This paper considers the existence of solutions for the following problem: where v(x) be a continuous function on R3,v(x) < 0, v(x) 0, (as x ); g(x) 0,g(x) 0 and g(x) E H-1 (R3). The author proves that there ...This paper considers the existence of solutions for the following problem: where v(x) be a continuous function on R3,v(x) < 0, v(x) 0, (as x ); g(x) 0,g(x) 0 and g(x) E H-1 (R3). The author proves that there exists a constant C, such that g(x) H-1 C,then there are at least two solutions for the above problem.展开更多
In this paper, we propose an accelerated search-extension method (ASEM) based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple...In this paper, we propose an accelerated search-extension method (ASEM) based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.展开更多
This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* su...This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.展开更多
In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2...In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2 < p < ∞. Under some assumptions on f and h, they prove that there exists a positive constant μ* <∞ such that problem (*)μ has at least one positive solution uμ if μ,∈ (0,μ*), there are no solutions for (*)μ if μ, > μ* and uμ is increasing with respect to μ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solution for μ ∈ (0,μ*) and a unique positive solution for μ, =μ* if p ≤2N/N-2.展开更多
In this paper,a research on the problem of multiple solutions of the three-coefficient low-spectrum model for the quasi-geostrophic ocean current equation with forcing and dissipation terms is carried out.The state of...In this paper,a research on the problem of multiple solutions of the three-coefficient low-spectrum model for the quasi-geostrophic ocean current equation with forcing and dissipation terms is carried out.The state of the ocean current under wind conditions such as those of typhoon is discussed carefully and the rela- tions between the multiple solutions and the coefficients R and ε are analyzed.It is seen that in an approxi- mate triangular region with the Rossby-coefficient R less than 0.5,and the friction-coefficient ε less than 0.22, there exist three equifibrium solutions,among which two are stable and one is unstable.For the former,the coefficient A or B in the expansion is rather large,while for the latter,A or B is relatively small.They respectively imply how much the ocean energy is fed back from the wind stress and the solution with a large A is much more stable than that with a larger B.展开更多
We are concerned with the nonlinear Schrödinger-Poisson equation{−Δu+(V(x)−λ)u+ϕ(x)u=f(u),−Δϕ=u^(2),lim|x|→+∞ϕ(x)=0,x∈R^(3),(P)whereλis a parameter,V(x)is an unbounded potential and f(u)is a general nonlin...We are concerned with the nonlinear Schrödinger-Poisson equation{−Δu+(V(x)−λ)u+ϕ(x)u=f(u),−Δϕ=u^(2),lim|x|→+∞ϕ(x)=0,x∈R^(3),(P)whereλis a parameter,V(x)is an unbounded potential and f(u)is a general nonlinearity.We prove the existence of a ground state solution and multiple solutions to problem(P).展开更多
In this article, we study the existence of multiple solutions for the following sys-tem driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions {(-△)p^su=a(x)|u|^q-2u+2α/α...In this article, we study the existence of multiple solutions for the following sys-tem driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions {(-△)p^su=a(x)|u|^q-2u+2α/α+βc(x)|u|^α-2u|u|^β,in Ω,(-△)p^su=a(x)|u|^q-2u+2α/α+βc(x)|u|^α|u|^β-2.inΩ,(0.1)u=v=0,in R^n/Ωwhere Ω is a smooth bounded domain in R^n, n 〉 ps with s ∈ (0, 1) fixed, a(x), b(x), c(x) 〉 0and a(x),b(x),c(x) ∈ L^∞(Ω), 1 〈 q 〈 p and α,β 〉 1 satisfy p 〈 α+β 〈 p^*, p^* = np/n-ps·By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity ofsolutions to problem (0.1).展开更多
I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= (x1, x2) and V satisfies (V. 1) V: R×R2→R is a C1-function, 1-periodic In t, (V.2) V...I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= (x1, x2) and V satisfies (V. 1) V: R×R2→R is a C1-function, 1-periodic In t, (V.2) V is periodic in x1 with the period T>0, (V. 3) V→O, Vx→O as |x2|→∞, uniformly in (t, x1).展开更多
Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number rang...Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.展开更多
In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:{-△pu-△qu=│u│^p*-2u+μ│u│^r-2u in Ω u...In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:{-△pu-△qu=│u│^p*-2u+μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^*=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ (0, μ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.展开更多
We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large...We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.展开更多
The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
In this paper,we use the ordinary differential equation theory of Banach spaces and minimax theory,and in particular,the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping n...In this paper,we use the ordinary differential equation theory of Banach spaces and minimax theory,and in particular,the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and get new multiple solutions and sign- changing solutions theorems,at last we get up to six nontrivial solutions.展开更多
The multiple solutions for one-dimensional cubic nonlinear problem u'+u^3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ'_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corr...The multiple solutions for one-dimensional cubic nonlinear problem u'+u^3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ'_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k.展开更多
In this paper, an algorithm is proposed to solve the 0(2) symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetr...In this paper, an algorithm is proposed to solve the 0(2) symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O(2) symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.展开更多
In this paper,based on the finite element formulation,we focus on multiple solutions and their evolution with time for a laminar flow in a permeable channel with expanding or contracting walls.Both Newtonian fluid and...In this paper,based on the finite element formulation,we focus on multiple solutions and their evolution with time for a laminar flow in a permeable channel with expanding or contracting walls.Both Newtonian fluid and micropolar fluid are consid-ered.For the Newtonian fluid model,we find that the profile of the unique solution in the case of injection remains the same for long time,which indicates that the solution may be stable.On the other hand,in the case of large suction,the profile of multiple solutions changes in time,which suggests that the multiple solutions may be unstable.Similar behaviors and conclusions are observed for the micropolar fluid model under different boundary parameters.展开更多
In this paper,we concern the Klein-Gordon-Maxwell system with steep potential well{-△u+(λa(x)+1)u-(2w+φ)φu=f(x,u),in R^3-△φ=-(w+)u^2,in R^3 Without global and local compactness,we can tell the difference of mult...In this paper,we concern the Klein-Gordon-Maxwell system with steep potential well{-△u+(λa(x)+1)u-(2w+φ)φu=f(x,u),in R^3-△φ=-(w+)u^2,in R^3 Without global and local compactness,we can tell the difference of multiple solutions from their norms in Lp(R3).展开更多
Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for ...Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for the case of the so-called resonance at infinity.展开更多
文摘This paper considers the existence of solutions for the following problem: where v(x) be a continuous function on R3,v(x) < 0, v(x) 0, (as x ); g(x) 0,g(x) 0 and g(x) E H-1 (R3). The author proves that there exists a constant C, such that g(x) H-1 C,then there are at least two solutions for the above problem.
基金supported by the National Natural Science Foundation of China (10571053, 10871066, 10811120282)Programme for New Century Excellent Talents in University(NCET-06-0712)
文摘In this paper, we propose an accelerated search-extension method (ASEM) based on the interpolated coefficient finite element method, the search-extension method (SEM) and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.
文摘This paper considers the quasilinear elliptic equation where , and 0 < m < p-1 < q < +∞, Ω is a bounded domain in RN(N 3).λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.
基金Research was supported by the Natural Science Foundation of China and the Excellent Teachers Foundation of Ministry of Education of China.
文摘In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2 < p < ∞. Under some assumptions on f and h, they prove that there exists a positive constant μ* <∞ such that problem (*)μ has at least one positive solution uμ if μ,∈ (0,μ*), there are no solutions for (*)μ if μ, > μ* and uμ is increasing with respect to μ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solution for μ ∈ (0,μ*) and a unique positive solution for μ, =μ* if p ≤2N/N-2.
基金The project partly supported by the national project of 75-76-01-03“Study on numerical prediction of the South China Sea current”
文摘In this paper,a research on the problem of multiple solutions of the three-coefficient low-spectrum model for the quasi-geostrophic ocean current equation with forcing and dissipation terms is carried out.The state of the ocean current under wind conditions such as those of typhoon is discussed carefully and the rela- tions between the multiple solutions and the coefficients R and ε are analyzed.It is seen that in an approxi- mate triangular region with the Rossby-coefficient R less than 0.5,and the friction-coefficient ε less than 0.22, there exist three equifibrium solutions,among which two are stable and one is unstable.For the former,the coefficient A or B in the expansion is rather large,while for the latter,A or B is relatively small.They respectively imply how much the ocean energy is fed back from the wind stress and the solution with a large A is much more stable than that with a larger B.
基金supported by NSFC(11871386 and12071482)the Natural Science Foundation of Hubei Province(2019CFB570)。
文摘We are concerned with the nonlinear Schrödinger-Poisson equation{−Δu+(V(x)−λ)u+ϕ(x)u=f(u),−Δϕ=u^(2),lim|x|→+∞ϕ(x)=0,x∈R^(3),(P)whereλis a parameter,V(x)is an unbounded potential and f(u)is a general nonlinearity.We prove the existence of a ground state solution and multiple solutions to problem(P).
基金supported by the National Natural Science Foundation of China(11761030)Hubei Provincial Natural Science Foundation of China(2017CFB352)+1 种基金Doctoral Science Research Foundation of Hubei University for Nationalities(MY2013B019)Youth Research Foundation of Hubei Institute for Nationalities(MY2017Q023)
文摘In this article, we study the existence of multiple solutions for the following sys-tem driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions {(-△)p^su=a(x)|u|^q-2u+2α/α+βc(x)|u|^α-2u|u|^β,in Ω,(-△)p^su=a(x)|u|^q-2u+2α/α+βc(x)|u|^α|u|^β-2.inΩ,(0.1)u=v=0,in R^n/Ωwhere Ω is a smooth bounded domain in R^n, n 〉 ps with s ∈ (0, 1) fixed, a(x), b(x), c(x) 〉 0and a(x),b(x),c(x) ∈ L^∞(Ω), 1 〈 q 〈 p and α,β 〉 1 satisfy p 〈 α+β 〈 p^*, p^* = np/n-ps·By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity ofsolutions to problem (0.1).
文摘I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= (x1, x2) and V satisfies (V. 1) V: R×R2→R is a C1-function, 1-periodic In t, (V.2) V is periodic in x1 with the period T>0, (V. 3) V→O, Vx→O as |x2|→∞, uniformly in (t, x1).
文摘Numerical solutions of the steady transonic small-disturbance(TSD) potential equation are computed using the conservative Murman-Cole scheme. Multiple solutions are discovered and mapped out for the Mach number range at zero angle of attack and the angle of attack range at Mach number 0.85 for the NACA 0012 airfoil. We present a linear stability analysis method by directly assembling and evaluating the Jacobian matrix of the nonlinear finite-difference equation of the TSD equation. The stability of all the discovered multiple solutions are then determined by the proposed eigen analysis. The relation of stability to convergence of the iterative method for solving the TSD equation is discussed. Computations and the stability analysis demonstrate the possibility of eliminating the multiple solutions and stabilizing the remaining unique solution by adding a sufficiently long splitter plate downstream the airfoil trailing edge. Finally, instability of the solution of the TSD equation is shown to be closely connected to the onset of transonic buffet by comparing with experimental data.
基金Supported by NSFC (10571069 and 10631030) the Lap of Mathematical Sciences, CCNU, Hubei Province, China
文摘In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:{-△pu-△qu=│u│^p*-2u+μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^*=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ (0, μ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.
基金supported by the State Committee for Scientific Research of Poland (KBN) under research grants nr 2 P03A 003 25 and nr 4T07A 027 26
文摘We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.
文摘The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.
基金Research supported by the National Natural Science Foundation of China and Postdoctoral Foundation of China
文摘In this paper,we use the ordinary differential equation theory of Banach spaces and minimax theory,and in particular,the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and get new multiple solutions and sign- changing solutions theorems,at last we get up to six nontrivial solutions.
基金Supported by The Special Funds of State Major Basic Research Projects (No.G1999032804)National Natural Science Foundation of China (No.19331021)Mathematical Tianyuan Youth Foundation of National Natural Science Foundation of China (No.10226016)
文摘The multiple solutions for one-dimensional cubic nonlinear problem u'+u^3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ'_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k.
基金Project supported by the National Natural Science Foundation of China (No. 10901106)the Shanghai Leading Academic Discipline Project (No. S30405)+2 种基金the Shanghai Normal University Academic Project (No. SK200936)the Natural Science Foundation of Shanghai (No. 09ZR1423200)the Innovation Program of Shanghai Municipal Education Commission (No. 09YZ150)
文摘In this paper, an algorithm is proposed to solve the 0(2) symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O(2) symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.
基金This work is partially supported by the National Natural Science Foundations of China(No.91430106)the Fundamental Research Funds for the Cen-tral Universities(No.06500073).
文摘In this paper,based on the finite element formulation,we focus on multiple solutions and their evolution with time for a laminar flow in a permeable channel with expanding or contracting walls.Both Newtonian fluid and micropolar fluid are consid-ered.For the Newtonian fluid model,we find that the profile of the unique solution in the case of injection remains the same for long time,which indicates that the solution may be stable.On the other hand,in the case of large suction,the profile of multiple solutions changes in time,which suggests that the multiple solutions may be unstable.Similar behaviors and conclusions are observed for the micropolar fluid model under different boundary parameters.
基金supported by the National Natural Science Foundation of China(Nos.11971393 and 11801465)by the China Postdoctoral Science Foundation(No.2020M683251)by the Graduate Student Scientific Research Innovation Projects in Chongqing(No.CYB18116)。
文摘In this paper,we concern the Klein-Gordon-Maxwell system with steep potential well{-△u+(λa(x)+1)u-(2w+φ)φu=f(x,u),in R^3-△φ=-(w+)u^2,in R^3 Without global and local compactness,we can tell the difference of multiple solutions from their norms in Lp(R3).
基金Supported by the National Natural Science Foundation of China (1072600311001151)+2 种基金the Natural Science Foundation of Shandong (Q2008A03)the Science Foundation of China Postdoctoral (201000481301)the Science Foundation of Shandong Postdoctoral
文摘Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for the case of the so-called resonance at infinity.