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1.
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
2.
91. IntroductionIn 1935, LandauLifshitz[1] proposed the fOllowing coupled system of the nonlinear evo-lution equationZr = --a,t x (2 x (b f H)) a,E x (b f A), (1.1)- 8E7 x H = -- aE, (1.2)0t- 0H 0ZV x E = ---- -- pfZ0t p7' (1'3)v. A gv. 2 = 0, v. E = 0, (l.4)where a1, a2, a, g are constants, cr1 2 0, a 2 0, Z(x,t) = (Z1(x,t), Z2(x,t), Z3(x,t))denotes the microscopic magnetization field, H = (H1 (x, f), H2(x, t), H3(x, t)) the magneticfield, E(x, t) = (E1(x, t), E2(x, t), E3(… 相似文献
3.
1 Illtroduction and Statement of the Main ResultIn this paper, we shall study the existence of periodic solutions for the twofOllowing differential delay equationsX'(t) = --f(x(t -- r1))g(x(t -- r2)) -- f(x(t -- r2))g(x(t -- r1)), (1)andX'(t) = f(x(t -- rl))g(x(t -- r2)) f(x(t -- r2))g(x(t -- rl)), (2)where ri (i = l,2) are positive constants. When the function g(x) = 1,equations (1) and (2) become respectivelyIn 1974, Kaplan and Yorke (see [101) proved the existence of periodic so1utions… 相似文献
4.
In this paper, we study the planar Hamiltonian system = J (A(θ)x + ▽f(x, θ)), θ = ω, x ∈ R2 , θ∈ Td , where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1-1 ) and ω is a Diophantine vector. Under the assumption that the unperturbed system = JA(θ)x, θ = ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system. 相似文献
5.
For x =(x1, x2, ···, xn) ∈ Rn+∪ Rn-, the symmetric functions Fn(x, r) and Gn(x, r) are defined by r1 + xFij n(x, r) = Fn(x1, x2, ···, xn; r) =x1≤iij1i2···ir ≤n j=1and r1- xGij n(x, r) = Gn(x1, x2, ···, xn; r) =,x1≤i1i2···ir ≤n j=1ij respectively, where r = 1, 2, ···, n, and i1, i2, ···, in are positive integers. In this paper,the Schur convexity of Fn(x, r) and Gn(x, r) are discussed. As applications, by a bijective transformation of independent variable for a Schur convex function, the authors obtain Schur convexity for some other symmetric functions, which subsumes the main results in recent literature; and by use of the theory of majorization establish some inequalities. In particular, the authors derive from the results of this paper the Weierstrass inequalities and the Ky Fan's inequality, and give a generalization of Safta's conjecture in the n-dimensional space and others. 相似文献
6.
A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS 总被引:32,自引:0,他引:32
We consider the following boundary value problem ill the unbounded donain
Liui = fi(x,u, Tu), i = 1, 2,' ! N,x E fl, (1)
olLi
"i0n Pi(x)t'i = gi(x,u), i = l, 2,',N,x E 0fl, (2)
where x = (x i,', x.), u = (u1,' f uN), Th = (T1tti,', TNi'N) and
[ n. 1
L, = -- I Z ajk(X)the i0j(X)C],
Li,k=1' j=1 J] l
Ltti = / K(x,y)ui(y)dy, x E n.
jn K(x, y)ui(y)dy, x E n.
Q denotes an unbounded dolllain in R", including the exterior of a boullded doinain and 0… 相似文献
7.
In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions. 相似文献
8.
Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y + εh_l~1(x) + ε~2h_l~2(x),y=-x- ε(f_n~1(x)y~(2p+1) + g_m~1(x)) + ∈~2(f_n~2(x)y~(2p+1) + g_m~2(x)),which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials h_l~1 and h_l~2 have degree l;f_n~1and f_n~2 have degree n;and g_m~1,g_m~2 have degree m.p ∈ N and[·]denotes the integer part function. 相似文献
9.
In this paper, we discuss the existence of the solution to the Cauchy problem: L_p~ku≡u_(xx)-x~(2k)u_(tt) Px~(k-1)u_t=f(x,t),t≥0, (1) (A_k) u(x,0)=ψ_1(x), u_t(x,0)=ψ_2(x), (2)where κ=2i 1, i=0,1,2,…,P R=(-∞, ∞), ψ_1(x),ψ_2(x)∈C~∞(R), f(x,t) ∈C~∞(R_2~ ),R_2~ ={(x, t)∈R~2|t≥0}.If κ=1, the uniqueness and existence of solution to the Cauchy problem(A_κ) was entirely solved by [1-3], if κ>1, A. Menikoff proved that the C~∞ 相似文献
10.
§ 1 IntroductionThis paper is concerned with the properties of the simple off-diagonal bivariatequadratic Hermite-Padé approximation.Thisapproximation may be defined asfollows(see,for example,[1 ] ) .Let f(x,y) be a bivariate function,analytic in some neighbourhood of the origin(0 ,0 ) ,whose series expansion about the origin is known.Let a0 (x,y) ,a1 (x,y) ,a2 (x,y) bebivariate polynomials,a0 (x,y) = ki,j=0 a(0 )ij xiyj,a1 (x,y) = ni,j=0 a(1 )ij xiyj,a2 (x,y) = mi,j=0 a(2 )ij xiyj,such th… 相似文献