共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper we consider the following problem
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Let r,s∈]1,2[ and λ,μ∈]0,+∞[. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0 boundary data for the semilinear elliptic equation −Δu=λus−1−ur−1 in Ω⊂RN, where N≥2. We prove that there exists a positive constant Λ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ, λ=Λ or λ<Λ. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given. 相似文献
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This paper concerns semilinear elliptic equations of the form – u+m(x)u=a(x)u
p
, wherea changes sign. We discuss the question of existence of positive solutions when the linear part is not coercive.This article was processed by the author using the LATEX style file pljourlm from Springer-Verlag. 相似文献
6.
The aim of this paper is to establish the existence of an unbounded sequence of weak solutions for a class of differential equations with p(x)-Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. The approach is based on variational methods. 相似文献
7.
Kaouther Ammar 《Journal of Differential Equations》2007,234(1):1-25
We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L1-setting. 相似文献
8.
In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented. 相似文献
9.
Alfredo Cano 《Journal of Differential Equations》2007,237(1):133-158
We consider the problem −Δu+a(x)u=f(x)|u|2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|2*−2u in Ω, u=0 on ∂Ω. 相似文献
10.
This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong. 相似文献
11.
Thomas Bartsch 《Journal of Differential Equations》2006,220(2):531-543
In this work, we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity. We establish conditions for existence and nonexistence of nontrivial solutions to these systems. 相似文献
12.
E. Colorado 《Journal of Functional Analysis》2003,199(2):468-507
This work deals with the analysis of problems
13.
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
14.
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
15.
Zhaoli Liu Jingxian Sun 《Calculus of Variations and Partial Differential Equations》2002,14(3):319-327
This paper concerns the existence of four (or six) solutions of semilinear elliptic boundary value problems provided that
two disorderly solutions are known. The results are obtained under very generic conditions.
Received: 26 August 2000 / Accepted: 23 February 2001 / Published online: 23 July 2001 相似文献
16.
Dumitru Motreanu 《Journal of Differential Equations》2007,232(1):1-35
In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation. 相似文献
17.
This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology. 相似文献
18.
We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem. 相似文献
19.
Dimitrios A. Kandilakis 《Journal of Differential Equations》2006,230(1):337-361
By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities. 相似文献