Indefinite Weight Research Articles

Linking solutions for p-Laplace equations with nonlinear boundary conditions and indefinite weight

We apply the linking method for cones in normed spaces to p-Laplace equations with various nonlinear boundary conditions. Some existence results are obtained.

Existence of nonnegative solutions for a class of semilinear elliptic systems with indefinite weight

We prove the existence of nonnegative solutions to the system − Δ u = λ a ( x ) f ( v ) in Ω , − Δ v = λ b ( x ) g ( u ) in Ω , u = 0 = v on ∂ Ω , where Ω is a bounded domain in R N with a smooth boundary ∂ Ω , λ is a positive parameter. By the method of monotone iteration and Schauder fixed point theorem, we prove the existence of a nonnegative solution to the above system for λ sufficiently small. In this study, we do not restrict the sign of a and b .

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces Lγ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in Lγ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Frechet derivative and weak* convergence.

An anti-maximum principle for degenerate elliptic boundary value problems with indefinite weights

This article deals with an anti-maximum principle to the degenerate case and to elliptic boundary value problems with indefinite weight function which include the Dirichlet and Robin boundary conditions as special cases. The result extends an earlier theorem by Clément and Peletier [An anti-maximum principle for second-order elliptic operators, J. Diff. Eqns. 34 (1979), pp. 218–229] to the degenerate case.

Minimization of eigenvalues for a quasilinear elliptic Neumann problem with indefinite weight

We investigate the minimization of the positive principal eigenvalue of the problem − Δ p u = λ m | u | p − 2 u in Ω, ∂ u / ∂ ν = 0 on ∂ Ω, over a class of sign-changing weights m with ∫ Ω m < 0 . It is proved that minimizers exist and satisfy a bang–bang type property. In dimension one, we obtain a complete description of the minimizers. This problem is motivated by applications from population dynamics.

PICONE INEQUALITIES FOR p-SUB-LAPLACIAN ON THE HEISENBERG GROUP AND ITS APPLICATIONS

In this paper, we establish the Picone inequalities for p-sub-Laplacian on the Heisenberg group. Then, we prove Hardy inequalities with weight functions involving partial variables and a comparison principle. As an immediate consequence, we obtain the uniqueness for p-sub-Laplace equation involving singular indefinite weights. Also we prove the existence of positive solutions.

A global multiplicity result for two-point boundary value problems of p-Laplacian systems

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for two-point boundary value problems of p-Laplacian systems which have a singular indefinite weight and real multiparameters. For proofs, we mainly make use of the upper and lower solution method and the fixed point index theorem. To obtain a global multiplicity result, we construct a weighted space to benefit richer topology of the solution space than C0-space.

Degenerate elliptic boundary value problems with asymmetric nonlinearity

This paper is devoted to the study of a class of semilinear degenerate elliptic boundary value problems with asymmetric nonlinearity which include as particular cases the Dirichlet and Robin problems. The most essential point is how to generalize the classical variational approach to eigenvalue problems with an indefinite weight to the degenerate case. The variational approach here is based on the theory of fractional powers of analytic semigroups. By making use of global inversion theorems with singularities between Banach spaces, we prove very exact results on the number of solutions of our problem. The results extend an earlier theorem due to Ambrosetti and Prodi to the degenerate case.

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by Lopez-Gomez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.

Multiplicity of solutions for gradient systems with strong resonance at higher eigenvalues

In this paper we establish existence and multiplicity of solutions for an elliptic system which has strong resonance at higher eigenvalues. We describe the resonance using an eigenvalue problem with indefinite weights. In all results we use Variational Methods and the Morse theory.

Multiplicity of Solutions for Gradient Systems Using Landesman‐Lazer Conditions

We establish existence and multiplicity of solutions for an elliptic system which presents resonance at infinity of Landesman‐Lazer type. In order to describe the resonance, we use an eigenvalue problem with indefinite weights. In all results, we use Variational Methods, Morse Theory and Critical Groups.

An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches

An indefinite weight eigenvalue problem characterizing the thresholdcondition for extinction of a population based on the single-speciesdiffusive logistic model in a spatially heterogeneous environment isanalyzed in a bounded two-dimensional domain with no-flux boundaryconditions. In this eigenvalue problem, the spatial heterogeneity ofthe environment is reflected in the growth rate function, which isassumed to be concentrated in $n$ small circular disks, or portions ofsmall circular disks, that are contained inside the domain. Theconstant bulk or background growth rate is assumed to be spatiallyuniform. The disks, or patches, represent either strongly favorable orstrongly unfavorable local habitats. For this class of piecewise constantbang-bang growth rate function, an asymptotic expansion for thepersistence threshold λ1, representing the positive principaleigenvalue for this indefinite weight eigenvalue problem, iscalculated in the limit of small patch radii by using the method ofmatched asymptotic expansions. By analytically optimizing thecoefficient of the leading-order term in the asymptotic expansion ofλ1, general qualitative principles regarding the effect ofhabitat fragmentation are derived. In certain degenerate situations,it is shown that the optimum spatial arrangement of the favorablehabit is determined by a higher-order coefficient in the asymptoticexpansion of the persistence threshold.

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm–Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.

On the nonreal eigenvalues of elliptic differential operators with indefinite weights on Lipschitz domains

AbstractThe nonreal spectrum of a second order elliptic differential operator with an indefinite weight function on a Lipschitz domain is investigated with the help of Krein space techniques and perturbation methods. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Definitizability of a Class of J‐Selfadjoint Operators with Applications

AbstractWe formulate an abstract result concerning the definitizability of J‐selfadjoint operators which, roughly speaking, differ by at most finitely many dimensions from the orthogonal sum of a J‐selfadjoint operator with finitely many negative squares and a semibounded selfadjoint operator in a Hilbert space. The general perturbation result is applied to a class of singular Sturm–Liouville operators with indefinite weight functions. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Singular Indefinite Sturm‐Liouville Operators with a Spectral Gap

AbstractSingular Sturm‐Liouville operators with the indefinite weight sgn(·) and a symmetric potential which has a positive limit at ∞ have a gap in the essential spectrum. Under an additional condition it is shown that in this gap are no eigenvalues. (© 2009 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the asymptotic behavior of the principal eigenvalues of some elliptic problems

This paper is concerned with nonself-adjoint elliptic problems involving indefinite weights and boundary conditions of the Dirichlet, Neumann or Robin type. We study the asymptotic behavior of the principal eigenvalues, when the first order term (drift term) becomes larger and larger. The basic results of Berestycki et al. (Commun. Math. Phys., 253:451–480, 2005) are extended to the present context. Moreover, answers are provided to some open problems raised in Berestycki et al. (Commun. Math. Phys., 253:451–480, 2005).

Schrödinger equations with indefinite weights in the whole space

We consider in this Note equations defined in R N involving Schrödinger operators with indefinite weight functions and with potentials which tend to infinity at infinity. We give some results for the existence of principal eigenvalues and for the maximum principle. We also obtain Courant–Fischer formulas for such eigenvalues. To cite this article: L. Cardoulis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Non-real eigenvalues of singular indefinite Sturm-Liouville operators

We study a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on \mathbb{R} and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at \pm \infty we prove that there are no real eigenvalues and the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to −d^2/dx^2 + V in L^2(\mathbb{R}). The general results are illustrated with examples.

Spectrum of the A_p-Laplacian Operator

This work deals with the nonlinear boundary eigenvalue problem: \begin{equation*}(V.P_{A,\rho,I}) \left\{ \begin{aligned} & A_p u = \lambda \rho(x) |u|^{p-2}u in I =]a; b[; & u(a) = u(b) = 0; \end{aligned} \right. \end{equation*} where A_p is called the A_p-Laplacian operator and defined by A_pu =(\Gamma(x) |u'|^{p-2}u')', p > 1, \gamma is a real parameter, \rho is an indefinite weight, a, b are real numbers and \Gamma\in C^1(I) \cap C^0 (\overline{I}) and it is nonnegative on \overline{I}. We prove in this paper that the spectrum of the A_p-Laplacian operator is given by a sequence of eigenvalues. Moreover, each eigenvalue is simple, isolated and verifies the strict monotonicity property with respect to the weight \rho and the domain I. The kith eigenfunction corresponding to the k-th eigenvalue has exactly k-1 zeros in (a; b). Finally, we give a simple variational formulation of eigenvalues.