In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <...In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.展开更多
In this paper,we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrodinger equation with aμ-Laplacian term(BNLS).Such BNLS models the propagation of intense laser beams in a bu...In this paper,we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrodinger equation with aμ-Laplacian term(BNLS).Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term.Denoting by Qpthe ground state for the BNLS withμ=0,we prove that in the mass-subcritical regime p∈(1,1+8/d),there exist orbit ally stable ground state solutions for the BNLS when p∈(-λ0,∞)for someλ0=λ0(p,d,‖Qp‖L2)>0.Moreover,in the mass-critical case p=1+8/d,we prove the orbital stability on a certain mass level below‖Q*‖L2,provided thatμ∈(-λ1,0),where■and Q*=Q1+8/d.The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality.Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when p is negative and p∈(1,1+8/d].展开更多
In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,w...In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].展开更多
In this paper,we investigate the following p-Kirchhoff equation{∫R^(N)|u|^(2)dx=ρ,(a+b)∫RN(|Δu}^(p)+|u|^(p))dx)(-Δpu+|u|^(p-2u)=|u|^(s-2)u+μu,x∈R^(N),where a>0,b≥0,p>0 are constants,constants,p*=N-P/Np i...In this paper,we investigate the following p-Kirchhoff equation{∫R^(N)|u|^(2)dx=ρ,(a+b)∫RN(|Δu}^(p)+|u|^(p))dx)(-Δpu+|u|^(p-2u)=|u|^(s-2)u+μu,x∈R^(N),where a>0,b≥0,p>0 are constants,constants,p*=N-P/Np is the critical Sobolev exponent,μis a Lagrange multiplier,-Δpu=-div(|Δu|_(p-2)u),2<p<N2p,μ∈R,and s∈(2N/N+2p-2,p*).We demonstratethat he p-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.展开更多
In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv fo...In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.展开更多
We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond t...We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint,namely,∫R2|u|2dx=c for c>0 given.Of particular interest is the competing and double L2-supercritical case,i.e.,µ=−1 and min{p,q}>4.We prove several existence and multiplicity results depending on the size of the exponents p and q.It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations.In addition,the asymptotic behavior of the solutions and the associated Lagrange multipliersλas c→0 is described.展开更多
In this paper,we study normalized solutions to a fourth-order Schrődinger equation with a positive second-order dispersion coefficient in the mass supercritical regime.Unlike the well-studied case where the second-ord...In this paper,we study normalized solutions to a fourth-order Schrődinger equation with a positive second-order dispersion coefficient in the mass supercritical regime.Unlike the well-studied case where the second-order term is zero or negative,the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer.Under suitable control of the second-order dispersion coefficient and mass,we find at least two radial normalized solutions,a ground state and an excited state,together with some asymptotic properties.It is worth pointing out that in the considered repulsive case,the compactness analysis of the related Palais-Smale sequences becomes more challenging.This forces the implementation of refined estimates of the Lagrange multiplier and the energy level to obtain normalized solutions.展开更多
In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),...In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).展开更多
In this paper we discuss the following Kirchhoff equation(−a+b R _(R3)|∇u|^(2) dx∆u+V(x)u+λu=µ|u|^(q−2)u+|u|^(p−2)u in R^(3),R R^(3) u^(2) dx=c^(2),where a,b,µ and c are positive numbers,λis unknown and a...In this paper we discuss the following Kirchhoff equation(−a+b R _(R3)|∇u|^(2) dx∆u+V(x)u+λu=µ|u|^(q−2)u+|u|^(p−2)u in R^(3),R R^(3) u^(2) dx=c^(2),where a,b,µ and c are positive numbers,λis unknown and appears as a Lagrange multiplier,143<q<p<6 and V is a continuous non-positive function vanishing at infinity.Under some mild assumptions on V,we prove the existence of a mountain pass normalized solution.Here we study the existence of normalized solution to mass supercritical Kirchhoff equation with potential via the minimax principle and Nehari-Pohozaev manifold.展开更多
In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At firs...In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .展开更多
In this paper, we are concerned with the the Schrödinger-Newton system with L^(2)-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical poin...In this paper, we are concerned with the the Schrödinger-Newton system with L^(2)-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical points of V(x) under certain assumptions on asymptotic behavior of V(x) and its first derivatives near these points. Especially, the critical points of V(x) in this paper must be degenerate.The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.展开更多
The main goal of this paper is to introduce necessary efficiency conditionsfor a class of multi-time vector fractional variational problems with nonlinear equal-ity and inequality constraints involving higher-order pa...The main goal of this paper is to introduce necessary efficiency conditionsfor a class of multi-time vector fractional variational problems with nonlinear equal-ity and inequality constraints involving higher-order partial derivatives.We considerthe multi-time multiobjective variational problem(MFP)of minimizing a vector ofpath-independent curvilinear integral functionals quotients subject to PDE and/or PDIconstraints,developing an optimization theory on the higher-order jet bundles.展开更多
基金supported by the BIT Research and Innovation Promoting Project(2023YCXY046)the NSFC(11771468,11971027,11971061,12171497 and 12271028)+1 种基金the BNSF(1222017)the Fundamental Research Funds for the Central Universities。
文摘In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2_(s)^(*),and 2_(s)^(*)=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.
基金partially supported by the National Natural Science Foundation of China(11501137)partially supported by the National Natural Science Foundation of China(11501395,12071323)the Guangdong Basic and Applied Basic Research Foundation(2016A030310258,2020A1515011019)。
文摘In this paper,we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrodinger equation with aμ-Laplacian term(BNLS).Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term.Denoting by Qpthe ground state for the BNLS withμ=0,we prove that in the mass-subcritical regime p∈(1,1+8/d),there exist orbit ally stable ground state solutions for the BNLS when p∈(-λ0,∞)for someλ0=λ0(p,d,‖Qp‖L2)>0.Moreover,in the mass-critical case p=1+8/d,we prove the orbital stability on a certain mass level below‖Q*‖L2,provided thatμ∈(-λ1,0),where■and Q*=Q1+8/d.The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality.Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when p is negative and p∈(1,1+8/d].
基金Supported by the National Natural Science Foundation of China (11971393).
文摘In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].
基金supported by Natural Science Foundation of Fujian Province(No.2022J013392020J01708)National Foundation Training Program of Jimei University(ZP2020057).
文摘In this paper,we investigate the following p-Kirchhoff equation{∫R^(N)|u|^(2)dx=ρ,(a+b)∫RN(|Δu}^(p)+|u|^(p))dx)(-Δpu+|u|^(p-2u)=|u|^(s-2)u+μu,x∈R^(N),where a>0,b≥0,p>0 are constants,constants,p*=N-P/Np is the critical Sobolev exponent,μis a Lagrange multiplier,-Δpu=-div(|Δu|_(p-2)u),2<p<N2p,μ∈R,and s∈(2N/N+2p-2,p*).We demonstratethat he p-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.
基金supported by National Natural Science Foundation of China(Grant No.11861053)supported by National Natural Science Foundation of China(Grant No.11831009)supported by National Natural Science Foundation of China(Grant No.11901582)。
文摘In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.
基金supported by National Natural Science Foundation of China (Grant No.12071486)supported by National Natural Science Foundation of China (Grant No.11671236)Shandong Provincial Natural Science Foundation (Grant No. ZR2020JQ01)。
文摘We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint,namely,∫R2|u|2dx=c for c>0 given.Of particular interest is the competing and double L2-supercritical case,i.e.,µ=−1 and min{p,q}>4.We prove several existence and multiplicity results depending on the size of the exponents p and q.It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations.In addition,the asymptotic behavior of the solutions and the associated Lagrange multipliersλas c→0 is described.
基金supported by National Natural Science Foundation of China (Grant No.11901147)the Fundamental Research Funds for the Central Universities of China (Grant No.JZ2020HGTB0030)。
文摘In this paper,we study normalized solutions to a fourth-order Schrődinger equation with a positive second-order dispersion coefficient in the mass supercritical regime.Unlike the well-studied case where the second-order term is zero or negative,the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer.Under suitable control of the second-order dispersion coefficient and mass,we find at least two radial normalized solutions,a ground state and an excited state,together with some asymptotic properties.It is worth pointing out that in the considered repulsive case,the compactness analysis of the related Palais-Smale sequences becomes more challenging.This forces the implementation of refined estimates of the Lagrange multiplier and the energy level to obtain normalized solutions.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371159 and 11771166)
文摘In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).
文摘In this paper we discuss the following Kirchhoff equation(−a+b R _(R3)|∇u|^(2) dx∆u+V(x)u+λu=µ|u|^(q−2)u+|u|^(p−2)u in R^(3),R R^(3) u^(2) dx=c^(2),where a,b,µ and c are positive numbers,λis unknown and appears as a Lagrange multiplier,143<q<p<6 and V is a continuous non-positive function vanishing at infinity.Under some mild assumptions on V,we prove the existence of a mountain pass normalized solution.Here we study the existence of normalized solution to mass supercritical Kirchhoff equation with potential via the minimax principle and Nehari-Pohozaev manifold.
文摘In this paper, we look for solutions to the following Schrödinger-Bopp-Podolsky system with prescribed L<sup>2</sup>-norm constraint , where q ≠ 0, a, ρ> 0 are constants. At first, by the classical minimizing argument, we obtain a ground state solution to the above problem for sufficiently small ρwhen . Secondly, in the case p = 6, we show the nonexistence of positive solutions by using a Liouville-type result. Finally, we argue by contradiction to investigate the orbital stability of standing waves for .
基金supported by the National Natural Science Foundation of China(No.11771469)Qing Guo is supported by National Natural Science Foundation of China(No.11771469)。
文摘In this paper, we are concerned with the the Schrödinger-Newton system with L^(2)-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical points of V(x) under certain assumptions on asymptotic behavior of V(x) and its first derivatives near these points. Especially, the critical points of V(x) in this paper must be degenerate.The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.
文摘The main goal of this paper is to introduce necessary efficiency conditionsfor a class of multi-time vector fractional variational problems with nonlinear equal-ity and inequality constraints involving higher-order partial derivatives.We considerthe multi-time multiobjective variational problem(MFP)of minimizing a vector ofpath-independent curvilinear integral functionals quotients subject to PDE and/or PDIconstraints,developing an optimization theory on the higher-order jet bundles.