共查询到20条相似文献,搜索用时 31 毫秒
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, limλ→0+‖uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1]. 相似文献
2.
A.G. Ramm 《Journal of Mathematical Analysis and Applications》2007,328(2):1290-1296
Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u0, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u0, where a>0 is a parameter and u0 is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen. 相似文献
3.
This paper is concerned with the existence and multiplicity of positive and sign-changing solutions of the fourth-order boundary value problem u (4)(t)=λ f(t,u(t),u ′′(t)), 0<t<1,?u(0)?=?u(1)=u ′′(0)=u ′′(1)?=0, where f:[0,1]×?→? is continuous, λ∈? is a parameter. By using the fixed-point index theory of differential operators, it is proved that the above boundary value problem has positive, negative and sign-changing solutions for λ being different intervals. As an example, the boundary value problem u (4)(t)+?η u ′′(t)??ζu(t)=?λ f(t,u(t)), ?0<t<1,?u(0)=?u(1)=?u ′′(0)=?u ′′(1)=0 is also considered and some obtained results are the complement of the known results. 相似文献
4.
We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ∗(y) or −φ∗(y) as s→∞, where φ∗ denotes the singular stationary solution; (c) u(x,T)/φ∗(x) tends to ±1 as x→0, where T is the blow-up time.Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1 continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t−T)1/(p−1)‖u(⋅,t)L∞‖ is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups. 相似文献
5.
Marcelo Montenegro 《Journal of Mathematical Analysis and Applications》2011,384(2):591-596
We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u0(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t0>0 such that the solution is identically equal to zero. 相似文献
6.
Zhijun Zhang Yiming Guo Huabing Feng 《Journal of Mathematical Analysis and Applications》2009,352(1):77-915
By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→0+g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary. 相似文献
7.
Svatoslav Staněk 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e153
The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u″))′=λf(t,u,u′,u″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y. 相似文献
8.
Jiangang Cheng 《Journal of Mathematical Analysis and Applications》2006,313(1):322-341
This paper is concerned with the exact number of positive solutions for the boundary value problem ′(|y′|p−2y′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf′(u) change sign exactly once from negative to positive on (0,∞). 相似文献
9.
Shuibo Huang Qiaoyu Tian Shengzhi Zhang Jinhua Xi 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(6):2342-2350
We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂RN. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory. 相似文献
10.
This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in RN, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory. 相似文献
11.
Jean-Christophe Bourin 《Linear algebra and its applications》2006,413(1):212-217
We review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, Z∗Z?I, and let f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms
‖f(Z∗AZ)‖?‖Z∗f(A)Z‖. 相似文献
12.
Jiangang Cheng 《Journal of Mathematical Analysis and Applications》2006,315(2):583-598
This paper is concerned with the exact number of positive solutions for boundary value problems ′(|y′|p−2y′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf′(u) changes sign from negative to positive. 相似文献
13.
C.A. Santos 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):6038-6043
We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈RN, where f:RN×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used. 相似文献
14.
Zongming Guo 《Journal of Differential Equations》2005,211(1):187-217
The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δpu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C1((0,∞)?{z2})∩C0[0,∞) satisfying f(0)=f(z1)=f(z2)=0 with 0<z1<z2, f<0 in (0,z1)∪(z2,∞), f>0 in (z1,z2). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2. 相似文献
15.
In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime. 相似文献
16.
Mohammed Guedda 《Journal of Mathematical Analysis and Applications》2009,352(1):259-270
A multiplicity result for the singular ordinary differential equation y″+λx−2yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ?. For 0<λγσ−1<Σ?, there are multiple positive solutions, while if γ=(λ−1Σ?)1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→0+ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d−2(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω). 相似文献
17.
18.
In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u(4)(t)-2u″(t)+u(t)=f(u(t)),0<t<1, u′(0)=u′(1)=u?(0)=u?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory. 相似文献
19.
A.G. Ramm 《Journal of Mathematical Analysis and Applications》2007,325(1):490-495
Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au=f is solvable, and ‖fδ−f‖?δ, then the following results are provided: Problem Fδ(u):=‖Au−fδ‖2+α‖u‖2 has a unique global minimizer uα,δ for any fδ, uα,δ=A*−1(AA*+αI)fδ. There is a function α=α(δ), limδ→0α(δ)=0 such that limδ→0‖uα(δ),δ−y‖=0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are given. Dynamical Systems Method (DSM) is justified for Eq. (1). 相似文献
20.
Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition
Arnaldo Simal do Nascimento Renato José de Moura 《Journal of Mathematical Analysis and Applications》2008,347(1):123-135
In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε2Δu+f(u) in a smooth bounded domain Ω⊂R3 under the boundary condition εν∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior. 相似文献