Abstract:
We study the best constant in Sobolev inequality with weights being powers of distance from the origin in . In this variational problem, the invariance of by the group of dilatations creates some possible loss of compactness. As a result we will see that the existence of extremals and the value of best constant essentially depends upon the relation among parameters in the inequality.

Abstract:
We study the best constant in Sobolev inequality with weights being powers of distance from the origin in ￠ n. In this variational problem, the invariance of ￠ n by the group of dilatations creates some possible loss of compactness. As a result we will see that the existence of extremals and the value of best constant essentially depends upon the relation among parameters in the inequality.

Abstract:
In this paper we discuss the applications of an abstract version of concentration compactness to minimax problems. In particular, we prove the existence of solutions to semilinear elliptic problems on unbounded subsets of the Heisenberg group.

Abstract:
We prove the existence of weak solutions for the semilinear elliptic problem $$ -Delta u=lambda hu+ag(u)+f,quad uin mathcal{D}^{1,2}({mathbb{R}^N}), $$ where $lambda in mathbb{R}$, $fin L^{2N/(N+2)}$, $g:mathbb{R} o mathbb{R}$ is a continuous bounded function, and $h in L^{N/2}cap L^{alpha}$, $alpha>N/2$. We assume that $a in L^{2N/(N+2)}cap L^{infty}$ in the case of resonance and that $a in L^1 cap L^{infty}$ and $fequiv 0$ for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem using the concentration-compactness lemma of Lions. Then we prove the existence of weak solutions by applying the saddle point theorem of Rabinowitz for the cases of non-resonance and resonance, and a linking theorem of Silva in the case of strong resonance. The main theorems in this paper constitute an extension to $mathbb{R}^N$ of previous results in bounded domains by Ahmad, Lazer, and Paul [2], for the case of resonance, and by Silva [15] in the strong resonance case.

Abstract:
In this article, we extend the well-known concentration - compactness principle by Lions to the variable exponent case. We also give some applications to the existence problem for the p(x)-Laplacian with critical growth.

Abstract:
We study the p-Laplace equation with Potentials $$ -hbox{div}(| abla u|^{p-2} abla u)+lambda V(x)|u|^{p-2}u=|u|^{q-2}u, $$ $uin W^{1,p}(mathbb{R}^N)$, $xin mathbb{R}^N$ where $2leq p$, $p

Abstract:
本文考虑如下形式的非线性Schr？dinger方程 (P)。利用有界区域逼近和集中紧致原理，当位势函数不恒等于常数，非线性项 不恒等于 ，本文证明了方程(P)存在最低能量解。
In this paper, we are concerned with the following nonlinear Schr？dinger equation
(P). By using the bounded domain approximate scheme and concen-tration compactness principle, we prove the existence of a ground state solution of (P) on the Nehari manifold when constant and .

Abstract:
To account for an external magnetic field in a Hamiltonian of a quantum system on a manifold (modelled here by a subelliptic Dirichlet form), one replaces the the momentum operator $frac 1i d$ in the subelliptic symbol by $frac 1i d-alpha$, where $alphain TM^*$ is called a magnetic potential for the magnetic field $eta=dalpha$. We prove existence of ground state solutions (Sobolev minimizers) for nonlinear Schrodinger equation associated with such Hamiltonian on a generally, non-compact Riemannian manifold, generalizing the existence result of Esteban-Lions [5] for the nonlinear Schr dinger equation with a constant magnetic field on $mathbb{R}^N$ and the existence result of [6] for a similar problem on manifolds without a magnetic field. The counterpart of a constant magnetic field is the magnetic field, invariant with respect to a subgroup of isometries. As an example to the general statement we calculate the invariant magnetic fields in the Hamiltonians associated with the Kohn Laplacian and for the Laplace-Beltrami operator on the Heisenberg group.

Abstract:
We study the perturbed equation $$displaylines{ -varepsilon^{p}hbox{div}(| abla u|^{p-2} abla u)+V(x)|u|^{p-2}u=h(x,u)+K(x)|u|^{p^*-2}u,quad xin mathbb{R}^Ncr u(x) o 0quad ext{as } |x| oinfty,. }$$ where $2leq p