u) |x|η in RN , (Pη) where 0 < η < N, M is a Kirchhoff-type function and V (x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity.Combining variational techniques with some estimates, they get the existence of ground state solution for (Pη). Moreover, they also get the same result without the A-R condition.     

具位势项的快扩散方程的第二临界指标

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.     

The critical case for a Berestycki-Lions theorem

We consider the existence of the ground states solutions to the following Schrdinger equation:△u+V(x)u=f(u),u∈H1(RN),where N 3 and f has critical growth.We generalize an earlier theorem due to Berestycki and Lions about the subcritical case to the current critical case.     

Existence of solutions for a critical fractional Kirchhoff type problem in(49) R~N

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity:(∫∫_(R~(2N)(|u(x)-u(y)|~2)/(|x-y|~(N+2s)dxdy)~(θ-1)(-?)~su = λh(x)u~(p-1)+u~(2_s*-1) in R~N,where(-?)~s is the fractional Laplacian operator with 0 s 1, 2_s~*= 2N/(N-2s), N 2s, p ∈(1, 2_s~*),θ∈ [1, 2_s~*/2), h is a nonnegative function and λ is a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.     

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR SOME DEGENERATE HYPERBOLIC EQUATION

The author considers the global existence and global nonexistence of the initial-boundary value problem for some degenerate hyperbolic equation of the form utt- div(|▽u|p-2▽u)= |u|mu, (x,t) ∈ [0, ∞) ×Ωwith p ＞ 2 and m ＞ 0. He deals with the global solutions by D.H.Sattinger's potential well ideas. At the same time, when the initial energy is positive, but appropriately bounded,the global nonexistence of solutions is verified by using the analysis method.     

Existence of ground state solutions to Hamiltonian elliptic system with potentials

In this paper, we investigate nonlinear Hamiltonian elliptic system{-?u + b(向量)(x) · ?u +(V(x) + τ)u = K(x)g(v) in R~N,-?v-(向量)b(x)·?v +(V(x) + τ)v = K(x)f(u) in R~N,u(x) → 0 and v(x) → 0 as |x| →∞,where N ≥ 3, τ 0 is a positive parameter and V, K are nonnegative continuous functions,f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.     

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {△2u- a+b∫R3|▽u|2dx △u+V(x)u=f(x, u), x∈ R3,u∈H2(R3),wherea, b 0 are constants and the primitive of the nonlinearityfis of superlinear growth near infinity inuand is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the AmbrosettiRabinowitz type condition.     

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→+∞.     

Ground state solutions for a class of fractional Kirchhoff equations with critical growth

In this paper, we study the effect of lower order perturbations in the existence of positive solutions to the fractional Kirchhoff equation with critical growth■ where a, b 0 are constants, μ 0 is a parameter,■ , and V : R~3→ R is a continuous potential function. For suitable assumptions on V, we show the existence of a positive ground state solution, by using the methods of the Pohozaev-Nehari manifold, Jeanjean's monotonicity trick and the concentration-compactness principle due to Lions(1984).     

SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH A GRADIENT

In this article, we consider existence and nonexistence of solutions to problem{ -?p u+g(x, u)|▽u|p=fin ?,u= 0 on??(0.1)with 1 p ∞, wherefis a positive measurable function which is bounded away from 0in ?, and the domain ? is a smooth bounded open set in RN(N≥ 2). Especially, under the condition thatg(x, s) = 1/|s|α(α 0) is singular ats= 0, we obtain thatα pis necessary and sufficient for the existence of solutions inW1,p0(?) to problem(0.1) whenfis sufficiently regular.