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1.
This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u(0)=u(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0<λ<λ*, λ=λ* and λ>λ*, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ0+uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].  相似文献   

2.
We study a priori estimates of positive solutions of the equation tuΔu=λu+a(x)up, xΩ, t>0, satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in Rn, λR, p>1 is subcritical, changes sign and a,p satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set is connected. Using our a priori bounds, we show that u blows up completely in Ω+ at t=T and the blow-up time T depends continuously on the initial data.  相似文献   

3.
A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three utt−Δxu+cut+λu=f(t,x), when c>0, λ∈(0,c2/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation utt−Δxu+cut=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.  相似文献   

4.
We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms (any ??1), the flux function f(u) being mth order growth at infinity. It is shown that if ε, δ=δ(ε) tend to 0, then the sequence {uε} of the smooth solutions converges to the unique entropy solution uL(0,T;Lq(R)) to the conservation law ut+fx(u)=0 in . The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ)=λ, ?=1 and LeFloch and Natalini when ?=1.  相似文献   

5.
In this paper we study the existence and multiplicity of the solutions for the fourth-order boundary value problem (BVP) u(4)(t)+ηu(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u(0)=u(1)=0, where is continuous, ζ,ηR and λR+ are parameters. By means of the idea of the decomposition of operators shown by Chen [W.Y. Chen, A decomposition problem for operators, Xuebao of Dongbei Renmin University 1 (1957) 95-98], see also [M. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostehizdat, Moscow, 1956], and the critical point theory, we obtain that if the pair (η,ζ) is on the curve ζ=−η2/4 satisfying η<2π2, then the above BVP has at least one, two, three, and infinitely many solutions for λ being in different interval, respectively.  相似文献   

6.
We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.  相似文献   

7.
Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u1,u2,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=Tu1u2,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.  相似文献   

8.
We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in ΩRn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ such that uλ vanishes on a set of positive measure for 0<λ<λ, and uλ>0 for λ>λ. If f is concave, for λ>λ we characterize uλ by its stability.  相似文献   

9.
By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, xRN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀xRN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.  相似文献   

10.
In this paper, for the fourth-order boundary value problem (BVP) ,0<t<1,u(0)=u(1)=u(0)=u(1)=0, where f:[0,1]×RR is continuous, η≤0 is a parameter, the existence of infinitely many mountain pass solutions are obtained with the variational methods and critical point theory. We prove the conclusion by combining sub-sup solution method, Mountain pass theorem in order intervals, Leray-Schauder degree theory and Morse theory.  相似文献   

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