The Schrodinger equation -△u＋λ2u=｜u｜2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1＋u1=｜u1｜2q-1u1-εb（x）｜u2...The Schrodinger equation -△u＋λ2u=｜u｜2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1＋u1=｜u1｜2q-1u1-εb（x）｜u2｜1｜u1｜q-1u1,-△u2＋u2=｜u2｜2q-2u2-εb（x）｜u1｜1｜u2｜q-1u2 has multiple-bump solutions which behave like Uλ in the neighborhood of some points. For u=（u1,u2）∈H1（R3）×H1（R3）, a nonlinear functional Iε（u）=I1（u1）＋I2（u2）-ε/q∫R3b（x）｜u1｜q｜u2｜qdx,is defined,where I1（u1）=1/2｜｜u1｜｜2-1/2q∫R3｜u1｜2qdx and I2（u2）=1/2｜｜u2｜｜2ω-1/2q∫R3｜u2｜2qdx. It is proved that the solutions of the system are the critical points of I,. Let Z be the smooth solution manifold of the unperturbed problem and TzZ is the tangent space. The critical point of I is rewritten as the form of z ＋ w, where w ∈ （TzZ）⊥. Using some properties of Iε, it is proved that there exists a critical point of I, close to the form which is a multi-bump solution.展开更多
In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,...In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,4},λis a positive parameter and the nonnegative potential function a(x)is continuous.Using variational methods,we prove that if the potential well int(a^-1(0))consists of k disjoint components,then there exist at least 2^k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed asλ→+∞.展开更多
This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation:{−Δu+λV(x)u=ulogu^(2)inRN,u∈H^(1)(R^(N)),where N≥1,⋋>0 is a parameter and the nonnega...This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation:{−Δu+λV(x)u=ulogu^(2)inRN,u∈H^(1)(R^(N)),where N≥1,⋋>0 is a parameter and the nonnegative continuous function V:ℝ^(N)→ℝhas potential wellΩ:=int V^(−1)(0)which possesses k disjoint bounded componentsΩ=∪^(k)_(j)=1Ω_(j).Using the variational methods,we prove that if the parameter⋋>0 is large enough,then the equation has at least 2^(k)−1 multi-bump positive solutions.展开更多
We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 ...We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 and suppose that G（x） has k local minimum points. For h 〉 0 small, we find multi-bump bound states ~bh （x, t） ---- e-iE~/huh （X） with Uh concentrating at the local minimum points of G（x） simultaneously as h ~ O. The potentials V（x） and K（x） are allowed to be either compactly supported or unbounded at infinity.展开更多
文摘In this paper,by using the method of Lyapunov-Schmidt reduction,we obtain the existence of multi-bump solutions for planar Schrödinger-Poisson system.
基金The National Natural Science Foundation of China(No.11171063)the Natural Science Foundation of Jiangsu Province(No.BK2010404)
文摘The Schrodinger equation -△u＋λ2u=｜u｜2q-2u has a unique positive radial solution Uλ, which decays exponentially at infinity. Hence it is reasonable that the Schrolinger system -△u1＋u1=｜u1｜2q-1u1-εb（x）｜u2｜1｜u1｜q-1u1,-△u2＋u2=｜u2｜2q-2u2-εb（x）｜u1｜1｜u2｜q-1u2 has multiple-bump solutions which behave like Uλ in the neighborhood of some points. For u=（u1,u2）∈H1（R3）×H1（R3）, a nonlinear functional Iε（u）=I1（u1）＋I2（u2）-ε/q∫R3b（x）｜u1｜q｜u2｜qdx,is defined,where I1（u1）=1/2｜｜u1｜｜2-1/2q∫R3｜u1｜2qdx and I2（u2）=1/2｜｜u2｜｜2ω-1/2q∫R3｜u2｜2qdx. It is proved that the solutions of the system are the critical points of I,. Let Z be the smooth solution manifold of the unperturbed problem and TzZ is the tangent space. The critical point of I is rewritten as the form of z ＋ w, where w ∈ （TzZ）⊥. Using some properties of Iε, it is proved that there exists a critical point of I, close to the form which is a multi-bump solution.
基金L.Guo is supported by the Fundamental Research Funds for the Central Universities(2662018QD039)T.Hu is supported by the Project funded by China Postdoctoral Science Foundation(2018M643389).
文摘In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-Δu+(λa(x)+1)u=(1/|x|α*F(u))f(u)in R^N,where N≥3,0<α<min{N,4},λis a positive parameter and the nonnegative potential function a(x)is continuous.Using variational methods,we prove that if the potential well int(a^-1(0))consists of k disjoint components,then there exist at least 2^k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed asλ→+∞.
基金supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq/Brazil(Grant No.304804/2017-7)supported by Natural Science Foundation of Shanghai(Grant Nos.20ZR1413900 and 18ZR1409100)。
文摘This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation:{−Δu+λV(x)u=ulogu^(2)inRN,u∈H^(1)(R^(N)),where N≥1,⋋>0 is a parameter and the nonnegative continuous function V:ℝ^(N)→ℝhas potential wellΩ:=int V^(−1)(0)which possesses k disjoint bounded componentsΩ=∪^(k)_(j)=1Ω_(j).Using the variational methods,we prove that if the parameter⋋>0 is large enough,then the equation has at least 2^(k)−1 multi-bump positive solutions.
基金supported by National Natural Science Foundation of China(11201132)Scientific Research Foundation for Ph.D of Hubei University of Technology(BSQD12065)the Scientific Research Project of Education Department of Hubei Province(Q20151401)
文摘We study the bound states to nonlinear Schrodinger equations with electro magnetic fields ihδψ/δt=（h/i -A（x））^2ψ＋V（x）ψ-K（x）｜ψ｜^p-1ψ=0,on R＋ ×R^N. Let G（x）=[V（x）p＋1/p-1-N/2][K（x）]-2/p-1 and suppose that G（x） has k local minimum points. For h 〉 0 small, we find multi-bump bound states ~bh （x, t） ---- e-iE~/huh （X） with Uh concentrating at the local minimum points of G（x） simultaneously as h ~ O. The potentials V（x） and K（x） are allowed to be either compactly supported or unbounded at infinity.
