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In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation u tt - a ∫ Ø m e g a | ∇ u | 2 d x + b Δ u = λ u + | u | p - 1 u , where a, b > 0 , p > 1 , λ ∈ R and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of a, b, λ and the initial data for the following cases: (i) 1 < p < 3 , (ii) p = 3 and a > 1 / Λ , (iii) p = 3 , a ≤ 1 / Λ and λ < b λ 1 , (iv) p = 3 , a < 1 / Λ and λ > b λ 1 , (v) p > 3 and λ ≤ b λ 1 , (vi) p > 3 and λ > b λ 1 , where λ 1 = inf ‖ ∇ u ‖ 2 2 : u ∈ H 0 1 (Ø m e g a) and ‖ u ‖ 2 = 1 , and Λ = inf ‖ ∇ u ‖ 2 4 : u ∈ H 0 1 (Ø m e g a) and ‖ u ‖ 4 = 1 . Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable a, b and λ , and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for a > 0 and a = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ (t , x , u) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ (t , x , v) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions (u ε , v ε) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , (u ε , v ε) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove some uniqueness and convergence results for a competing system and its singular limit, and an interior measure estimate of the free boundary for the singular limit. [Copyright &y& Elsevier]

Abstract: We study the boundary value problems for Monge–Ampère equations: in , , . First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter in Ω, , . By using Lyapunov–Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray–Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. [Copyright &y& Elsevier]

Abstract: We obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow. [Copyright &y& Elsevier]

Abstract: Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems. [Copyright &y& Elsevier]

In this paper, we use a bifurcation approach, Morse theory to study asymptotically linear and superlinear elliptic boundary value problems, and obtain new results on the exact multiplicity of solutions and sign-changing solutions to asymptotically linear and superlinear problems and the properties of the solutions. [Copyright &y& Elsevier]

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, in particular, the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero, and get new multiple solutions and sign-changing solutions theorems. Moreover, the sign-changing solutions change sign exactly once under some conditions. [Copyright &y& Elsevier]

Abstract 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. [ABSTRACT FROM AUTHOR]