共查询到20条相似文献,搜索用时 31 毫秒
1.
Massimo Grossi Angela Pistoia Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2000,11(2):143-175
We study a perturbed semilinear problem with Neumann boundary condition
where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function
generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero.
Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000 相似文献
2.
Futoshi Takahashi 《Calculus of Variations and Partial Differential Equations》2007,29(4):509-520
We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E
p
): as p gets large, where Ω is a smooth bounded domain in R
4
. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly
in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc.
AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates
of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky
pattern, under the assumption that the domain is convex. 相似文献
3.
Robert Dalmasso 《Mathematische Annalen》2000,316(4):771-792
We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on .
Received: 4 February 1999 相似文献
4.
We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ2 and f grows superlinearly. (A typical f(u) is f(u)= a1 u+p – a1 u-p, a1, a2 >0, p, q>1.) More precisely, for any positive integer K, there exists εK>0 such that for 0<ε<εK, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed. 相似文献
5.
Mauricio Bogoya 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):143-150
We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈RN with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions. 相似文献
6.
We use a method recently devised by Bolle to establish the existence of an infinite number of solutions for various non-homogeneous
boundary value problems. In particular, we consider second order systems, Hamiltonian systems as well as semi-linear partial
differential equations. The non-homogeneity can originate in the equation but also from the boundary conditions. The results
are more satisfactory than those obtained by the standard “Perturbation from Symmetry” method that was developed – in various forms – in the early eighties by Bahri–Berestycki, Struwe and Rabinowitz.
Received: 13 August 1998 / Revised version: 6 July 1999 相似文献
7.
We consider the Allen–Cahn equation
where Ω is a smooth and bounded domain in
such that the mean curvature is positive at each boundary point. We show that there exists a sequence ε j → 0 such that the Allen–Cahn equation has a solution
with an interface which approaches the boundary as j → + ∞. 相似文献
8.
A regularity result for solutions to boundary blow-up problems for the complex Monge–Ampère operator in balls in is proved. For certain boundary blow-up problems on bounded, strongly pseudoconvex domains in with smooth boundary an estimate of the blow-up rate of solutions are given in terms of the distance to the boundary and the product of the eigenvalues of the Levi form. 相似文献
9.
An elliptic system is considered in a smooth bounded domain, subject to Dirichlet boundary conditions of three different types. Based on the construction of certain upper and sub-solutions, we obtain some conditions on the parameters ai,bi,ci (i=1,2) and the exponents m,n,p,q to ensure the existence of positive solutions. Furthermore, uniqueness and boundary behavior of positive solutions is also discussed. 相似文献
10.
Hans-Christoph Kaiser Hagen Neidhardt Joachim Rehberg 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(3):287-310
Using results on abstract evolutions equations and recently obtained results on elliptic operators with discontinuous coefficients
including mixed boundary conditions we prove that quasilinear parabolic systems admit a local, classical solution in the space
of p–integrable functions, for some p greater than 1, over a bounded two dimensional space domain. The treatment of such equations in a space of integrable functions
enables us to define the normal component of the current across the boundary of any Lipschitz subset. As applications we have
in mind systems of reaction diffusion equations, e.g. van Roosbroeck’s system. 相似文献
11.
In this paper, an elliptic system with boundary blow-up is considered in a smooth bounded domain. By constructing certain upper solution and subsolution, we show the existence of positive solutions and give a global estimate. Furthermore, the boundary behavior of positive solutions is also discussed. 相似文献
12.
We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition. 相似文献
13.
We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1. 相似文献
14.
Mohamed Ben Ayed Khalil El Mehdi 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(4):485-509
This paper is concerned with a biharmonic equation under the Navier boundary condition
, u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in
, n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P
−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point
x
0 ∈Ω as ε → 0, moreover x
0 is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x
0 of the Robin’s function, there exist solutions of (P
−ε) concentrating around x
0 as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P
−ε), the supercritical problem (P
+ε) has no solutions which concentrate around a point of Ω as ε → 0.
Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like
to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena
Pacella for their constant support. 相似文献
15.
The Neumann problem for nonlocal nonlinear diffusion equations 总被引:1,自引:0,他引:1
Fuensanta Andreu José M. Mazón Julio D. Rossi Julián Toledo 《Journal of Evolution Equations》2008,8(1):189-215
We study nonlocal diffusion models of the form
Here Ω is a bounded smooth domain andγ is a maximal monotone graph in . This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove existence
and uniqueness of solutions with initial conditions in L
1 (Ω). Moreover, when γ is a continuous function we find the asymptotic behaviour of the solutions, they converge as t → ∞ to the mean value of the initial condition.
Dedicated to I. Peral on the Occasion of His 60th Birthday 相似文献
16.
《偏微分方程通讯》2013,38(3-4):809-845
17.
A. Haraux 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(4):435-445
We examine the rate of decay to 0, as t → +∞., of the projection on the range of A of the solutions of an equation of the form u′ + Au + |u|
p−1
u = 0 or u′′ + u′ + Au + |u|
p−1
u = 0 in a bounded domain of
N
, where A = −Δ with Neumann boundary conditions or A = −Δ − λ1
I with Dirichlet boundary conditions. In general this decay is much faster than the decay of the projection on the kernel;
it is often exponential, but apparently not always. 相似文献
18.
We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on
bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the
time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity
of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
19.
Manuel del Pino Jean Dolbeault Monica Musso 《Journal de Mathématiques Pures et Appliquées》2004,83(12):1405-1456
We consider the problem of finding positive solutions of Δu+λu+uq=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ1. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small. 相似文献
20.
Manuel del Pino Patricio L. Felmer Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2000,10(2):119-134
We consider the problem
where is a smooth domain in , not necessarily bounded, is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as approaches zero, at a maximum of the function , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly
degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the
critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We
allow a degenerate distance function and a more general nonlinearity.
Received September 3, 1998 / Accepted February 29, 1999 相似文献