In this paper,we obtain a lower semicontinuity result with respect to the strong L^1-convergence of the integral functionals F(u,Ω)=∫_Ωf(x,u(x),εu(x))dx defined in the space SBD of special functions with bounded deformation.Hereεu represents the absolutely continuous part of the symmetrized distributional derivative Eu.The integrand f satisfies the standard growth assumptions of order p>1 and some other conditions.Finally,by using this result,we discuss the existence of an constrained variational problem.
In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear Schrdinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0≤β =μ1=μ2=μ3 or β >max{μ1, μ2, μ3}.
In this article,the authors obtain an integral representation for the relaxation of the functional F(x,u,Ω):=intergral from n=Ω to ∞(f(x,u(x),E u(x))dx ) if u∈W 1,1(Ω,RN), +∞ otherwise, in the space of functions of bounded deformation,with respect to L1-convergence.Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu.f(x,p,ξ)satisfying weak convexity assumption.
