Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

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 \(\mathbb R ^2\) .     

Homogenization of Fučík Eigencurves

In this work we study the convergence of an homogenization problem for half-eigenvalues and Fu?ík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.     

Basis Properties of Fučík Eigenfunctions

We establish sufficient assumptions on sequences of Fu?ík eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fu?ík eigenfunctions form a Riesz basis in L²(0,π).

     

Discrete analogue of Fučik spectrum of the Laplacian

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.     

On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae

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     

Nonresonance conditions on the potential with respect to the Fučík spectrum for semilinear Dirichlet problems

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.     

On the Fučik spectrum of non-local elliptic operators

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.     

Nonresonance conditions on the potential with respect to the Fu?ík spectrum for semilinear Dirichlet problems

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.     

On the Fučík Spectrum of the <Emphasis Type="Italic">p</Emphasis>-Laplacian

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.     

On the Fu?ik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation

utt−uss=au⁺−bu⁻+p(s,t,u),     

New approach to calculating the spectrum of a quantum space–time

We study the dynamics of a massive pointlike particle coupled to gravity in four space–time dimensions. It has the same degrees of freedom as an ordinary particle: its coordinates with respect to a chosen origin (observer) and the canonically conjugate momenta. The effect of gravity is that such a particle is a black hole: its momentum becomes spacelike at a distances to the origin less than the Schwarzschild radius. This happens because the phase space of the particle has a nontrivial structure: the momentum space has curvature, and this curvature depends on the position in the coordinate space. The momentum space curvature in turn leads to the coordinate operator in quantum theory having a nontrivial spectrum. This spectrum is independent of the particle mass and determines the accessible points of space–time.