On the antimaximum principle for the p-Laplacian with indefinite weight

Godoy, T.; Gossez, Jean–Pierre; Paczka, S.

doi:10.1016/s0362-546x(01)00839-2

Cited by 52 publications

(18 citation statements)

References 13 publications

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“…By taking in (3.10) ε > 0 even smaller if necessary, we can apply Theorem 5.1 of Godoy, Gossez and Paczka [13] (the antimaximum principle) and conclude thatx ∈ int C + . 2…”

Section: Solutions Of Constant Signmentioning

confidence: 98%

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

Filippakis

Papageorgiou

2008

Journal of Differential Equations

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We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.

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“…By taking in (3.10) ε > 0 even smaller if necessary, we can apply Theorem 5.1 of Godoy, Gossez and Paczka [13] (the antimaximum principle) and conclude thatx ∈ int C + . 2…”

Section: Solutions Of Constant Signmentioning

confidence: 98%

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

Filippakis

Papageorgiou

2008

Journal of Differential Equations

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“…(However, see Godoy, Gossez and Paczka [24] for an anti-maximum principle for the p-Laplacian operator.) The proof of Theorem 2.5 is based on an indirect argument due to Birindelli [3].…”

Section: Theorem 23 (See Ishii and Yoshimuramentioning

confidence: 99%

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Armstrong

2009

Journal of Differential Equations

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We study the fully nonlinear elliptic equationin a smooth bounded domain Ω, under the assumption that the nonlinearity F is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum principle of Clément and Peletier [P. Clément, L.A. Peletier, An antimaximum principle for second-order elliptic operators, J. Differential Equations 34 (2) (1979) 218-229] to homogeneous, fully nonlinear operators.

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“…The next proposition is again proved in [9] and will be an important key in the study of the existence of positive solutions of problem (P ). Now we introduce the one-parameter eigenvalue problem…”

Section: Preliminaries and Motivationmentioning

confidence: 85%

“…This shows that if m has a sign, there are no nonzero principal eigenvalues. For signchanging weight functions m, the study of (E) m was first investigated in [11] and later carried on in [9]. In particular, the following result holds (cf.…”

Section: Preliminaries and Motivationmentioning

confidence: 96%

“…In particular, the following result holds (cf. [9]): This theorem applies also to the weight −m, and so λ = −λ * (−m) is the unique nonzero principal eigenvalue of (E) m when m dx > 0 and m changes sign. Let us further recall that the principal eigenvalues of (E) m are simple and that ϕ ∈ C 1,a (Ω) for every associated eigenfunction ϕ.…”

Section: Preliminaries and Motivationmentioning

confidence: 96%

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

Derlet¹,

Gossez²,

Takáč³

2010

Journal of Mathematical Analysis and Applications

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We investigate the minimization of the positive principal eigenvalue of the problem − p u = λm|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.

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On the antimaximum principle for the p-Laplacian with indefinite weight

Cited by 52 publications

References 13 publications

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

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

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