MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRODINGER EQUATIONS

In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^(N),(0.1)where N≥4,2≤p<2^(*),2_α^(*)=(2N-α)/(N-2)with 0<α<4,λ>0,μ∈R,A(x)=(A_(1)(x),A_(2)(x),…,A_(N)(x))is a real local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on R^(N).Supposing thatΩ=int V^(-1)(0)■R^(N)is bounded,we show that problem(0.1)possesses at least cat_(Ω)(Ω)nontrivial solutions ifλis large.

Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with p-Laplacian Operator

This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.

Multiple solutions and fermion mass effect in QED_3

Due to the negligible non-perturbation effect in the low-energy region, quantum chromodynamics （QCD） is limited to be applied to hadron problems in particle physics. However, QED has mature non-perturbation models which can be applied to Fermi acting-energy between quark and gluon. This paper applies quantum electrodynamics in 2 ＋ 1 dimensions （QED3） to the Fermi condensation problems. First, the Dyson-Schwinger equation which the fermions satisfy is constructed, and then the Fermi energy gap is solved. Theoretical calculations show that within the chirality limit, there exist three solutions for the energy gap; beyond the chirality limit, there are two solutions; all these solutions correspond to different fermion condensates. It can be concluded that the fermion condensates within the chirality limit can be used to analyze the existence of antiferromagnetic, pseudogap, and superconducting phases, and two fermion condensates are discovered beyond the chirality limit.

MULTIPLE SOLUTIONS OF NONHOMOGENEOUS FOR RELATED CHOQUARD'S EQUATION

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.

Existence of multiple solutions for the laminar flow in a porous channel with suction at both slowly expanding and contracting walls

The asymptotic behavior of solutions of a similarity equation for the laminar flow in a porous channel with suction at both expanding and contracting walls has been obtained by using a singular perturbation method.However,in the matching process,this solution neglects exponentially small terms.To take into account these exponentially small terms,a method involving the inclusion of exponentially small terms in a perturbation series was used to find two of the solutions analytically.The series involving the exponentially small terms and expansion ratio predicts dual solutions.Furthermore,the result indicates that the expansion ratio has much important influence on the solutions.

THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

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.

THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION

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.

MULTIPLE SOLUTIONS FOR AN INHOMOGENEOUS SEMILINEAR ELLIPTIC EQUATION IN R^N

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.

ON THE MULTIPLE SOLUTIONS TO A THREE-COEFFICIENT SPECTRAL MODEL OF WIND-DRIVING OCEAN CURRENT EQUATION

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.

MULTIPLE SOLUTIONS FOR THE SCHR?DINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY

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).

EXISTENCE OF MULTIPLE SOLUTIONS FOR A FRACTIONAL p-LAPLACIAN SYSTEM WITH CONCAVE-CONVEX TERM

Abstract 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）.

MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

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）.

Multiple solutions and stability of the steady transonic small-disturbance equation

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.

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

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.

MULTIPLE SOLUTIONS FOR NONAUTONOMOUS SECOND ORDER PERIODIC SYSTEMS

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.

EXISTENCE OF MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

The paper is concerned with the multiplicity of solutions for some nonlinear elliptic equations involving critical Sobolev exponents and mixed boundary conditions.

A scheme to fix multiple solutions in amplitude analyses

Decays of unstable heavy particles usually involve the coherent sum of several amplitudes,like in a multiple slit experiment.Dedicated amplitude analysis techniques have been widely used to resolve these amplitudes for better understanding of the underlying dynamics.In special cases where two spin-1/2 particles and two(pseudo-)scalar particles are present in the process,multiple equivalent solutions are found owing to intrinsic symmetries in the summed probability density function.In this study,the problem of multiple solutions is discussed,and a scheme to overcome this problem is proposed by fixing some free parameters.Toys are generated to validate the strategy.A new approach to align the helicities of initial-and final-state particles in different decay chains is also introduced.

Multiple solutions for a Schrdinger-Poisson-Slater equation with external Coulomb potential

Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suitably large. When β 〉 0, -β/｜x｜ is an important potential in physics, which is called external Coulomb potential. In this paper, we find that （P） with β 〉 0 has totally different properties from that of β = 0. For β 〉 0, we prove that （P） has a ground state and multiple solutions if λ 〉 cp,ω, where cp,ω 〉 0 is a constant which can be expressed explicitly via ω and p.

Sign-Changing and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

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.

Multiple Solutions and Its Morse Index for One-Dimensional Nonlinear Problem

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.