For any s∈(0,1),let the nonlocal Sobolev space X^(s)(R^(N))be the linear space of Lebesgue measure functions from R^(N) to R such that any function u in X^(s)(R^(N))belongs to L2(R^(N))and the function(x,y)→(u(x)-u...For any s∈(0,1),let the nonlocal Sobolev space X^(s)(R^(N))be the linear space of Lebesgue measure functions from R^(N) to R such that any function u in X^(s)(R^(N))belongs to L2(R^(N))and the function(x,y)→(u(x)-u(y)√K(x-y)is in L^(2)(R^(N),R^(N)).First,we show,for a coercive function V(x),the subspace E:={u∈X^s(R^N):f_(R)^N}V(x)u^(2)dx<+∞}of X^(s)(R^(N))is embedded compactly into L^(p)(R^(N))for p\in[2,2_(s)^(*)),where 2_(s)^(*)is the fractional Sobolev critical exponent.In terms of applications,the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation-L_(k)u+V(x)u=f(x,u),x∈R^N are obtained,where-L_(K)is an integro-differential operator and V is coercive at infinity.展开更多
In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a ste...In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.展开更多
In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlin...In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.展开更多
In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a sm...In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .展开更多
In this article, we give a new proof on the existence of infinitely many sign- changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -△u-μ(u/|x|^2)=λu+|u...In this article, we give a new proof on the existence of infinitely many sign- changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -△u-μ(u/|x|^2)=λu+|u|^2*-2u inΩ, u=0 on eΩ,where Ω is a smooth open bounded domain of R^N which contains the origin, 2*=2N/n-2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7, μ ∈ [0, μ- 4), and μ=(N-2)^2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ 〉 0. Our proof is based on a combination of invariant sets method and Lj usternik-Schnirelman theory.展开更多
In this paper, we study the existence of least energy sign-changing solutions for aKirchhoff-type problem involving the fractional Laplacian operator. By using the constraintvariation method and quantitative deformati...In this paper, we study the existence of least energy sign-changing solutions for aKirchhoff-type problem involving the fractional Laplacian operator. By using the constraintvariation method and quantitative deformation lemma, we obtain a least energy nodal solu-tion ub for the given problem. Moreover, we show that the energy of ub is strictly larger thantwice the ground state energy. We also give a convergence property of ub as b O, where bis regarded as a positive parameter.展开更多
In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.
In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(...In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, f : R → R is continuous, strictly increasing and f(0) = 0.展开更多
In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈...In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].展开更多
We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^...We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^(N),N≥1.By using a special function space with the constraint ∫_(Ω)udx=0,under suitable assumptions on f and g(x,u),we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem.Recent results from the literature are extended.展开更多
In this paper,we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics.More precisely,for any given nonnegat...In this paper,we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics.More precisely,for any given nonnegative integer k,by using a minimization argument,we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem,and show,by a deformation lemma and Miranda's theorem,that the minimizer is the desired solution.展开更多
The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions. Based on this, this paper intends to study nodal solutions for a kind of Schr<span st...The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions. Based on this, this paper intends to study nodal solutions for a kind of Schr<span style="white-space:nowrap;">ö</span>dinger-Poisson equation. We consider a class of Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>dinger-Poisson equation with variable potential under weaker conditions in this paper. By introducing some new techniques and using truncated functional, Hardy inequality and Poho<span style="white-space:nowrap;"><span style="white-space:nowrap;">ž</span></span>aev identity, we obtain an existence result of a least energy sign-changing solution and a ground state solution for this kind of Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>dinger-Poisson equation. Moreover, the energy of the sign-changing solution is strictly greater than the ground state energy.展开更多
Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinea...Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.展开更多
By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive soluti...By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.展开更多
In this paper,we consider the fractional critical Schrödinger equation(FCSE)(-Δ)^(s)u-|u|2^(*)s-2 u=0,where u∈˙H^(s)(R^(N)),N≥4,0<s<1 and 2^(*)s=2 N/N-2 s is the critical Sobolev exponent of order s.By ...In this paper,we consider the fractional critical Schrödinger equation(FCSE)(-Δ)^(s)u-|u|2^(*)s-2 u=0,where u∈˙H^(s)(R^(N)),N≥4,0<s<1 and 2^(*)s=2 N/N-2 s is the critical Sobolev exponent of order s.By virtue of the variational method and the concentration compactness principle with the equivariant group action,we obtain some new type of nonradial,sign-changing solutions of(FCSE)in the energy space˙H^(s)(R^(N)).The key component is that we take the equivariant group action to construct several subspace of˙H^(s)(R^(N))with trivial intersection,then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of(FCSE)in˙H^(s)(R^(N)).展开更多
This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones a...This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.展开更多
This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depen...This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.展开更多
Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-...Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-speed rail contact wires,electronic component connectors,and other devices.Overcoming the contradiction between low alloying and high performance is an important challenge in the development of solid solution-strengthened copper alloys.Taking the typical solid solution-strengthened alloy Cu-4Zn-1Sn as the research object,we proposed using the element In to replace Zn and Sn to achieve low alloying in this work.Two new alloys,Cu-1.5Zn-1Sn-0.4In and Cu-1.5Zn-0.9Sn-0.6In,were designed and prepared.The total weight percentage content of alloying elements decreased by 43%and 41%,respectively,while the product of ultimate tensile strength(UTS)and electrical conductivity(EC)of the annealed state increased by 14%and 15%.After cold rolling with a 90%reduction,the UTS of the two new alloys reached 576 and 627MPa,respectively,the EC was 44.9%IACS and 42.0%IACS,and the product of UTS and EC(UTS×EC)was 97%and 99%higher than that of the annealed state alloy.The dislocations proliferated greatly in cold-rolled alloys,and the strengthening effects of dislocations reached 332 and 356 MPa,respectively,which is the main reason for the considerable improvement in mechanical properties.展开更多
In the present paper, the author studies the existence of sign-changing solutions for nonlinear elliptic equations, which have jumping nonlinearities, and may or may not be resonant with respect to Fucik spectrum, via...In the present paper, the author studies the existence of sign-changing solutions for nonlinear elliptic equations, which have jumping nonlinearities, and may or may not be resonant with respect to Fucik spectrum, via linking methods under Cerami condition.展开更多
基金supported by the NSFC(12261107)Yunnan Key Laboratory of Modern Analytical Mathematics and Applications(202302AN360007).
文摘For any s∈(0,1),let the nonlocal Sobolev space X^(s)(R^(N))be the linear space of Lebesgue measure functions from R^(N) to R such that any function u in X^(s)(R^(N))belongs to L2(R^(N))and the function(x,y)→(u(x)-u(y)√K(x-y)is in L^(2)(R^(N),R^(N)).First,we show,for a coercive function V(x),the subspace E:={u∈X^s(R^N):f_(R)^N}V(x)u^(2)dx<+∞}of X^(s)(R^(N))is embedded compactly into L^(p)(R^(N))for p\in[2,2_(s)^(*)),where 2_(s)^(*)is the fractional Sobolev critical exponent.In terms of applications,the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation-L_(k)u+V(x)u=f(x,u),x∈R^N are obtained,where-L_(K)is an integro-differential operator and V is coercive at infinity.
基金the National Natural Science Foundation of China (11971393)。
文摘In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.
基金supported by the National NaturalScience Foundation of China(12071170,11961043,11931012,12271196)supported by the excellent doctoral dissertation cultivation grant(2022YBZZ034)from Central China Normal University。
文摘In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.
文摘In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on R<sup>N</sup> × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, I<sub>α</sub> is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .
基金supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China
文摘In this article, we give a new proof on the existence of infinitely many sign- changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -△u-μ(u/|x|^2)=λu+|u|^2*-2u inΩ, u=0 on eΩ,where Ω is a smooth open bounded domain of R^N which contains the origin, 2*=2N/n-2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7, μ ∈ [0, μ- 4), and μ=(N-2)^2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ 〉 0. Our proof is based on a combination of invariant sets method and Lj usternik-Schnirelman theory.
基金supported by the NSFC(11501231)the "Fundamental Research Funds for the Central Universities"(WUT2017IVA077,2018IB014)
文摘In this paper, we study the existence of least energy sign-changing solutions for aKirchhoff-type problem involving the fractional Laplacian operator. By using the constraintvariation method and quantitative deformation lemma, we obtain a least energy nodal solu-tion ub for the given problem. Moreover, we show that the energy of ub is strictly larger thantwice the ground state energy. We also give a convergence property of ub as b O, where bis regarded as a positive parameter.
基金Supported by NSFC 11361077Young Academic and Technical Leaders Program(2015HB028)Yunnan Normal University,Lian Da Scholar Program
文摘In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.
基金Supported by the Foundation of the Office of Science and Technology of Henan(122102310373)Supported by the NSF of Education Department of Henan Province(12B110025)
文摘In this paper, by using the fixed-point index theory, we study the existence of sign-changing solution of some three-point boundary value problems {y ''(t) + f(y) = 0, t ∈ [0, 1], y' (0) = 0, y(1) = αy(η), where 0 < α < 1, 0 < η < 1, f : R → R is continuous, strictly increasing and f(0) = 0.
基金supported by the Fundamental Research Funds for the Central Universities(2014QNA67)
文摘In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u + λV(x)u = K(x)|u|^p-2u x∈R^N, u→0 as |x|→ +∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].
文摘We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^(N),N≥1.By using a special function space with the constraint ∫_(Ω)udx=0,under suitable assumptions on f and g(x,u),we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem.Recent results from the literature are extended.
基金supported by National Natural Science Foundation of China(11971147)China Postdoctoral Science Foundation(2019M662475)Henan Postdoctoral Research Grant(201902026).
文摘In this paper,we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics.More precisely,for any given nonnegative integer k,by using a minimization argument,we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem,and show,by a deformation lemma and Miranda's theorem,that the minimizer is the desired solution.
文摘The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions. Based on this, this paper intends to study nodal solutions for a kind of Schr<span style="white-space:nowrap;">ö</span>dinger-Poisson equation. We consider a class of Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>dinger-Poisson equation with variable potential under weaker conditions in this paper. By introducing some new techniques and using truncated functional, Hardy inequality and Poho<span style="white-space:nowrap;"><span style="white-space:nowrap;">ž</span></span>aev identity, we obtain an existence result of a least energy sign-changing solution and a ground state solution for this kind of Schr<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>dinger-Poisson equation. Moreover, the energy of the sign-changing solution is strictly greater than the ground state energy.
文摘Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.
文摘By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.
基金supported by National Key Research and Development Program of China(No.2020YFA0712900)NSFC(No.112371240 and No.12431008)supported by NSFC(No.12001284)。
文摘In this paper,we consider the fractional critical Schrödinger equation(FCSE)(-Δ)^(s)u-|u|2^(*)s-2 u=0,where u∈˙H^(s)(R^(N)),N≥4,0<s<1 and 2^(*)s=2 N/N-2 s is the critical Sobolev exponent of order s.By virtue of the variational method and the concentration compactness principle with the equivariant group action,we obtain some new type of nonradial,sign-changing solutions of(FCSE)in the energy space˙H^(s)(R^(N)).The key component is that we take the equivariant group action to construct several subspace of˙H^(s)(R^(N))with trivial intersection,then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of(FCSE)in˙H^(s)(R^(N)).
文摘This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique.
基金supported by the Key Project of the NSFC(12131010)the NSFC(11771155,12271032)+1 种基金the NSF of Guangdong Province(2021A1515010249,2021A1515010303)supported by the NSFC(11971179,12371205)。
文摘This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.
基金financially supported by the National Key Research and Development Program of China(No.2021YFB3803101)the National Natural Science Foundation of China(Nos.52022011,51974028,and 52090041)+1 种基金the Xiaomi Young Scholars ProgramChina National Postdoctoral Program for Innovative Talents(No.BX20230042)。
文摘Solid solution-strengthened copper alloys have the advantages of a simple composition and manufacturing process,high mechanical and electrical comprehensive performances,and low cost;thus,they are widely used in high-speed rail contact wires,electronic component connectors,and other devices.Overcoming the contradiction between low alloying and high performance is an important challenge in the development of solid solution-strengthened copper alloys.Taking the typical solid solution-strengthened alloy Cu-4Zn-1Sn as the research object,we proposed using the element In to replace Zn and Sn to achieve low alloying in this work.Two new alloys,Cu-1.5Zn-1Sn-0.4In and Cu-1.5Zn-0.9Sn-0.6In,were designed and prepared.The total weight percentage content of alloying elements decreased by 43%and 41%,respectively,while the product of ultimate tensile strength(UTS)and electrical conductivity(EC)of the annealed state increased by 14%and 15%.After cold rolling with a 90%reduction,the UTS of the two new alloys reached 576 and 627MPa,respectively,the EC was 44.9%IACS and 42.0%IACS,and the product of UTS and EC(UTS×EC)was 97%and 99%higher than that of the annealed state alloy.The dislocations proliferated greatly in cold-rolled alloys,and the strengthening effects of dislocations reached 332 and 356 MPa,respectively,which is the main reason for the considerable improvement in mechanical properties.
基金Project supported by the National Natural Science Foundation of China(No.10571123)the Shandong Provincial Natural Science Foundation of China(No.Y2006A04).
文摘In the present paper, the author studies the existence of sign-changing solutions for nonlinear elliptic equations, which have jumping nonlinearities, and may or may not be resonant with respect to Fucik spectrum, via linking methods under Cerami condition.