NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS

In the present paper,we prove the existence,non-existence and multiplicity of positive normalized solutions(λ_(c),u_(c))∈ℝ×H^(1)(ℝ^(N))to the general Kirchhoff problem-M\left(\int_{\mathbb{R}^N}\vert\nabla u\vert^2{\rm d}x\right)\Delta u+\lambda u=g(u)~\hbox{in}~\mathbb{R}^N,u\in H^1(\mathbb{R}^N),N\geq 1,satisfying the normalization constraint\int_{\mathbb{R}^N}u^2{\rm d}x=c,where M∈C([0,∞))is a given function satisfying some suitable assumptions.Our argument is not by the classical variational method,but by a global branch approach developed by Jeanjean et al.[J Math Pures Appl,2024,183:44–75]and a direct correspondence,so we can handle in a unified way the nonlinearities g(s),which are either mass subcritical,mass critical or mass supercritical.

Pseudo S-Asymptotically(ω,c)-Periodic Solutions to Fractional Differential Equations of Sobolev Type

In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.

Oscillation for Solutions of Systems of High Order Partial Differential Equations of Neutral Type

In this paper, sane sufficient conditions are obtained for the oscillation for solutions of systems of high order partial differential equations of neutral type.

OSCILLATION CRITERIA OF NEUTRAL TYPE IMPULSIVE HYPERBOLIC EQUATIONS

In this paper, oscillatory properties of all solutions for neutral type impulsive hyperbolic equations with several delays under the Robin boundary condition are investigated and several new sufficient conditions for oscillation are presented.

Rational solutions of Painlevé-Ⅱequation as Gram determinant

Under the Flaschka-Newell Lax pair,the Darboux transformation for the Painlevé-Ⅱequation is constructed by the limiting technique.With the aid of the Darboux transformation,the rational solutions are represented by the Gram determinant,and then we give the large y asymptotics of the determinant and the rational solutions.Finally,the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION

In the present paper, we consider the problem {-△u=u^(β_(1))|■u|^(β_(2)),in Ω,u=0,on ■Ω,u>0,in Ω,(0.1) where β_(1), β_(2) > 0 and β_(1) + β_(2) < 1, and Ω is a convex domain in R~n. The existence, uniqueness,regularity and (2-β_(2))/(1-β_(1)-β_(2))-concavity of the positive solutions of the problem(0.1) are proven.

Energy decay for a class of nonlinear wave equations with a critical potential type of damping

The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions are obtained by using multiplier techniques to establish identity ddtE（t）＋F（t）=0 and skillfully selecting f（t）,g（t）,h（t）when the initial data have a compact support.Using the similar method,the Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│＋t）-1 and a nonlinearity │u│p-1u is studied,similar solutions are obtained when the initial data have a compact support.

PROPERTIES OF DIFFERENCE PAINLEV IV EQUATIONS

In this paper, we investigate difference Painleve IV equations, and obtain some results on Nevanlinna exceptional values of transcendental meromorphic solutions w（z） with finite order, their differences △w（z） = w（z ＋ 1） - w（z） and divided differences △w（z）/w（z）.

A class of coupled nonlinear Schrdinger equations:Painlev'e property,exact solutions,and application to atmospheric gravity waves

The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger （CNLS） equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.

The Method of Solutions for some Kinds of Singular Integral Equations of Convolution Type with Both Reflection and Translation Shift

In this paper,some kinds of singular integral equations of convolution type with reflection and translation shift are discussed and they are turned into Riemann boundary value problems with both discontinuous coefficients and reflection by using the Fourier transform.In spite of the classical method for solution,we are to give another method,therefore the general solution and condition of solvability are obtained in class{0}.

Similarity Reductions and Painlevé Property of Coupled KdV Equations

Using Ablowitz-Ramani Segur algorithm, the coupled KdV systems are reclassified under the Painlevé integrable sense while the similarity reductions of the model are obtained by using the Clarkson and Kruskal＇s direct method. Some new types of Painlevé integrable models including a model with different dispersion relations for two layer fluids are found.

CLASSIFICATION OF POSITIVE SOLUTIONS TO ASYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).

Invariance of Painlev property for some reduced (1+1)-dimensional equations

We study the Painlevé property of the （1＋1）-dimensional equations arising from the symmetry reduction for the （2＋1）- dimensional ones. Firstly, we derive the similarity reduction of the （2＋1）-dimensional potential Calogero-Bogoyavlenskii- Schiff （CBS） equation and Konopelchenko-Dubrovsky （KD） equations with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced CBS, KD, and Burgers equations with Painlevé test, re-spectively, we find both the Painlevé integrability, and the number and location of resonance points are invariant, if the similarity variables include all of the independent variables.

Solitons and Bifurcations for the Generalized Tzitzéica Type Equation in Nonlinear Fiber Optics

Solitons and bifurcations for the generalized Tzitzéica type equation are studied by using the theory of dynamical systems and Hamilton function. With the help of Maple and bifurcation theory of differential equations, the bifurcation parameter conditions and all the bifurcation phase portraits are obtained. Because the same energy value of the Hamiltonian function is corresponding to the same orbit, thus the periodic wave solutions, bright soliton and dark soliton solutions are defined.

S-ASYMPTOTICALLY BLOCH TYPE PERIODIC SOLUTIONS TO SOME SEMI-LINEAR EVOLUTION EQUATIONS IN BANACH SPACES

This paper is mainly concerned with the S-asymptotically Bloch type periodicity.Firstly,we introduce a new notion of S-asymptotically Bloch type periodic functions,which can be seen as a generalization of concepts of S-asymptoticallyω-periodic functions and S-asymptoticallyω-anti-periodic functions.Secondly,we establish some fundamental properties on S-asymptotically Bloch type periodic functions.Finally,we apply the results obtained to investigate the existence and uniqueness of S-asymptotically Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces.