AN IMPROVEMENT ON THE CONCENTRATION-COMPACTNESS PRINCIPLE

In this paper we first improve the concentration- compactness lemma by proving that the vanishing case is a special case of dichotomy, then we apply this improved concentration- compactness lemma to a typical restrcted minimization problem, and get some new results.

Solutions for Schrodinger-Poisson system involving nonlocal term and critical exponent

In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev critical exponent.By combining the variational method with the concentration-compactness principle of Lions,we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions.The nonlinear term includes the nonlinearity f(u)~|u|^(q-2)u for the well-studied case q∈[4,6),and the less-studied case q∈(2,3),we adopt two different strategies to handle these cases.Our result improves and extends some related works in the literature.

Infinitely many solutions of p-Laplacian equations with limit subcritical growth

We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified. We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.

EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES

In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△)_(p)^(s)u+λV(x)|u|^(p-2)u=(∫_(R^(N))|U|^(P_(μ,S)^(*))/|x-y|^(μ)dy)|u|^(P_(μ,S)^(*))^(-2)u,x∈R^(N),where(-△)_(p)^(s)is the fractional p-Laplacian with 0<s<1<p,0<μ<N,N>ps,a,b>0,λ>0 is a parameter,V:R^(N)→R^(+)is a potential function,θ∈[1,2_(μ,s)^(*))and P_(μ,S)^(*)=pN-pμ/2/N-ps is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory.To the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.

Scattering for Focusing Combined Power-type NLS

We consider the scattering of Cauchy problem for the focusing combined power-type Schroodinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi.

Existence of Solutions to Nonlinear Schr?dinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem.

Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n.

Quantization of the Blow-Up Value for the Liouville Equation with Exponential Neumann Boundary Condition

In this paper,we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition.In particular,we will obtain a sharp mass quantization result for the solution sequences at a blow-up point.

Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4 s,andλ>0 is a parameter,2s*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0<λ*<+∞such that the problem has exactly two positive solutions ifλ∈(0,λ*),no positive solutions forλ>λ*,a unique solution(λ*,uλ*)ifλ=λ*,which shows that(λ*,uλ*)is a turning point in Hs(RN)for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.

EXISTENCE OF POSITIVE SOLUTION FOR A BIHARMONIC EQUATION

In this paper we consider the biharmonic equation (?) in a smooth bounded domain Ω (?) RN with boundary condition (?) where N ≥5, 1 < q < 2, λ>0 and 2= (2N)/(N-4). We prove the existence of λ such that for 0 < λ < λ the above problem has a positive solution.