We consider the following quasilinear elliptic problem ⎧ ⎨ ⎩ −Δ p u ± u q = λ u p−1 |x| p + h in Ω, u 0 and u = 0 on∂ Ω, where, 1 < p < N, Ω ⊂ R N is a bounded regular domain such that 0 ∈ Ω , q > p − 1 and h is a nonnegative measurable function with suitable hypotheses. The main goal of this paper is to analyze the interaction between the Hardy potential, and the term u q , in order to get existence and non existence of positive solution. We can summarize our main results, in the two following points: (i) If u q appears as a reaction term, then we show the existence of a critical exponent q + (λ) , such that for q > q + , the considered problem has no positive distributional solution. If q < q + we find solutions under suitable hypothesis on h. (ii) If u q appears as an absorption term, then there exists q * such that if q > q * , the problem under consideration has a positive solution for all λ > 0 and for all h ∈ L 1 (Ω). The optimality of q * is proved in the sense that if q < q * , then nonexistence holds if λ > Λ N,p .
In this paper we deal with the following quasilinear parabolic problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (u θ)t − Δpu = λ u p−1 |x| p + u q + f, u ≥ 0 in Ω× (0, T), u(x, t) = 0 on ∂Ω × (0, T), u(x, 0) = u0(x), x ∈ Ω, where θ is either 1 or (p − 1), N ≥ 3, Ω ⊂ I R N is either a bounded regular domain containing the origin or Ω ≡ I R N , 1 < p < N, q > 0 and u0 ≥ 0, f ≥ 0 with suitable hypotheses. The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, we analyze blowup phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle. We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power.
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