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We study spectral properties of the discrete Laplacian L?=??Δ?+?V on ? with finitely supported potential V. We give sufficient and necessary conditions for L to satisfy that the number of negative (resp. positive) eigenvalues is equal to one of the points x on which V(x) is negative (resp. positive). In addition, we prove that L has at least one discrete eigenvalue. If ∑ x∈? V(x)?=?0, then L has both negative and positive discrete eigenvalues.  相似文献   

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The discrete analog of the Fučik spectrum for elliptic equations, namely M-matrices, is shown to have properties analogous to the continuum. In particular, the Fučik spectrum of a M-matrix contains a continuous and decreasing curve which is symmetric with respect to the diagonal.  相似文献   

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In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fuík spectrum and (b) possible variants and extensions of the monotonicity formula by Alt-Caffarelli-Friedman [1]. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this approach, we give a new variational characterization of the first curve of the Fuík spectrum for the Laplacian, promptly adapted to more general operators. In the second part we prove a monotonicity formula in the case of many subharmonic components and we give an extension to solutions of a class of reaction-diffusion equation, providing some Liouville-type theorems.Received: 27 December 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000): 35J65 (58E05)Work partially supported by MIUR project Metodi Variazionali ed Equazioni Differenziali Non Lineari  相似文献   

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In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum.  相似文献   

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We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.  相似文献   

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We study generalized solutions of the nonlinear wave equation
uttuss=au+bu+p(s,t,u),  相似文献   

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The Hamiltonian of a system of four arbitrary quantum particles with three-particle short-range interaction potentials on a three-dimensional lattice is examined. The location of the essential spectrum of this Hamiltonian is described by Faddeev’s equations.  相似文献   

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The existence of solutions for semilinear equations with Dirichlet condition are established under the assumption that the nonlinearity is of linear growth and the asymptotic behavior of its primitive at infinity stays away from the Fučík spectrum.  相似文献   

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By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .  相似文献   

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We construct and variationally characterize by a min-max procedure an unbounded sequence of continuous and strictly decreasing curves in the Fuík spectrum of the p-Laplacian. Applications to quasilinear elliptic boundary value problems are given.  相似文献   

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The existence of solutions for semilinear equations with Dirichlet condition are established under the assumption that the nonlinearity is of linear growth and the asymptotic behavior of its primitive at infinity stays away from the Fu?ík spectrum.  相似文献   

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