A fractional Laplacian problem with mixed singular nonlinearities and nonregular data

Abstract

In this note, we study on the existence and uniqueness of a positive solution to the following doubly singular fractional problem: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u = \dfrac{K(x)}{u^q} + \dfrac{f(x)}{u^{\gamma }}+\mu &{} \mathrm {in} \,\, \Omega ,\\ u>0 &{} \mathrm {in} \,\, \Omega , \\ u=0 &{} \mathrm {in} \,\, \big ({\mathbb {R}}^N \setminus \Omega \big ). \end{array}\right. } \end{aligned}$$Here \(\Omega \subset {\mathbb {R}}^N\) (\(N > 2s\)) is an open bounded domain with smooth boundary, \( s \in (0,1)\), \(q>0\), \(\gamma >0\), and K(x) is a positive Hölder continuous function in which behaves as \(\mathrm {dist}(x, \partial \Omega )^{-\beta }\) near the boundary with \(0 \le \beta <2s\). Also, \(0 \le f, \mu \in L^1(\Omega )\), or non-negative bounded Radon measures in \(\Omega \). Moreover, we assume that \(0<\frac{\beta }{s}+q < 1\), or \(\frac{\beta }{s}+q > 1\) with \(2\beta +q(2s-1)<(2s+1)\). For \(s \in (0,\frac{1}{2})\), we take advantage of the convexity of \(\Omega \). For any \(\gamma >0\), we will prove the existence of a positive weak (distributional) solution to the above problem. Besides, for the case \( 0 < \gamma \le 1\), \(\mu \in L^1(\Omega )+X^{-s}(\Omega )\), and some weighted integrable functions f, we will show the existence and uniqueness of another notion of a solution, so-called entropy solution. Also, we will discuss the uniqueness of the weak solution for the case \(\gamma >1\), and also the equivalence of entropy and weak solutions for the case \(0< \gamma \le 1\). Finally, we will have some relaxation on the assumptions of f in order to prove the existence of solutions.