共查询到20条相似文献,搜索用时 31 毫秒
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We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)2 on a bounded domain Ω ? ?N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc. 相似文献
3.
Florica Corina Cîrstea Cristina Trombetti 《Calculus of Variations and Partial Differential Equations》2008,31(2):167-186
We consider the Monge–Ampère equation det D 2 u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω. 相似文献
4.
In this note, we consider semilinear equations , with zero Dirichlet boundary condition, for smooth and nonnegative f, in smooth, bounded, strictly convex domains of . We study positive classical solutions that are semi-stable. A solution u is said to be semi-stable if the linearized operator at u is nonnegative definite. We show that in dimension two, any positive semi-stable solution has a unique, nondegenerate, critical
point. This point is necessarily the maximum of u. As a consequence, all level curves of u are simple, smooth and closed. Moreover, the nondegeneracy of the critical point implies that the level curves are strictly
convex in a neighborhood of the maximum of u. Some extensions of this result to higher dimensions are also discussed. 相似文献
5.
In this note, we investigate the regularity of the extremal solution u? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two. 相似文献
6.
In this paper we study the existence of nontrivial solutions for the periodic discrete nonlinear equation Lun ωun = fn(un), where Lun = a n+1un+1 + an1un1+ bnun is the discrete Laplacian in one spatial dimension. The given real-valued sequences an , bn are assumed to be N -periodic in n, i.e., an+N = an , b n+N = bn . The nonlinearity fn(t) is N-periodic in n and asymptotically linear at infinity. We show that, if ω is in the spectrum gap of L, there is a nontrivial solution. The proof is based on the strongly indefinite functional critical points theorem developed recently. 相似文献
7.
S. A. Nazarov 《Functional Analysis and Its Applications》2006,40(2):97-107
For the Helmholtz equation Δu + k 2 u = 0 in a domain Ω with a cylindrical outlet Q + = ω × ?+ to infinity, we construct a fictitious scattering operator $\mathfrak{S}$ that is unitary in L 2(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of $\mathfrak{S}$ corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λn ≤ k 2 of the Dirichlet problem for the Laplace operator on the cross-section ω. 相似文献
8.
Let T = (V, E) be a tree whose vertices are properly 2-colored. A bipartite labeling of T is a bijection f: V ← {0, 1, ?, | E |} for which there is a k such that whenever f(u) ≤ k < f(v), then u and v have different colors. The α-size of the tree T is the maximum number of distinct values of the induced edge labels |f(u) - f(v)|, uv ? E, taken over all bipartite labelings f of T. We investigate the asymptotic behavior of the α-size of trees. Let α(n) be the smallest α-size among all the trees with n edges. As our main result we prove that 5(n + 1)/7 ≤ α(n) ≤ (5n + 9)/6. A connection with the graceful tree conjecture is established, in that every tree with n edges is shown to have “gracesize” at least 5n/7. © 1995 John Wiley & Sons, Inc. 相似文献
9.
On sparse reconstruction from Fourier and Gaussian measurements 总被引:1,自引:0,他引:1
This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size A random set Ω satisfies this with high probability. This estimate is optimal within the log logn and log3r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n/r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short. © 2007 Wiley Periodicals, Inc. 相似文献
10.
We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n?9 (for every g), and belongs to H3=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent. 相似文献
11.
P. Lesky 《Mathematical Methods in the Applied Sciences》1990,12(4):275-291
Consider the polyharmonic wave equation ?u + (? Δ)mu = f in ?n × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order tα or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u0 among the solutions of the polyharmonic equation ( ? Δ)mu = f in ?n. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation. 相似文献
12.
We investigate the nonnegative solutions of the system involving the fractional Laplacian: where 0 < α < 1, n > 2, f i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p i ≥ 0 and nondecreasing with respect to the independent variables u 1, u 2,..., u m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x 0 if p i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.
相似文献
$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$
13.
Bryan P. Rynne 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1996,47(5):730-739
Semilinear elliptic equations of the form $$\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^n {(a_{ij} (x)u_{xi} (x))_{x_j } + } f(\lambda ,x,u(x)) = 0,} & {x \in \Omega ,} \\ {u(x) = 0,} & x \\ \end{array} $$ are considered, where λ ε ? is a parameter, Ω ? ? n is a bounded domain andf is a smooth non-linear function. It is shown that for ‘generic’ functionsf, the set of non-trivial solutions (λ,u) consists of a finite, or countable, collection of smooth, 1-dimensional curves and any such solution is either hyperbolic or is a saddle-node bifurcation point of the curve. 相似文献
14.
We prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition withA function \({u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}\) is called parabolic minimizer if it satisfies the minimality conditionfor every \({\varphi \in C^\infty_0(\Omega_T)}\). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition.
相似文献
$$1 < p \leq q \leq p \tfrac{n+2}{n}.$$
$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z$$
15.
Martin Fuchs Guo Zhang 《Calculus of Variations and Partial Differential Equations》2012,44(1-2):271-295
If ${u: \mathbb{R}^{n} \to \mathbb{R}^{M}}$ locally minimizes the energy with density ${|\nabla u|\ln (1 + |\nabla u|)}$ , then we show that the boundedness of the function u already implies its constancy. The same is true in case n = M = 2 for entire solutions of the equations modelling the stationary flow of a so-called Prandtl-Eyring fluid. Moreover, in the variational setting we will present various extensions of the above mentioned Liouville theorem for entire local minimizers valid in any dimensions n and M. 相似文献
16.
Hong Huang 《Potential Analysis》2014,41(3):771-782
Let u be a positive solution of the ultraparabolic equation $$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$ where 1 ≤ k ≤ n and 0 < T ≤ + ∞. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) ? H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that \(\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0\) , where l = log u, and \(\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2} \) . 相似文献
17.
A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f = g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C 1,1 functions and lower-C 2 functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry. 相似文献
18.
Long Wei 《Mathematica Slovaca》2014,64(2):379-390
We investigate the regularity of extremal solutions to some p-Laplacian Dirichlet problems with strong nonlinearities. Under adequate assumptions we prove the smoothness of the extremal solutions for some classes of nonlinearities. Our results suggest that the extremal solution’s boundedness for some range of dimensions depends on the nonlinearity f. 相似文献
19.
Let n ≥ 2 and let Ω ? ? n be an open set. We prove the boundedness of weak solutions to the problem where ? is a Young function such that the space W 0 1 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ? n .
相似文献
$$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}}{{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u}{{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$
20.
Zhuoran Du Zheng Zhou Baishun Lai 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(1):101-114
We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian
-Dpu=f(u) \textin Rn, -\Delta_{p}u=f(u) \quad \text{in}\,\,R^n, 相似文献
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