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.

In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)