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■,satisfying the normalization constraint f_(R)^N u^2dx=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.

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.

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation−(a+b∫_(R^(3))|∇u|^(2))Δu+V(x)u=f(u)in R^(3),with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions.Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity,especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

The Existence of Periodic and Almost Periodic Solutions of Logistic Type Equation

In this paper, we study the Logistic type equation x= a（t）x -b（t）x^2＋ e（t）. Under the assumptions that e（t） is small enough and a（t）, b（t） are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a（t）, b（t） and e（t） are periodic functions, and this solution is an almost periodic solution if a（t）, b（t） and e（t） are almost periodic functions.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

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.

Stability of Schrdinger-Poisson type equations

Variational methods are used to study the nonlinear SchrSdinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Hamiltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.

A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION

In this paper the Tricomi problem for a nonlinear mixed type equation is studied. The coefficients of the mixed type equation are discontinuous on the line, where the equation changes its type. The existence of solution to this problem is proved. The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.

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.

Approximation and Superconvergence Analysis for a New Higher Order Wilson Element to Sobolev Type Equations

In this paper, a new higher order Wilson element is presented, and the convergence is proved. Then the interpolation postprocessing technique is used to obtain the global superconvergence and posterior error estimate of higher accuracy of this new element for the Sobolev type equations.

Time-delay effects on the dynamics of Linard type equation with fast and slow variables

Time-delay effects on the dynamics of Li^nard type equation with one fast variable and one slow variable are investigated in the present paper. By using the methods of stability switch and geometric singular perturbation, time-delay-induced complex oscillations and bursting are investigated, and in several case studies, the mechanism of the generation of the complex oscillations and bursting is illuminated. Numerical results demonstrate the validity of the theoretical results.

The Sufficient and Necessary Condition of Lagrange Stability of Quasi-periodic Pendulum Type Equations

The quasi-periodic pendulum type equations are considered. A sufficient and necessary condition of Lagrange stability for this kind of equations is obtained. The result obtained answers a problem proposed by Moser under the quasi-periodic case.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

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.

The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.

Double-Peakon Solutions of Two Four-Component Camassa-Holm Type Equations

This paper is contributed to study two new integrable four-component systems reduced from a multi-component generation of Camassa-Holm equation. Some double peakon solutions of both systems are described in an explicit formula by the method of variation of constant for ordinary differential equations. These double peakon solutions are established in weak sense. The dynamic behaviors of the obtained double peakon solutions are illustrated by some figures.

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

A non-autonomous Logistic type equation with infinite delay is investigated. For general nonautonomous case, sufficient conditions which guarantee the uniform persistence and globally attractivity of the system arc obtained; For almost periodic case, by means of a suitable Lyapunov functional, sufficient conditions are derived for the existence and uniqueness of almost periodic solution of the system. Some new results are obtained.

A-high-order Accuraqcy Implicit Difference Scheme for Solving the Equation of Parabolic Type

In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method.

Removable Singularities of Solutions of A-harmonic Type Equations

A Caccioppoli type estimate is established for a class of second order PDEs of divergence type, and its removable singularities of Hausdorff dimension greater than zero is obtained.

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem.The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem.Compared to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).