共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equationswhere N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C 1 nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).
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$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$
2.
Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X: and we will obtain a variational formula for this multifractal spectrum.
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$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$
3.
In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions. 相似文献
4.
Jessica Angélica Jaurez-Rosas 《Journal of Dynamical and Control Systems》2017,23(1):89-109
For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem. 相似文献
5.
In this paper, we consider the memory-type elasticity system \(\boldsymbol {u}_{tt}-\upmu {\Delta }{\boldsymbol {u}}-(\upmu +\lambda )\nabla (\text {div}\boldsymbol {u})+{{\int }^{t}_{0}}g(t-\tau ){\Delta }{\boldsymbol {u}}(s)ds=0,\) with nonhomogeneous boundary control condition and establish the uniform stability result of the solution. The exponential decay result and polynomial decay result in some literature are the special cases of this paper. 相似文献
6.
In this paper, we study the existence of viable solutions to the differential inclusionwhere f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? c V of an uniformly regular function V.
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$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $
7.
In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR N , the distributed control u belongs to L 2([0,τ];U) (U?=?L 2(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ]. 相似文献
8.
Tomoo Yokoyama 《Journal of Dynamical and Control Systems》2017,23(2):197-212
Consider the set \(\chi ^{0}_{\text {nw}}\) of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M?Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in \(\chi ^{0}_{\text {nw}}\); (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in \(\chi ^{0}_{\text {nw}}\) if and only if the surface is homeomorphic to either the sphere \(\mathbb {S}^{2}\), the projective plane \(\mathbb {P}^{2}\), or the Klein bottle \(\mathbb {K}^{2}\). 相似文献
9.
In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrödinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor \( \mathcal{A} \) for the nonlocal equation in the strong topology of H 1(Ω). We also prove that the global attractor is regular, i.e., \( \mathcal{A} \subset {H^2}\left( \Omega \right) \), assuming that f(x) is of class C 2. Furthermore, we estimate the number of the determining modes for this equation. 相似文献
10.
Fengping Yao 《Journal of Dynamical and Control Systems》2017,23(2):337-348
In this paper, we obtain the following global L q estimates in a convex domain Ω of weak solutions for nonlinear elliptic equations of p-Laplacian type with vanishing Neumann data where ν is the outwardpointing unit normal to ?Ω. Our argument is based on the works of Banerjee and Lewis (Nonlinear Anal 100:78–85, 2014), Kinnunen and Zhou (Comm Partial Differential Equations 24(11&12):2043–2068, 1999, Differential and Integral Equations 14(4):475–492, 2001), and Byun, Wang, and Zhou (Comm Pure Appl Math 57(10):1283–1310, 2004, J Funct Anal 20(3):617–637, 2007). In the proof of the above result, we only focus on the boundary case while the interior case can be obtained as a corollary.
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$$\left|\mathbf{f}\right|^{p } \in L^{q}({\Omega}) \Rightarrow \left|\nabla u\right|^{p } \in L^{q}({\Omega}) \quad \text{for any} ~~q\ge 1 $$
$$\begin{array}{@{}rcl@{}} \text{div} \left( \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \right) & =& \text{div} \left( | \mathbf{f}|^{p-2} \mathbf{f} \right) \quad\text{in} ~~{\Omega},\\ \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \cdot \mathbf{\nu} &=& | \mathbf{f}|^{p -2} \mathbf{f}\cdot \mathbf{\nu} \quad \quad \text{on}~~ \partial{\Omega}, \end{array} $$
11.
In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h 1(x)|u| m?2 u + λ h 2(x)|u| r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h 1 (x), h 2(x) ∈ C(?N) satisfy some conditions. The potential V(x) > 0 is continuous in ? N and V(x)→0 as |x|→+∞. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem. 相似文献
12.
R. Duits A. Ghosh T. C. J. Dela Haije A. Mashtakov 《Journal of Dynamical and Control Systems》2016,22(4):771-805
We consider the problem P c u r v e of minimizing \(\int \limits _{0}^{L} \sqrt {\xi ^{2} + \kappa ^{2}(s)} \, \mathrm {d}s\) for a curve x in \(\mathbb {R}^{3}\) with fixed boundary points and directions. Here, the total length L≥0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ>0 is constant. We lift problem P c u r v e on \(\mathbb {R}^{3}\) to a sub-Riemannian problem P m e c on SE(3)/({0}×SO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P c u r v e . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions. 相似文献
13.
Shun-Tang Wu 《Journal of Dynamical and Control Systems》2018,24(2):287-295
This paper is concerned with the study of the nonlinear damped wave equation in a bounded domain with smooth boundary. The blow-up of solutions are investigated under some conditions. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.
相似文献
$$u_{tt}+{\Delta}^{2}u-{\Delta} u-\omega{\Delta} u_{t}+\alpha(t)u_{t}=\left\vert u\right\vert^{p-2}u, $$
14.
Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation
Gabriel Pietrzkowski 《Journal of Dynamical and Control Systems》2012,18(4):551-571
For a general differential system $ \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} $ , where X d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u d ’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures. 相似文献
15.
Let X be a real Banach space and I a nonempty interval. Let \(K:I\rightsquigarrow X\) be a multi-function with the graph \(\mathcal {K} \). We give here a characterization for \(\mathcal {K} \) to be approximate/near weakly invariant with respect to the differential inclusion \(x^{\prime }(t)\in F(t, x(t))\) by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârj? et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârj? et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion \(x^{\prime }(t)\in F(t,x(t))\). 相似文献
16.
Bo Liu 《Journal of Dynamical and Control Systems》1998,4(1):29-47
We use here HUM (cf. Lions [9]–[l0]) to study the Neumann controllability of a two-dimensional hybrid system membrane with strings on general convex polygon domains (cf. Lee and You [1], Littman [11] for a related version of this model). This system is governed by u
tt
– u = 0 in
on
on 2 × (0,T), u = 0 on 3 × (0,T); u(A
j
) = 0 if
if e
j
2 and e
j+1 1, 0<t<T, and
if e
j
1 and e
j+1 2, 0<t<T (see Sec. 1 for notations). An inverse inequality of the energy has been derived when satisfies certain geometric conditions and T is sufficiently large. As a consequence, an exact control in
or
is respectively obtained. Some other interesting properties (such as the uniqueness of the solution and a Carleman type inequality) of the above problems are also presented. 相似文献
17.
We prove a version of the Gauss-Bonnet theorem in sub-Riemannian Heisenberg space \(\mathbb H^{1}\). The sub-Riemannian distance makes \(\mathbb H^{1}\) a metric space that consequently has a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and, for compact surfaces (which are topologically a torus), we obtain \({\int }_{S}K=0\). 相似文献
18.
We prove that for every pair of nonzero complex numbers λ 1 and λ 2 with \(\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}\) there is an embedding \(S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}\) transverse to the linear holomorphic vector field \(Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}\) . This extends a previous result by Ito (1989). 相似文献
19.
20.
Let G be a nilpotent Lie group and let = {X
1,X
2} be a bracket generating left invariant distribution on G. In this paper we study the left invariant optimal control problem on G defined by the differential equation
the cost functional
and the family of measurable and bounded control functions t u = (u
1(t), u
2(t)). We use the Pontryagin maximum principle and the associated Hamiltonian formalism to obtain the optimal controls of the system. Optimal solutions are necessarily projections of trajectories of a Hamiltonian system which in the normal case can be explicitly integrated in terms of hyperelliptic functions. Abnormal extremals (those which do not depend on the cost functional) turn out to be not strictly abnormal. 相似文献