S. Paczka

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

This paper is concerned with the antimaximum principle for the quasilinear prob-

Positive solutions for sublinear periodic parabolic problems

1\mathrm{e}\mathrm{m}-\Delta_{p}u=\lambda m(x)|u|^{p-2}u+h(x)

Antimaximum principle for elliptic problems with weight

,

A minimax formula for principal eigenvalues and application to an antimaximum principle

\Delta_{p}

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

is the -laplacian and

On some singular periodic parabolic problems

m(x)

Minimax formula for the principal eigenvalue and application to the antimaximum principle

is aweight function which may change sign. We will in particular investigate the question of the uniformity of this principle and provide avariational characterization for the interval of uniformity. An identity of Picones type for the

The periodic parabolic eigenvalue problem with

\mathrm{p}

L^\infty

-laplacian plays an important role in our approach.

weight.

Abstract Let Ω be a bounded domain in R N . We characterize the set of positive principal eigenvalues for the Dirichlet periodic parabolic problem Lu=λg(x,t,u) in Ω× R , under the assumptions that g is a function such that ξ→g(x,t,ξ)/ξ is continuously differentiable and nonincreasing in [0,∞) a.e. (x,t)∈Ω× R satisfying some integrability and positivity conditions. We also prove the uniqueness of the positive solution. As a consequence we obtain a necessary and sufficient condition for the existence of a (unique) positive solution for the periodic parabolic logistic equation with unbounded weights. Existence of positive solutions for a generalization of the logistic equation is also shown.