共查询到19条相似文献,搜索用时 140 毫秒
1.
MOUSSAOUI Abdelkrim KHODJA Brahim 《偏微分方程(英文版)》2009,22(2):111-126
In this paper, we study the existence of nontrivial solutions for the problem
{-△u=f(x,u,v)+h1(x)in Ω
-△v=g(x,u,v)+h2(x)inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 (Ω). The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g:
{lim s,|t|→+∞f(x,s,t)/s=lim |s|,t→+∞g(x,s,t)/t=λ+1 uniformly on Ω,
lim -s,|t|→+∞f(x,s,t)/s=lim |s|,-t→+∞g(x,s,t)/t=λ-,uniformly on Ω,
where λ+,λ-∈(0)∪σ(-△),σ(-△)denote the spectrum of -△. The cases (i) where λ+ = λ_ and (ii) where λ+≠λ_ such that the closed interval with endpoints λ+,λ_ contains at most one simple eigenvatue of -△ are considered. 相似文献
{-△u=f(x,u,v)+h1(x)in Ω
-△v=g(x,u,v)+h2(x)inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 (Ω). The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g:
{lim s,|t|→+∞f(x,s,t)/s=lim |s|,t→+∞g(x,s,t)/t=λ+1 uniformly on Ω,
lim -s,|t|→+∞f(x,s,t)/s=lim |s|,-t→+∞g(x,s,t)/t=λ-,uniformly on Ω,
where λ+,λ-∈(0)∪σ(-△),σ(-△)denote the spectrum of -△. The cases (i) where λ+ = λ_ and (ii) where λ+≠λ_ such that the closed interval with endpoints λ+,λ_ contains at most one simple eigenvatue of -△ are considered. 相似文献
2.
Let G =(V,E) be a locally finite graph,whose measure μ(x) has positive lower bound,and A be the usual graph Laplacian.Applying the mountain-pass theorem due to Ambrosetti and Rabinowitz(1973),we establish existence results for some nonlinear equations,namely △u+hu=f(x,u),x∈V.In particular,we prove that if h and f satisfy certain assumptions,then the above-mentioned equation has strictly positive solutions.Also,we consider existence of positive solutions of the perturbed equation △u+hu=f(x,u)+∈g.Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds. 相似文献
3.
In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)spu+aup-1=f(u,v),(-△)tpv+bv(p-1)=g(u,v),where 0N(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively. 相似文献
4.
In this paper, the authors consider the positive solutions of the system of the evolution p-Laplacian equations
with nonlinear boundary conditions
and the initial data (u0, v0), where Ω is a bounded domain in Rn with smooth boundary δΩ, p 〉 2, h(·,·) and s(·,· ) are positive C1 functions, nondecreasing in each variable. The authors find conditions on the functions f, g, h, s that prove the global existence or finite time blow-up of positive solutions for every (u0, v0). 相似文献
with nonlinear boundary conditions
and the initial data (u0, v0), where Ω is a bounded domain in Rn with smooth boundary δΩ, p 〉 2, h(·,·) and s(·,· ) are positive C1 functions, nondecreasing in each variable. The authors find conditions on the functions f, g, h, s that prove the global existence or finite time blow-up of positive solutions for every (u0, v0). 相似文献
5.
MaLi SuNing 《偏微分方程(英文版)》2004,17(1):49-56
In this note, we establish some estimates of solutions of the scalar Ginzburg-Landau equation and other nonlinear Laplacian equation △u =f(x, u). This will give an estimate of the Hausdorff dimension for the free boundary of the obstacle problem. 相似文献
6.
In this paper, we are concerned with the elliptic system of
{ -△u+V(x)u=g(x,v), x∈R^N,
-△v+V(x)v=f(x,u), x∈R^N,
where V(x) is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods. 相似文献
{ -△u+V(x)u=g(x,v), x∈R^N,
-△v+V(x)v=f(x,u), x∈R^N,
where V(x) is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods. 相似文献
7.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
8.
In this paper, we prove the existence of at least one positive solution pair (u, v)∈ H1(RN) × H1(RN) to the following semilinear elliptic system {-△u+u=f(x,v),x∈RN,-△u+u=g(x,v),x∈RN (0.1),by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0(RN× R1) are that, f(x, t) and g(x, t) are superlinear at t = 0 as well as at t =+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u+u=f(x,u),x∈Ω,u∈H0^1(Ω) where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5& 6.pp.925-954, 2004] concerning (0.1) when f and g are asymptotically linear. 相似文献
9.
The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″(t)=f(t,u(t)),a.e,t∈[0,1],u(0)=u′(η)=u″(1)=0, where the nonlinear term f(t, u) is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f(t, u) 〉 ≥-h(t). The existence of n positive solutions is proved by considering the integrations of "height functions" and applying the Krasnosel'skii fixed point theorem on cone. 相似文献
10.
Marian Slodicka Jan Busa Jr. 《计算数学(英文版)》2008,(5):677-688
This paper is devoted to the study of a nonlinear evolution eddy current model of the type δtB(H) + △↓× ( △↓ × H) =0 subject to homogeneous Dirichlet boundary conditions H × v = 0 and a given initial datum. Here, the magnetic properties of a soft ferromagnet are linked by a nonlinear material law described by B(H). We apply the backward Euler method for the time discretization and we derive the error estimates in suitable function spaces. The results depend on the nonlinearity of B(H). 相似文献
11.
The Existence of Solutions of Elliptic Equations
with Neumann Boundary Condition for Superlinear Problems 总被引:1,自引:0,他引:1
ChongLI 《数学学报(英文版)》2004,20(6):965-976
In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):{-△u αu=f(u),x∈Ω, x∈Ω,δu/δr=0,x∈δΩ,is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow. 相似文献
12.
In the “lost notebook”, Ramanujan recorded infinite product expansions for
$\frac{1}
{{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }}
{2}} \right)\sqrt r and \frac{1}
{{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }}
{2}} \right)\sqrt r ,$\frac{1}
{{\sqrt r }} - \left( {\frac{{1 - \sqrt 5 }}
{2}} \right)\sqrt r and \frac{1}
{{\sqrt r }} - \left( {\frac{{1 + \sqrt 5 }}
{2}} \right)\sqrt r , 相似文献
13.
In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f
1(z), f
2(z), …, f
n
(z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ
n
and
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