A Pohozaev Identity for the Fractional Hnon System

In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-shaped and bounded domain Ω for s E （0, 1）. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritieal cases.

A Pohozaev Identity and Critical Exponents of Some Complex Hessian Equations

In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of Kahler-Einstein metric. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don＇t know if a priori radially symmetric property holds in the complex case.

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■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>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-Sobolev exponent,respectively.

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRODINGER SYSTEMS

In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation−(a+b∫_(R^(3))|∇u|^(2))Δu+V(x)u=f(u)in R^(3),with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions.Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity,especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).

Non-existence of Multi-peak Solutions to the Schrödinger-Newton System with L2-constraint

In this paper, we are concerned with the the Schrödinger-Newton system with L^(2)-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical points of V(x) under certain assumptions on asymptotic behavior of V(x) and its first derivatives near these points. Especially, the critical points of V(x) in this paper must be degenerate.The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.

New existence of multi-spike solutions for the fractional Schrodinger equations

We consider the following fractional Schr¨odinger equation:(-Δ)^(s)u+V(y)u=u^(p);u>0 in R^(N);(0.1)where s ∈(0,1),1<p<N+2s/N-2s,and V(y)is a positive potential function and satisfies some expansion condition at infinity.Under the Lyapunov-Schmidt reduction framework,we construct two kinds of multi-spike solutions for(0.1).The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the(y1;y2)-plane with k and the radius large enough.Then we show that uk is non-degenerate in our special symmetric workspace,and glue it with an n-spike solution,whose centers lie in another circle in the(y3;y4)-plane,to construct infinitely many multi-spike solutions of new type.The nonlocal property of(-Δ)^(s)is in sharp contrast to the classical Schr¨odinger equations.A striking difference is that although the nonlinear exponent in(0.1)is Sobolev-subcritical,the algebraic(not exponential)decay at infinity of the ground states makes the estimates more subtle and difficult to control.Moreover,due to the non-locality of the fractional operator,we cannot establish the local Pohozaev identities for the solution u directly,but we address its corresponding harmonic extension at the same time.Finally,to construct new solutions we need pointwise estimates of new approximate solutions.To this end,we introduce a special weighted norm,and give the proof in quite a different way.

On Global Existence for Mass-supercritical Nonlinear Fractional Hartree Equations

In this paper, we consider the nonlinear fractional Schr6dinger equations with Hartree type nonlin- earity in mass-supercritical and energy-subcritical case. Pohozaev identity, we established a threshold condition space. By sharp Hardy-Littlewood-Sobolev inequality and the which leads to a global existence of solutions in energy

Flat Solutions of Some Non-Lipschitz Autonomous Semilinear Equations May be Stable for N≥3

The authors prove that flat ground state solutions(i.e. minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions N = 1, 2 and they can be stable for N ≥ 3 for suitable values of the involved exponents.

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper,the authors study the asymptotically linear elliptic equation on manifold with conical singularities-ΔBu+λu=a(z)f(u),u≥0 in R+N,where N=n+1≥3,λ>0,z=(t,x_(1),…,x_(n)),and ΔB=(t■t)^(2)+■^(2)x_(1)+…+■^(2)x_(n).Combining properties of cone-degenerate operator,the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation,we obtain a positive solution under some suitable conditions on a and f.