Let 0 < s < 1 and p > 1 be such that ps < N . Assume that Ω is a bounded domain containing the origin. Starting from the ground state inequality by R. Frank and R. Seiringer in [16] to obtain:(1) The following improved Hardy inequality for p 2: For all q < p, there exists a positive constant C ≡ C(Ω, q, N, s) such that
Abstract. In this article the problem to be studied is the followingwhere Ω is a bounded domain, and (−∆ s p ) is the fractional p-Laplacian operator defined by|x − y| N+ps dy with 1 < p < N , s ∈ (0, 1) and f, u 0 are measurable functions. The main goal of this work is to prove, problem (P ) has a weak solution with suitable regularity. In addition, if f 0 , u 0 are nonnegative, we show that the problem above has a nonnegative entropy solution.In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of p.
Abstract. The aim of this paper is to study the following problemwhere Ω is a smooth bounded domain of IR N containing the origin,with 0 ≤ β < N−ps 2 , 1 < p < N , s ∈ (0, 1) and ps < N . The main purpose of this work is to prove the existence of a weak solution under some hypotheses on f . In particular, we will consider two cases:(1) f (x, σ) = f (x), in this case we prove the existence of a weak solution, that is in a suitable weighted fractional Sobolev spaces for all f ∈ L 1 (Ω). In addition, if f 0, we show that problem (P ) has a unique entropy positive solution.(2) f (x, σ) = λσ q + g(x), σ ≥ 0, in this case, according to the values of λ and q, we get the largest class of data g for which problem (P ) has a positive solution. In the case where f 0, then the solution u satisfies a suitable weak Harnack inequality.
In this paper, we investigate the existence of solutions to a nonlinear parabolic system, which couples a non-homogeneous reaction-diffusion-type equation and a non-homogeneous viscous Hamilton-Jacobi one. The initial data and right-hand sides satisfy suitable integrability conditions and non-negative. In order to simplify the presentation of our results, we will consider separately two simplified models : first, vanishing initial data, and then, vanishing right-hand sides.
The aim goal of this paper is to treat the following problemwhere Ω is a bounded domain containing the origin,N+ps dy with 1 < p < N, s ∈ (0, 1) and f, u 0 are non negative functions. The main goal of this work is to discuss the existence of solution according to the values of p and λ.1 0 σ ps−1 |1 − σ N −ps p | p K(σ)dσ Key words and phrases. Nonlinear nonlocal parabolic problems, Hardy potential, Caffarelli-Khon-Nirenberg inequality for degenerate weights, finite time extension, non existence result.
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