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1.
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   

2.
We consider the nonlinear eigenvalue problem −Δuf(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in RN (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution uλ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of uλ with the exact second term and the ‘optimal’ estimate of the third term.  相似文献   

3.
In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.  相似文献   

4.
Summary We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD d (d3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary ofD. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality forW 1,2(D). Special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics.Research supported in part by NSF Grants DMS 8657483 and GER 9023335This article was processed by the authors using the style filepljourlm from Springer-Verlag.  相似文献   

5.
Let Ω be a bounded domain in RN,N≥2, with C2 boundary. In this work, we study the existence of multiple positive solutions of the following problem:
  相似文献   

6.
Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ1(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ1(Ω)), in the case Ω=2(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ1(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.  相似文献   

7.
8.
We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.  相似文献   

9.
In this paper we consider the following problem
(?)  相似文献   

10.
We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.  相似文献   

11.
Research supported in part by NSF Grants DMS 8657483 and GER 9023335  相似文献   

12.
We investigate entire radial solutions of the semilinear biharmonic equation Δ2u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.  相似文献   

13.
In this paper, we extend the result of Donig[Do] and show the finiteness of the lower spectrum of the uniformly elliptic operators–·(A(x))+q with singular potentialsq which belong to the Kato class. Even in the case of the Schrödinger operator–+q, our proof simplifies the one in [Do].  相似文献   

14.
In this paper we consider a nonlinear eigenvalue problem driven by the pp-Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ>0λ>0 varies in a half-line.  相似文献   

15.
16.
We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999  相似文献   

17.
In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC 2-function at both isolated critical point and infinity.  相似文献   

18.
Suppose that E is a bounded domain of class C2,λ in and L is a uniformly elliptic operator in E. The set of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions uΓ?wΓ. We also introduced a class of solutions uν in 1-1 correspondence with a certain class of σ-finite measures ν on ∂E. With every we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and . We called this pair the fine boundary trace of u and we denoted in tr(u).Let uv stand for the maximal solution dominated by u+v. We say that u belongs to the class if the condition tr(u)=(Γ,ν) implies that u?wΓuν and we say that u belongs to if the condition tr(u)=(Γ,ν) implies that u?uΓuν.It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class contains all bounded domains. It follows from results of Mselati (Thése de Doctorat de l'Université Paris 6, 2002; C.R. Acad. Sci. Paris Sér. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C4 belong to where Δ is the Laplacian and ψ(u)=u2. [Mselati proved that, in his case, uΓ=wΓ and therefore the condition tr(u)=(Γ,ν) implies u=uΓuν=wΓuν.]By modifying Mselati's arguments, we extend his result to ψ(u)=uα with 1<α?2 and all bounded domains of class C2,λ.We start from proving a general localization theorem: under broad assumptions on L, ψ if, for every y∂E there exists a domain such that E′⊂E and ∂E∂E′ contains a neighborhood of y in ∂E.  相似文献   

19.
20.
Under some non-degeneracy condition we show that sequences of entropy solutions of a semi-linear elliptic equation are strongly pre-compact in the general case of a Carathéodory flux vector. The proofs are based on localization principles for H-measures corresponding to sequences of measure-valued functions.  相似文献   

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