共查询到20条相似文献,搜索用时 343 毫秒
1.
Reika Fukuizumi Tohru Ozawa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):1000-1011
Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in
where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific
rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in
the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|2.
Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday
Received: May 4, 2004 相似文献
2.
T. del Vecchio 《Potential Analysis》1995,4(2):185-203
In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results. 相似文献
3.
We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator
to a general quasilinear operator and remove a strong ellipticity assumption in Philippin (Maximum principles and eigenvalue
problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow,
175, pp. 34–48, 1988) and a growth assumption in Garofalo and Lewis (A symmetry result related to some overdetermined boundary
value problems, Am. J. Math. 111, 9–33, 1989) on the diffusion coefficient A, as well as a starshapedness assumption on Ω in Fragalà et al. (Overdetermined boundary value problems with possibly degenerate
ellipticity: a geometric approach. Math. Zeitschr. 254, 117–132, 2006). 相似文献
4.
We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ2ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method. 相似文献
5.
We study uniformly elliptic fully nonlinear equations of the type F(D2u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations. 相似文献
6.
Thomas Bartsch Zhi-Qiang Wang Zhitao Zhang 《Journal of Fixed Point Theory and Applications》2009,5(2):305-317
We investigate the Fučik point spectrum of the Schr?dinger operator when the potential Vλ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that Sλ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties
of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions
for nonlinear Schr?dinger equations with jumping nonlinearity.
相似文献
7.
Jinggang Tan 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2098-2110
In this paper, we establish the existence of ground state solutions and bound state solutions for fractional field equations
(0.1) 相似文献
8.
The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in
ℝ3. In ℝn,n≥4, the Agmon-Miranda maximum principle andL
p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL
p Dirichlet problem for Δ
m
fails to be solvable forp>2(n−1)/(n−3).
Supported in part by the NSF. 相似文献
9.
In continuation with [17], we investigate the asymptotic behavior of weighted eigenfunctions in two half-spaces connected by a thin tube. We provide several improvements about some convergences stated in [17]; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in [17] and thus describe the (N−1)-order singularity developed at a junction of the tube (where N is the space dimension). 相似文献
10.
In this paper, we are concerned with the existence of solutions to the N-dimensional nonlinear Schrödinger equation −ε2Δu+V(x)u=K(x)up with u(x)>0, u∈H1(RN), N?3 and . When the potential V(x) decays at infinity faster than −2(1+|x|) and K(x)?0 is permitted to be unbounded, we will show that the positive H1(RN)-solutions exist if it is assumed that G(x) has local minimum points for small ε>0, here with denotes the ground energy function which is introduced in [X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633-655]. In addition, when the potential V(x) decays to zero at most like (1+|x|)−α with 0<α?2, we also discuss the existence of positive H1(RN)-solutions for unbounded K(x). Compared with some previous papers [A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006) 317-348; A. Ambrosetti, D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907; A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321-1332] and so on, we remove the restrictions on the potential function V(x) which decays at infinity like (1+|x|)−α with 0<α?2 as well as the restrictions on the boundedness of K(x)>0. Therefore, we partly answer a question posed in the reference [A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, preprint, 2006]. 相似文献
11.
Sandra Cerrai 《Probability Theory and Related Fields》1999,113(1):85-114
In the present paper we consider the transition semigroup P
t
related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in
the Banach space of continuous functions , where ⊂ℝ
d
is a bounded open set. In L
2() the only result proved is the strong Feller property, for d=1. Here we are able to prove that if f∈C
∞(ℝ) and d≤3, then for any and t>0. An important application is to the study of the ergodic properties of the system. These results are also of interest for
some problem in stochastic control.
Received: 20 August 1997 / Revised version: 27 May 1998 相似文献
12.
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
13.
《Mathematische Nachrichten》2018,291(4):632-651
In this paper we analyze an eigenvalue problem related to the nonlocal p‐Laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenfunctions, simplicity and isolation) we investigate the dependence of the first eigenvalue on the potential function and establish the existence of some optimal potentials in some admissible classes. 相似文献
14.
We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H . Moreover, we prove that if u?0, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set. 相似文献
15.
Marc Briane Juan Casado-Díaz 《Calculus of Variations and Partial Differential Equations》2007,29(4):455-479
In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F
n
, , defined in L
2(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F
n
is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains
the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by
some minimizers of the equicoercive sequence F
n
, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density
is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question
of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional
conductivity. 相似文献
16.
Jorge García-Melián José C. Sabina De Lis Julio D. Rossi 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):499-525
We deal with positive solutions of Δu = a(x)u
p
in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile
of the solution as λ → σ1.
Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D.
Rossi). J.D. Rossi is a member of CONICET. 相似文献
17.
Pei-Yong Wang 《Journal of Geometric Analysis》2003,13(4):715-738
In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear
elliptic equations and also shown the free boundary has finite Hn−1 Hausdorff measure and a normal in a measuretheoretic sense Hn−1 almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact
that the free boundary is locally a C1,α surface near Hn−1-a.e. point. 相似文献
18.
Jesus Garcia Azorero Andrea Malchiodi Luigi Montoro Ireneo Peral 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in [J. Garcia Azorero, A. Malchiodi, L. Montoro, I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Existence results, Arch. Ration. Mech. Anal., in press]. 相似文献
19.
Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension
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In this paper, we study the following Kirchhoff‐type elliptic problem where is a bounded domain with smooth boundary ?Ω, a,b,λ,μ > 0 and 1 < q < 2?=2N/(N ? 2). When N = 4, we obtain that there is a ground state solution to the problem for q∈(2,4) by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for q∈(1,2) by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for N≥5 under some suitable conditions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
20.
We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming
normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second
order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution
for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary
integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze
an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order
in essentially work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain.
This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work
initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during
the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the
C.I.R.M., Marseille, France. 相似文献