Existence and asymptotics of normalized solutions for the logarithmic Schrödinger system

This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)|u_(2)|^(q-2)u_(2),∫_(Ω)|u_(i)|^(2)dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2)),where Ω=R^(N)or Ω■R^(N)(N≥3)is a bounded smooth domain,andω_(i)R,μ_(i),ρ_(i)>0 for i=1,2.Moreover,p,q≥1,and 2≤p+q≤2^(*),where 2^(*):=2N/N-2.By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u^(2),firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q≤2^(*).Secondly,we consider the stability of normalized ground state solutions.Finally,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2^(*).Notably,the uncertainty of the sign of u log u^(2)in(0,+∞)is one of the difficulties of this paper,and also one of the motivations we are interested in.In particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations(i.e.,p+q=2^(*)).In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L^(2)-mass.constraint ∫_(Ω)|u_(i)|^(2)dx=ρ_(i)(i=1,2)by two different methods,respectively.Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrodinger system with logarithmic perturbations.

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).

非线性Schrödinger方程的正规化解

由于非线性Schrödinger方程在众多物理问题中有着十分重要的应用,其正规化解问题在近年来逐渐引起大批学者的关注:{-Δu+λu=g(u),x∈R^N,u∈H^1(R^N),∫R^N|u|^2=c,其中正规化条件c∈R^+是给定的,而Lagrange乘子λ是未知的.本文首先介绍单个方程在不同条件下正规化解的存在性、多解性及其他一些性质,然后介绍非线性Schrödinger方程组正规化解的相关新结果,并介绍一些与正规化解有关的待解决问题.

On the existence and regularity of vector solutions for quasilinear systems with linear coupling

We study the following coupled system of quasilinear equations:Under some assumptions on the nonlinear terms f and g, we establish some results about the existence and regularitl of vector solutions for the p-Laplacian systems by using variational methods. In particular, we get two pairs of nontrivial solutions. We also study the different asymptotic behavior of solutions as the coupling parameter λ tends to zero.

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.