This paper is concerned with the existence of nodal solutions for the following quasilinear Schrödinger equation with a cubic term■where N≥3,λ>0,the function V(|x|)is a radially symmetric and positive poten...This paper is concerned with the existence of nodal solutions for the following quasilinear Schrödinger equation with a cubic term■where N≥3,λ>0,the function V(|x|)is a radially symmetric and positive potential.By using the variational method and energy comparison method,for any given integer k≥1,the above equation admits a radial nodal solution U_(k,4)^(λ)having exactly k nodes via a limit approach.Furthermore,the energy of U_9k,4)^(λ)is monotonically increasing in k and for any sequence{λ_n},up to a subsequence,■converges strongly to some■asλ_(n)→+∞,which is a radial nodal solution with exactly k nodes of the classical Schrödinger equation■Our results extend the existing ones in the literature from the super-cubic case to the cubic case.展开更多
In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x...In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x|→∞.We establish,for smallε,the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method,and employ some new analytical skills to overcome the obstacles caused by the nonlocal term φu(x)=1/4π∫R^(3)u^(2)(y)/|x−y|dy.Our results improve and extend related ones in the literature.展开更多
By bifurcation and topological methods,we study the global structure of a radial nodal solutions set of the mean curvature equation in a standard static spacetime div {a∇u√1−a^(2)|∇u|^(2)+g(∇u,∇a)/√1−a^(2)|∇u|^(2)=...By bifurcation and topological methods,we study the global structure of a radial nodal solutions set of the mean curvature equation in a standard static spacetime div {a∇u√1−a^(2)|∇u|^(2)+g(∇u,∇a)/√1−a^(2)|∇u|^(2)=λNH,with a 0-Dirichlet boundary condition on the unit ball.According to the behavior of H near 0,we obtain the global structure of sign-changing radial spacelike graphs for this problem.展开更多
We consider a nonlinear Robin problem driven by the(p,q)-Laplacian plus an indefinite potential term and with a parametric reaction term.Under minimal conditions on the reaction function,which concern only its behavio...We consider a nonlinear Robin problem driven by the(p,q)-Laplacian plus an indefinite potential term and with a parametric reaction term.Under minimal conditions on the reaction function,which concern only its behavior near zero,we show that,for all λ>0 small,the problem has a nodal solution y_(λ)∈C^(1)(Ω)and we have y_(λ)→0 in C^(1)(Ω)asλ→0^(+).展开更多
In this paper, we study the existence of nodal solutions for the following problem： -（φp（x＇））＇=α（t）φp（x＋）＋β（t）φp（x-）＋ra（t）f（x）,0〈t〈1,x（0）=x（1）=0,where φp（s）=｜s｜p-2x,α∈ C（[0,1],...In this paper, we study the existence of nodal solutions for the following problem： -（φp（x＇））＇=α（t）φp（x＋）＋β（t）φp（x-）＋ra（t）f（x）,0〈t〈1,x（0）=x（1）=0,where φp（s）=｜s｜p-2x,α∈ C（[0,1],（0,∞））,x＋=max{x,0},x-=-min{x,0},α（t）,β（t）∈C[0,1];f∈C（■,■）,sf（s）〉0 for s≠0,and f0,f∞∈（0,∞）,where f0=lim f（s）/φ p（s）,f∞=lim｜s｜→＋∞f（s）/φp（s）.We use bifurcation techniques and the approximation of connected components to prove our main results.展开更多
In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near...In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near ∞ more detail than that of [3]-[5].展开更多
We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely ma...We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely many nodal solutions.Our approach is based on variational methods combining upper-lower solutions and truncation techniques,and flow invariance arguments.展开更多
We consider the boundary value problem△u＋︳x︳^2α︳u︳p-1u=0,-1〈α≠0.in the unit ballB with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, weprove that for any po...We consider the boundary value problem△u＋︳x︳^2α︳u︳p-1u=0,-1〈α≠0.in the unit ballB with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, weprove that for any positive integer m, there exists a multi-peak nodal solution vp whose maxima and minima arelocated alternately near the origin and the other m points q1=（λcos^2Л（1-1）/m,λsin 2Л（1-1）/m,1=2,…,m＋1such that as p goes to ＋∞ ,p︳x︳2α︳up︳p-1 up→8Лe（1＋α）（1＋α）δ0＋∑^m＋1δ_1=28Лe（-1）l-1δql,whereλ∈（0, 1）, m is an odd number with（1＋α）（m＋2） -- 1 〉 0, or m is an even number. The same techniqueslead also to a more general result on general domains.展开更多
1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1...1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.展开更多
In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained mini...In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.展开更多
We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1<q<p).The reaction is(p-1)-linear near±∞and the problem is noncoercive.Using variational tools and truncation and c...We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1<q<p).The reaction is(p-1)-linear near±∞and the problem is noncoercive.Using variational tools and truncation and comparison techniques together with critical groups,we produce five nontrivial smooth solutions all with sign information and ordered.In the particular case when q=2,we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.展开更多
In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=...In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=△u(N+2)with the weight function m:[2,N+1]Z→(0,∞),[2,N+1]_(Z)={2,3,...,N+1}.As an application,we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.展开更多
Let Ω ? ? N be a ball centered at the origin with radius R > 0 and N ? 7, 2* = $ \frac{{2N}} {{N - 2}} $ . We obtain the existence of infinitely many radial solutions for the Dirichlet problem ?Δu = $ \frac{\mu }...Let Ω ? ? N be a ball centered at the origin with radius R > 0 and N ? 7, 2* = $ \frac{{2N}} {{N - 2}} $ . We obtain the existence of infinitely many radial solutions for the Dirichlet problem ?Δu = $ \frac{\mu } {{|x|^2 }}u + |u|^{2^* - 2} u + \lambda u $ in Ω, u = 0 on ?Ω for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.展开更多
A non lipschitzian nonlinear elliptic equation is reviewed and results of existence, uniqueness, positivity and classification are proved using direct methods derived from the equation.
The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a...The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=2m/2m-1∈（1,2].The following five problems are studied： （I） A sharp asymptotic behaviour of solutions as t → ＋∞ is governed by a discrete spectrum and a countable set Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same spectrum σ（B^＊）=σ（B）.Applications of these spectral results also include： （Ⅲ） a unique continuation theorem, and （IV） boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜u｜^nu）,in R^N×R＋,where n〉0.For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n → 0^＋ and to use spectral theory of the pair {B,B^＊}.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12001188)the Natural Science Foundation of Hunan Province(Grant No.2022JJ30235)。
文摘This paper is concerned with the existence of nodal solutions for the following quasilinear Schrödinger equation with a cubic term■where N≥3,λ>0,the function V(|x|)is a radially symmetric and positive potential.By using the variational method and energy comparison method,for any given integer k≥1,the above equation admits a radial nodal solution U_(k,4)^(λ)having exactly k nodes via a limit approach.Furthermore,the energy of U_9k,4)^(λ)is monotonically increasing in k and for any sequence{λ_n},up to a subsequence,■converges strongly to some■asλ_(n)→+∞,which is a radial nodal solution with exactly k nodes of the classical Schrödinger equation■Our results extend the existing ones in the literature from the super-cubic case to the cubic case.
文摘In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x|→∞.We establish,for smallε,the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method,and employ some new analytical skills to overcome the obstacles caused by the nonlocal term φu(x)=1/4π∫R^(3)u^(2)(y)/|x−y|dy.Our results improve and extend related ones in the literature.
基金Research supported by NNSF of China(11871129)Xinghai Youqing funds from Dalian University of Technology+1 种基金NSF of Liaoning Province(2019-MS-109)HSSF of Chinese Ministry of Education(20YJA790049).
文摘By bifurcation and topological methods,we study the global structure of a radial nodal solutions set of the mean curvature equation in a standard static spacetime div {a∇u√1−a^(2)|∇u|^(2)+g(∇u,∇a)/√1−a^(2)|∇u|^(2)=λNH,with a 0-Dirichlet boundary condition on the unit ball.According to the behavior of H near 0,we obtain the global structure of sign-changing radial spacelike graphs for this problem.
基金supported by Piano della Ricerca di Ateneo 2020-2022-PIACERIProject MO.S.A.I.C"Monitoraggio satellitare,modellazioni matematiche e soluzioni architettoniche e urbane per lo studio,la previsione e la mitigazione delle isole di calore urbano",Project EEEP&DLaD.S。
文摘We consider a nonlinear Robin problem driven by the(p,q)-Laplacian plus an indefinite potential term and with a parametric reaction term.Under minimal conditions on the reaction function,which concern only its behavior near zero,we show that,for all λ>0 small,the problem has a nodal solution y_(λ)∈C^(1)(Ω)and we have y_(λ)→0 in C^(1)(Ω)asλ→0^(+).
基金Supported by the National Natural Science Foundation of China(Grant No.11561038)the Natural Science Foundation of Gausu Province(Grant No.145RJZA087)
文摘In this paper, we study the existence of nodal solutions for the following problem： -（φp（x＇））＇=α（t）φp（x＋）＋β（t）φp（x-）＋ra（t）f（x）,0〈t〈1,x（0）=x（1）=0,where φp（s）=｜s｜p-2x,α∈ C（[0,1],（0,∞））,x＋=max{x,0},x-=-min{x,0},α（t）,β（t）∈C[0,1];f∈C（■,■）,sf（s）〉0 for s≠0,and f0,f∞∈（0,∞）,where f0=lim f（s）/φ p（s）,f∞=lim｜s｜→＋∞f（s）/φp（s）.We use bifurcation techniques and the approximation of connected components to prove our main results.
文摘In this paper,we have investigated the asymptotic behavior of nodal solutions of semilinear elliptic equations in R n. We conclude more precise and extensive results and give the expression of asymptotic behavior near ∞ more detail than that of [3]-[5].
基金Guangdong Basic and Applied Basic Research Foundation(2023A1515010603)。
文摘We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely many nodal solutions.Our approach is based on variational methods combining upper-lower solutions and truncation techniques,and flow invariance arguments.
基金Supported by the National Natural Science Foundation of China(No.11171214)
文摘We consider the boundary value problem△u＋︳x︳^2α︳u︳p-1u=0,-1〈α≠0.in the unit ballB with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, weprove that for any positive integer m, there exists a multi-peak nodal solution vp whose maxima and minima arelocated alternately near the origin and the other m points q1=（λcos^2Л（1-1）/m,λsin 2Л（1-1）/m,1=2,…,m＋1such that as p goes to ＋∞ ,p︳x︳2α︳up︳p-1 up→8Лe（1＋α）（1＋α）δ0＋∑^m＋1δ_1=28Лe（-1）l-1δql,whereλ∈（0, 1）, m is an odd number with（1＋α）（m＋2） -- 1 〉 0, or m is an even number. The same techniqueslead also to a more general result on general domains.
文摘1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.
文摘In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.
文摘We consider a nonlinear Dirichlet problem driven by a(p,q)-Laplace differential operator(1<q<p).The reaction is(p-1)-linear near±∞and the problem is noncoercive.Using variational tools and truncation and comparison techniques together with critical groups,we produce five nontrivial smooth solutions all with sign information and ordered.In the particular case when q=2,we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1190146411801453)the Young Teachers’Scientific Research Capability Upgrading Project of Northwest Normal University(Grant No.NWNULKQN2020-20).
文摘In this article,we established the structure of all eigenvalues and the oscillation property of corresponding eigenfunctions for discrete clamped beam equation △^(4)u(k-2)=λm(k)u(k),k∈[2,N+1]Z,u(0)=△u(0)=0=u(N+2)=△u(N+2)with the weight function m:[2,N+1]Z→(0,∞),[2,N+1]_(Z)={2,3,...,N+1}.As an application,we obtain the global structure of nodal solutions of the corresponding nonlinear problems based on the nonlinearity satisfying suitable growth conditions at zero and infinity.
基金supported by the National Natural Science Foundation of China (Grant No. 10526008)
文摘Let Ω ? ? N be a ball centered at the origin with radius R > 0 and N ? 7, 2* = $ \frac{{2N}} {{N - 2}} $ . We obtain the existence of infinitely many radial solutions for the Dirichlet problem ?Δu = $ \frac{\mu } {{|x|^2 }}u + |u|^{2^* - 2} u + \lambda u $ in Ω, u = 0 on ?Ω for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.
文摘A non lipschitzian nonlinear elliptic equation is reviewed and results of existence, uniqueness, positivity and classification are proved using direct methods derived from the equation.
文摘The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=2m/2m-1∈（1,2].The following five problems are studied： （I） A sharp asymptotic behaviour of solutions as t → ＋∞ is governed by a discrete spectrum and a countable set Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same spectrum σ（B^＊）=σ（B）.Applications of these spectral results also include： （Ⅲ） a unique continuation theorem, and （IV） boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜u｜^nu）,in R^N×R＋,where n〉0.For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n → 0^＋ and to use spectral theory of the pair {B,B^＊}.
