Quasilinear elliptic problem with Hardy potential and a reaction-absorbtion term

Abstract

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 .

Some remarks on quasilinear parabolic problems with singular potential and a reaction term

“…Next, we will give the results that we need about the stationary problem, we refer to [36] for more details.…”

“…If p = 2 a stronger non existence result is proved in [16]. For p = 2 the existence results and the behavior of the positive solution near the singular point can be seen in [36].…”

Biharmonic elliptic problems with second Hessian and gradient nonlinearities

“…There have been several works on second-order elliptic equations with gradient nonlinearity, see for instance [2, 9, 10, 20-22, 31, 33, 34] and references cited therein. For 𝑝-Laplace equation with gradient nonlinearity, we refer to [25,26] and references cited therein. Chapiro et al [7] established the existence of a solution to fourth-order ordinary differential equations, using the iteration method introduced by de Figueiredo et al [11].…”

“…There have been several works on second‐order elliptic equations with gradient nonlinearity, see for instance [2, 9, 10, 20–22, 31, 33, 34] and references cited therein. For p ‐Laplace equation with gradient nonlinearity, we refer to [25, 26] and references cited therein. Chapiro et al.…”