共查询到20条相似文献,搜索用时 420 毫秒
1.
Xi-Nan Ma 《Mathematische Zeitschrift》2002,240(1):1-11
We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty.
The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and
a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to
certain semilinear boundary value problems in convex domains.
Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002 相似文献
2.
We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary. 相似文献
3.
In the present paper we consider the Dirichlet problem in a convex domain for the multidimensional p-Laplace equation with nonlinear source. We prove the existence of the unique continuous viscosity solution under quite general
assumptions on the structure of the source.
Received: 18 January 2006 相似文献
4.
Dongho Chae Oleg Y. Imanuvilov 《Calculus of Variations and Partial Differential Equations》2003,16(1):47-61
In this paper we construct a non-topological multivortex solution of a generalized version of the relativistic self-dual
Chern-Simons-Higgs system in that makes the energy functional finite. Our method of proof is an extension of the previous argument used by the authors
to prove the existence of general type of non-topological multivortex solutions of the relativistic Chern-Simons-Higgs system,
using an implicit function theorem argument with features similar to the Liapunov-Schmidt decomposition.
Received: 20 January 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002
This research supported partially by BSRI-MOE, KOSEF(2000-2-10200-002-5). 相似文献
5.
The equation with boundary Dirichlet zero data is considered in a bounded domain . Under the assumption that concentrates, as , round a manifold and that f is a superlinear function, satisfying suitable growth assumptions, the existence of multiple distinct positive solutions
is proved.
Received: 19 December 2000 / Accepted: 8 May 2001 / Published online: 5 September 2002 相似文献
6.
M. Flucher A. Garroni S. Müller 《Calculus of Variations and Partial Differential Equations》2002,14(4):483-516
We study the variational problem
where , is a bounded domain, , F satisfies $0\leq F|t|\leq \alpha |t|^{2^*}$ and is upper semicontinuous. We show that to second order in the value only depends on two ingredients. The geometry of enters through the Robin function (the regular part of the Green's function) and F enters through a quantity which is computed from (radial) maximizers of the problem in . The asymptotic expansion becomes
Using this we deduce that a subsequence of (almost) maximizers of must concentrate at a harmonic center of : i.e., , where is a minimum point of .
Received: 24 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001 相似文献
7.
Ryuji Kajikiya 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(7):2117-2131
In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to . 相似文献
8.
Dmitry Golovaty Leonid Berlyand 《Calculus of Variations and Partial Differential Equations》2002,14(2):213-232
Consider a class of Sobolev functions satisfying a prescribed degree condition on the boundary of a planar annular domain.
It is shown that, within this class, the Ginzburg-Landau functional possesses the unique, radially symmetric minimizer, provided
that the annulus is sufficiently narrow. This result is known to be false for wide annuli where vortices are energetically
feasible. The estimate for the critical radius below which the uniqueness of the minimizer is guaranteed is obtained as well.
Received: 19 July 2000 / Accepted: 23 February 2001/ Published online: 23 July 2001 相似文献
9.
Robert L. Jerrard Halil Mete Soner 《Calculus of Variations and Partial Differential Equations》2002,14(2):151-191
We study the Ginzburg-Landau functional
for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation
of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional.
Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001 相似文献
10.
Jan Chabrowski Zhi-Qiang Wang 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):683-697
We consider the solvability of the Neumann problem for equation (1.1) in exterior domains in both cases: subcritical and critical.
We establish the existence of least energy solutions. In the subcritical case the coefficient
b(x) is allowed to have a potential well whose steepness is controlled by a parameter λ > 0. We show that least energy solutions
exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions. 相似文献
11.
J. Chabrowski M. Willem 《Calculus of Variations and Partial Differential Equations》2002,15(4):421-431
We investigate the effect of the coefficient of the critical nonlinearity for the Neumann problem on the existence of least
energy solutions. As a by-product we establish a Sobolev inequality with interior norm.
Received: 26 April 2000 / Accepted: 25 February 2001 / Published online: 5 September 2002 相似文献
12.
Nils Ackermann 《Journal of Differential Equations》2009,246(4):1470-1499
The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials. 相似文献
13.
We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem. 相似文献
14.
N. I. Karachalios N. B. Zographopoulos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):11-30
We study a real Ginzburg-Landau equation, in a bounded domain of
with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.Received: May 6, 2002; revised: October 3, 2002 相似文献
15.
When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the other part blows up. Using the decomposition, we derive the best possible estimates for the blow-up of the gradient. We then consider the case when the inclusions have positive permittivities. We show quantitatively that in this case the size of the blow-up is reduced. 相似文献
16.
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. In this paper, we prove a general, abstract theorem ( Theorem 26) which allows to prove the existence of a class of solitons. Such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons. Then we apply the abstract theory to problems related to the nonlinear Schrödinger equation (NSE) and to the nonlinear Klein–Gordon equation (NKG). 相似文献
17.
Toru Kan 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):2941-2956
We study a semilinear elliptic problem on thin domains with a bifurcation parameter. It is shown that the set of solutions is upper semicontinuous as the thickness of a domain tends to 0, and that solution branches including bifurcation points persist near those of a one-dimensional limiting equation. 相似文献
18.
We use variational methods to study problems in nonlinear 3-dimensional elasticity where the deformation of the elastic body
is restricted by a rigid obstacle. For an assigned variational problem we first verify the existence of constrained minimizers
whereby we extend previous results. Then we rigorously derive the Euler-Lagrange equation as necessary condition for minimizers,
which was possible before only under strong smoothness assumptions on the solution. The Lagrange multiplier corresponding
to the obstacle constraint provides structural information about the nature of frictionless contact. In the case of contact
with, e.g., a corner of the obstacle, we derive a qualitatively new contact condition taking into account the deformed shape
of the elastic body. By our analysis it is shown here for the first time rigorously that energy minimizers really solve the
mechanical contact problem.
Received: 20 October 2000 / Accepted: 7 June 2001 / Published online: 5 September 2002 相似文献
19.
Given p∈[2,+∞), we obtain the global W1,p estimate for the weak solution of a boundary-value problem for an elliptic equation with BMO nonlinearity in a Reifenberg domain, assuming that the nonlinearity has sufficiently small BMO seminorm and that the boundary of the domain is sufficiently flat. 相似文献
20.
The predator–prey system with non-monotonic functional response is an interesting field of theoretical study. In this paper we consider a strongly coupled partial differential equation model with a non-monotonic functional response—a Holling type-IV function in a bounded domain with no flux boundary condition. We prove a number of existence and non-existence results concerning non-constant steady states (patterns) of the underlying system. In particular, we demonstrate that cross-diffusion can create patterns when the corresponding model without cross-diffusion fails. 相似文献