The aim of this paper is to study the following problem: { ( - Δ ) p , β s ⁢ u = f ⁢ ( x , u ) in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p,\beta}u&\displaystyle=f(x% ,u)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is a smooth bounded domain of ℝ N {\mathbb{R}^{N}} containing the origin, ( - Δ ) p , β s ⁢ u ⁢ ( x ) := PV ⁢ ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p - 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ d ⁢ y | x | β ⁢ | y | β (-\Delta)^{s}_{p,\beta}u(x):=\mathrm{PV}\int_{\mathbb{R}^{N}}\frac{\lvert u(x)% -u(y)\rvert^{p-2}(u(x)-u(y))}{\lvert x-y\rvert^{N+ps}}\frac{dy}{\lvert x\rvert% ^{\beta}\lvert y\rvert^{\beta}} with 0 ≤ β < N - p ⁢ s 2 {0\leq\beta<\frac{N-ps}{2}} , 1 < p < N {1<p<N} , s ∈ ( 0 , 1 ) {s\in(0,1)} , and p ⁢ s < N {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: (i) f ⁢ ( x , σ ) = f ⁢ ( x ) {f(x,\sigma)=f(x)} ; in this case we prove the existence of a weak solution, that is, in a suitable weighted fractional Sobolev space for all f ∈ L 1 ⁢ ( Ω ) {f\in L^{1}(\Omega)} . In addition, if f ⪈ 0 {f\gneq 0} , we show that the problem above has a unique entropy positive solution. (ii) f ⁢ ( x , σ ) = λ ⁢ σ q + g ⁢ ( x ) {f(x,\sigma)=\lambda\sigma^{q}+g(x)} , σ ≥ 0 {\sigma\geq 0} ; in this case, according to the values of λ and q , we get the largest class of data g for which the problem above has a positive solution.

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem where Ω⊂ℝ^N, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u₀≥0, f≥0 are in a suitable class of functions.There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.The Cauchy problem, Ω=ℝ^N, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).