共查询到20条相似文献,搜索用时 31 毫秒
1.
We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u
1(x),...,u
N
(x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be
, a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14]. 相似文献
2.
Alexander Mielke 《Journal of Dynamics and Differential Equations》1992,4(3):419-443
We consider the equation a(y)uxx+divy(b(y)yu)+c(y)u=g(y, u) in the cylinder (–l,l)×, being elliptic where b(y)>0 and hyperbolic where b(y)<0. We construct self-adjoint realizations in L2() of the operatorAu= (1/a) divy(byu)+(c/a) in the case ofb changing sign. This leads to the abstract problem uxx+Au=g(u), whereA has a spectrum extending to + as well as to –. For l= it is shown that all sufficiently small solutions lie on an infinite-dimensional center manifold and behave like those of a hyperbolic problem. Anx-independent cross-sectional integral E=E(u, ux) is derived showing that all solutions on the center manifold remain bounded forx ±. For finitel, all small solutionsu are close to a solution on the center manifold such that u(x)-(x)
Ce
-(1-|x|) for allx, whereC and are independent ofu. Hence, the solutions are dominated by hyperbolic properties, except close to the terminal ends {±1}×, where boundary layers of elliptic type appear. 相似文献
3.
For elliptic equations ε2Δu − V(x) u + f(u) = 0, x ∈ R
N
, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component
of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal.
An erratum to this article can be found at 相似文献
4.
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u
tt
− c(u)(c(u)u
x
)
x
= 0. We allow for initial data u|
t = 0 and u
t
|
t=0 that contain measures. We assume that
0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby
singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times
of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples. 相似文献
5.
J. B. Diaz 《Archive for Rational Mechanics and Analysis》1957,1(1):357-390
This paper develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x
0)=y
0) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation u
xy
=f(x, y, u, u
x
, y
y
), u(x, y
0)=(x), u(x
0, y)=(y), where (x
0)=(y
0). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions (x) and (y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.
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6.
S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428
Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional
dynamic equation
( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0, t ? [ t0,¥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}}, 相似文献
7.
We study the nonnegative solutions of the initial-value problem ut=(ur|ux|p-1ux)x,u(x, 0)L
1(), where p>0, r+p>0. The local velocity of propagation of the solutions is identified as V = -vx| vx|p-1 where v =cu (with r +p - 1)/p and c (r +p/(r +p- 1)) is the nonlinear potential. Our main result is the a priori estimate (vx|vx|p-1)x-
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