Let Ω ⊂ ℝN be a bounded regular domain of ℝN and 1 < p < ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all [Formula: see text], we have [Formula: see text] where d(x) = dist (x, ∂Ω), [Formula: see text] and C is a positive constant depending only on p, N and Ω. The optimality of the exponent of the logarithmic term is also proved. In the second part, we consider the following class of elliptic problem [Formula: see text] where 0 < q ≤ 2* - 1. We investigate the question of existence and nonexistence of positive solutions depending on the range of the exponent q.

In this research, we investigate the influence of predator harvesting on the predator–prey interaction in the presence of prey social behavior using a reaction–diffusion system subject to the Neumann boundary conditions. It has been proved that the investigated model can undergo Hopf, Turing–Hopf bifurcation, which indicates the possibility of having a homogenous/nonhomogeneous periodic solution under some conditions on the model parameters. The stability of these periodic solutions is studied using the normal form on the center of the manifold theory. The obtained mathematical results are checked numerically.

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.More precisely, we consider the problemwhere is a bounded regular domain of IR N , d(x) = dist(x, ∂ ), p > 0, and λ > 0 is a positive constant. We prove that 1. If 0 < p < 1, then (P) has no positive very weak solution. 2. If p = 1, then (P) has a positive very weak solution under additional hypotheses on λ and u 0 . 3. If p > 1, then, for all λ > 0, the problem (P) has a positive very weak solution under suitable hypothesis on u 0 . Moreover, we consider also the concave-convex-related case.

Let Ω ⊂ R N be a bounded regular domain, 0 < s < 1 and N > 2s. We considerThe main goal of this paper is to analyze existence and non existence of solution to problem (P ) according to the values of s and q.