Abstract. Let B be a ball in R N , N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form −Δu = m (x) u p in B, u = 0 on ∂B.
Mathematics Subject Classification (2000). 35J25, 35J61, 35B09, 35J65.
Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k
Abstract. We consider the Heisenberg group H n = C n ×R. Let ν be the Borel measure on H n defined by ν(E) = C n χ E (w, ϕ(w)) η(w)dw, where ϕ(w) = n j=1 a j |w j | 2 , w = (w 1 , ..., wn) ∈ C n , a j ∈ R, and η(w) = η 0 |w| 2 with η 0 ∈ C ∞ c (R). In this paper we characterize the set of pairs (p, q) such that the convolution operator with ν isWe also obtain L p -improving properties of measures supported on the graph of the function ϕ(w) = |w| 2m .
We give necessary and sufficient conditions for the existence of positive solutions for sublinear Dirichlet periodic parabolic problems Lu = g (x,t,u) in Ω × R (where Ω ⊂ R N is a smooth bounded domain) for a wide class of Carathéodory functions g : Ω × R × [0,∞) → R satisfying some integrability and positivity conditions.
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