1.1 Understanding the notion of a solution

We start by understanding the meaning of a ‘weak’ solution for problems (1.2)–(1.3). An elementary way to define the notion of a solution of problem (1.2) is given by: a function $u$ such that

1. (i) $u>0$ a.e. in $\Omega$,

2. (ii) the following weak formulation equality holds:

   (1.4)\begin{align} & \int_{\Omega} \nabla u \cdot \nabla \psi\,{\rm d}x + \frac{C(N,s)}{2} \int_{\mathbb{R}^N} \int_{\mathbb{R}^N} \frac{(u(x)- u(y))(\psi(x)- \psi(y))}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y\nonumber\\ & \quad = \int_{\Omega} \frac{f(x)}{u^\gamma}\psi\,{\rm d}x, \end{align}
   for a class of test functions $\psi \in \mathcal {T}(\Omega )$ and a function $u$ ‘regular’ enough so that all the integrals are well defined.

The first condition $(i)$ is imposed to give a meaning to the term $u^{-\gamma }$ known as ‘singular nonlinearities’ and the second condition $(ii)$ is motivated and obtained by multiplying a smooth function $\psi$ to equation (1.2) and using standard integration by parts formula for a smooth function $u$. Since solutions to equations involving a fractional Laplacian and singular nonlinearities generally are not of class $C^2$ therefore a solution to (1.2) has to be understood in the ‘weak’ sense via (1.4).

A ‘$natural$’ space to look for the solution $u$ of problems (1.2)–(1.3), more accurately, to define the integrals on the left-hand side of (1.4), is as follows:

\[ \mathbb{H}(\Omega):= \{u \in H^1(\mathbb{R}^N): \ u=0 \ \text{in}\ \mathbb{R}^N {\setminus} \Omega\}. \]

In the light of the boundary regularity of $\Omega$, it is well known that the space $\mathbb {H}(\Omega )$ can be identified with $H_0^1(\Omega ).$ Precisely, we understand the identification (see [Reference Brezis19, Proposition 9.18]) in the following way: the function $u \in H_0^1(\Omega )$ as a zero extension of the function $\tilde {u}:= u \cdot {\bf 1}_{\Omega } \in \mathbb {H}(\Omega )$ and

\[ u \in \mathbb{H}(\Omega) \Longrightarrow u|_\Omega \in H_0^1(\Omega). \]

In view of the above identification and employing the classical embedding theorem [Reference Di Nezza, Palatucci and Valdinoci46, Proposition 2.2], both the integrals on the left-hand side of (1.4) are well defined in $H_0^1(\Omega ).$ Due to the fact that $\gamma \geq 0$, the nonlinearity $u^{-\gamma }$ in our problems (1.2)–(1.3) may blow up near the boundary and this is the reason why we regard (1.2) as an equation with ‘singular nonlinearities’. By taking into account the singular nature of the nonlinearities and the regularity of the datum $f$, specifically, when the nature of the singularity is strong i.e. $\gamma \gg 1$ (see [Reference Lazer and McKenna40]) or $f \in L^r(\Omega )$ when $r$ is close to 1 (see [Reference Boccardo and Orsina16]), we cannot always expect our solution $u$ in $H_0^1(\Omega )$ and in place of that, either we have $u^\alpha \in H_0^1(\Omega )$ for some $\alpha \geq 1$ or $u \in W_0^{1,q}(\Omega )$ for some $q \in [1,2)$. For these reasons, as customary in the literature, we adopt the following definition to understand the Dirichlet datum in a generalized sense (see [Reference Arora, Giacomoni and Warnault8, Reference Canino, Montoro, Sciunzi and Squassina21]):

Definition 1.1 A function $u \leq 0$ on $\partial \Omega$, if $u=0$ in $\mathbb {R}^N {\setminus} \Omega$ and for any $\epsilon >0$, we have

\[ (u-\epsilon)^+{\in} H_0^1(\Omega). \]

We say that $u=0$ on $\partial \Omega$, if $u \geq 0$ and $u \leq 0$ on $\partial \Omega.$

To provide a meaningful interpretation for the integral in equation (2.5), we carefully select a suitable class of test functions denoted $\mathcal {T}(\Omega )$. The choice of this class depends on crucial factors, such as the exponent value $\gamma \geq 0$ and the regularity of both the datum $f$ and the solution $u$. This selection is pivotal for the rigorous analysis and understanding of the problem at hand. Motivating from the above discussion, the exact notion of a weak solution to our main problems (1.2)–(1.3) with different choices of a class of test functions and sufficient regularity of the solution $u$, is detailed in § 2 while stating the main results.

1.2 Previous work

One of the seminal breakthroughs in the study of singular nonlinearities was the work of Crandall et al. [Reference Crandall, Rabinowitz and Tartar29], which majorly set-up this direction of research. Afterwards, a large number of publications has been devoted to investigate a diverse spectrum of issues rotating around local/nonlocal elliptic equations involving the singular nonlinearities (see, e.g. [Reference Díaz, Morel and Oswald30, Reference Fulks and Maybee31, Reference Stuart48] and monographs [Reference Ghergu and Rădulescu32, Reference Hernández and Mancebo34]). Let us recall some known results in the literature for both local and nonlocal elliptic equations with singular nonlinearities.

In the local case, Crandall et al. [Reference Crandall, Rabinowitz and Tartar29] studied the singular boundary value problems (1.2)–(1.3) with Laplace operator and $f=1$. By using the classical method of sub-supersolutions on the nonsingular approximating problem, they proved the existence and uniqueness results of the classical solution of our original problem. In addition, by exploiting the second-order ordinary differential equation techniques and localization near the boundary, the boundary behaviour of the solution is deduced. By Stuart [Reference Stuart49], similar results on the existence of solutions were obtained using, this time, an approximation argument with respect to the boundary condition. Actually, both papers [Reference Crandall, Rabinowitz and Tartar29, Reference Stuart49] provide results for more general differential operators with smooth coefficients, not necessarily in divergence form, and for nonmonotone nonlinearities as well. The same model of elliptic equations with singular nonlinearities and $f \in C^\alpha (\overline {\Omega })$, was considered by Lazer and McKenna [Reference Lazer and McKenna40] in which they simplified the proof of boundary behaviour of classical by constructing appropriate sub and super solutions. In addition to that, they also obtained the optimal power related to the existence of finite-energy solutions. In fact, a solution $u \in H_0^1(\Omega )$ exists if and only if $\gamma <3.$ In [Reference Yijing and Zhang50], Yijing and Zhang analysed the threshold value $3$ when the datum $f \in L^1(\Omega )$ and a positive function, and provided a classical Lazer–Mckenna obstruction. In [Reference Boccardo and Orsina16, Reference Canino, Sciunzi and Trombetta23], authors studied the existence and uniqueness results when $f \in L^r(\Omega )$ for $r \geq 1$ and showed how the regularity of this solution depends upon the summability of the datum and the singular datum. In particular, Boccardo and Orsina [Reference Boccardo and Orsina16] proved the existence and regularity of distributional solution:

\[ \left\{\begin{array}{@{}ll} u \in W_0^{1,({Nr(1+\gamma)})/({N-r(1-\gamma)})}(\Omega) & \text{ if } \ 0<\gamma <1 \ \text{and} \ f \in L^r(\Omega) \ \text{with} \ r \\ & \quad\in [1,(2^\ast{/}(1-\gamma))'),\\ u \in H_0^{1}(\Omega) & \text{ if } \ 0<\gamma <1 \ \text{and} \ f \in L^r(\Omega) \ \text{with} \ r\\ & \quad =(2^\ast{/}(1-\gamma))',\\ u \in H_0^{1}(\Omega) & \text{ if } \ \gamma =1 \ \text{and} \ f \in L^1(\Omega),\\ u^{{(1+\gamma)}/{2}} \in H_0^{1}(\Omega) & \text{ if } \ \gamma >1 \ \text{and} \ f \in L^1(\Omega), \end{array}\right. \]

Extending the work of [Reference Boccardo and Orsina16], Arcoya and Moreno-Mérida in [Reference Arcoya and Moreno-Mérida5] studied some particular cases of strongly singular elliptic equations, i.e. $1< \gamma < ({3r-1})/({r+1})$ and $f \in L^r(\Omega )$, $r>1$ and $f$ is strictly far away from zero on $\Omega$ and proved the power-type Sobolev regularity:

\[ u^\alpha \in H_0^1(\Omega) \ \text{for all} \ \alpha \in \left(\frac{(r+1)(1+\gamma)}{4r}, \frac{1+\gamma}{2}\right] \]

Moreover, in connection with the same problem, in [Reference Boccardo and Casado-Díaz15], Boccardo and Casado-Díaz proved the uniqueness of finite-energy solution by extending the set of admissible test functions. For similar works concerning the local elliptic or integral equations with purely singular nonlinearities, we refer to [Reference Brandolini, Chiacchio and Trombetti18, Reference Díaz, Morel and Oswald30, Reference Gomes33, Reference Stuart48, Reference Zhang and Cheng52] and for singular nonlinearities with source terms or absorption terms, we refer to [Reference Canino and Degiovanni20, Reference Coclite and Palmieri28, Reference Oliva and Petitta45, Reference Stuart49] with no intent to furnish an exhaustive list.

Turning to the nonlocal case, the singular problems have been investigated more recently and there are few works in the literature, in particular, with the fractional Laplacian $(-\Delta )^s$ and related to Lazer–Mckeena-type problem (see, for instance, [Reference Adimurthi, Giacomoni and Santra4, Reference Arora, Giacomoni, Goel and Sreenadh7, Reference Arora, Giacomoni and Warnault8, Reference Barrios, De Bonis, Medina and Peral10]). In [Reference Barrios, De Bonis, Medina and Peral10], Barrios et al. studied the solvability of the nonlocal problem in the presence of singular nonlinearities and weight function $f$. In particular, they proved the existence and regularity of solution in a very weak sense depending upon the regularity of the datum $f$ and the singular exponent $\gamma$:

\[ \left\{\begin{array}{ll} u \in H_0^{s}(\Omega) & \text{if} \ 0<\gamma \leq 1 \ \text{and} \ f \in L^r(\Omega) \ \text{with} \ r=(2_s^\ast{/}(1-\gamma))',\\ u^{{(1+\gamma)}/{2}} \in H_0^{s}(\Omega) & \text{ if } \ \gamma >1 \ \text{and} \ f \in L^1(\Omega), \end{array}\right. \]

and very recently in [Reference Youssfi and Mahmoud51], Youssfi and Mahmoud extended the result of [Reference Arcoya and Moreno-Mérida5] to the fractional Laplacian and established the following:

\begin{align*} & u \in W_0^{1,{(Nr(1+\gamma))}/{(N-rs(1-\gamma))}}(\Omega) \ \text{if} \ 0<\gamma <1 \ \text{and} \ f\\ & \quad\in L^r(\Omega) \ \text{with} \ r \in [1,(2_s^\ast{/}(1-\gamma))'), \end{align*}

and for a non-negative datum $f \in L^r(\Omega )$, $r>1$ and $\gamma >1$:

\[ u^\alpha \in H_0^s(\Omega) \ \text{for all}\ \alpha \in \left(\max\left(\frac{1}{2},\frac{sr(1+\gamma)-r+1}{2rs}\right), \frac{1+\gamma}{2}\right]. \]

In the case of $f \sim \delta ^{-\zeta }$ for some $\zeta \in [0,2s)$, Adimurthi et al. [Reference Adimurthi, Giacomoni and Santra4] and Arora et al. [Reference Arora, Giacomoni, Goel and Sreenadh7] studied the same problem with singular nonlinearities in case of $N >2s$ and $N=2s$, respectively, and discussed the existence and uniqueness results of the classical solutions with respect to the singular parameters. Moreover, using the integral representation via the Green function and maximum principle, they proved the sharp boundary behaviour of the weak solution. For further issues on nonlocal and nonlinear singular problems, the interested reader can consult to the bibliographic references in [Reference Arora6, Reference Arora, Giacomoni and Warnault8, Reference Canino, Montoro, Sciunzi and Squassina21, Reference Canino, Sciunzi and Trombetta23, Reference Giacomoni, Mukherjee and Sreenadh38].

Notations: Throughout the paper, we assume that $\Omega \subset \mathbb {R}^N$ ($N \geq 2$) is a bounded domain with $C^{1,1}$ boundary. Set $\delta (x) := \operatorname {dist}(x, \partial \Omega )$ and $\mathcal {D}_{\Omega }=\operatorname {diam}(\Omega ).$ For $i \in \mathbb {N}$, we denote by $C_i, c_i, d_i$ positive constants that may vary from line to line. If necessary, we will write $C = C(a, b)$ to emphasize the dependence of $C$ on $a, b.$ For a number $q \in (1, \infty )$, we denote by $q'$ the conjugate exponent of $q$, namely $q'={q}/{(q-1)}$. For two functions $f, g$, we write $f \lesssim g$ or $f \gtrsim g$ if there exists a constant $C > 0$ such that $f \leq Cg$ or $f \geq Cg$. We write $f \sim g$ if $f \lesssim g$ and $g \gtrsim f$. For $\eta >0$, $\chi _{\Omega _\eta }$ denotes the characteristic function of $\Omega _\eta := \{x \in \Omega : \delta (x) < \eta \}.$ Denote

\[ r^\sharp:= \left(\frac{2^*}{1-\gamma}\right)' \ \text{for} \ 0\leq \gamma< 1\ \text{and}\ 2^* := \frac{2N}{N-2} \]

\[ \mathcal{P}_{r, \gamma}:= \{(r, \gamma): \ \text{$1 \leq r \leq{+} \infty, \ \gamma \geq 0\ \text{and} \ (r, \gamma) \neq (1,0)$}\}, \]

and

\[ L^r_c(\Omega):= \{f \in L^r(\Omega): \operatorname{supp}(f) \Subset \Omega\}. \]

For $\gamma >0$ and $\zeta \geq 0$, we define a class of functions

\[ \mathcal{A}_\zeta(\Omega):=\left\{f:\Omega \to \mathbb{R}^+{\cup} \{0\}: f \asymp \delta^{-\zeta} \right\} \]

and for $\zeta \neq 2$, denote

\[ \mathfrak{L}^*:= \frac{\gamma + \zeta-1}{2-\zeta}. \]

1.3 Description of main results

In the present work, we derive the qualitative properties of the weak solution to a mixed-type elliptic problem for two different classes of weight functions $f$, and in both the presence or absence of singular nonlinearities, i.e. $\gamma >0$ or $\gamma =0$, respectively. In this section, we give a short description of our main results and for a detailed presentation, we refer the reader to § 2.

For the first class of weight function $f$, i.e. $f \in L^r(\Omega )$ for $1 \leq r \leq \infty$, we show:

• • Existence results: For this, we use the classical approach of regularizing the singular nonlinearities $u^{-\gamma }$ by $(u+{1}/{n})^{-\gamma }$ and derive uniform a priori estimates for the weak solution of the regularized problem. The crucial step here is to choose an appropriate test function in the energy space. By taking into account the combined interaction of the summability of the datum $f$ and the singular exponent $\gamma$, we obtain our existence results in two disjoint subsets of our admissible set $\mathcal {P}_{r, \gamma }$ with different Sobolev regularity:

  \[ u \in\left\{\begin{array}{@{}ll} W_0^{1,q}(\Omega) & \text{if} \ (r,\gamma) \in \mathcal{P}_{r, \gamma} \cap \{(r, \gamma): r \in [1, r^\sharp), 0 \leq \gamma <1\},\\ H^1_{loc}(\Omega) & \text{if}\ (r,\gamma) \in \mathcal{P}_{r, \gamma} {\setminus} \{(r, \gamma): r \in [1, r^\sharp), 0 \leq \gamma < 1\}, \end{array}\right. \]
  with $q:= ({Nr(1+\gamma )})/({N-r(1-\gamma )}).$ Here, the notion of solution in two disjoint subsets may differ due to different Sobolev regularity of the solution. Moreover, we observe that there is a kind of continuity in the summability exponents in the sense that as $r \to (r^\sharp )^-$, $q \to 2$ when $0 \leq \gamma < 1.$ These existence results for mixed-type operator also complement the results for classical Laplacian in [Reference Boccardo and Orsina16] and fractional Laplacian in [Reference Barrios, De Bonis, Medina and Peral10, Reference Youssfi and Mahmoud51].

• • Power- and exponential-type Sobolev regularity of the weak solution: Here, the term power-type Sobolev regularity means that $u^\alpha \in H_0^1(\Omega )$ for some $\alpha >0$ and the exponential-type Sobolev regularity means that there exists $\beta >0$ such that

  \[ \exp(\beta u)-1 \in H_0^1(\Omega)\ \text{when} \ 0 \leq \gamma \leq 1, \]
  and for $\tau \geq \gamma$:
  \[ \left(\exp(\beta u)-1\right)^\tau \in H_0^1(\Omega) \ \text{when} \ \gamma >1. \]
  For this, we prove two types of power-type Sobolev regularity result, depending upon the value of $\alpha,$ one with the help of an appropriate choice of test functions when $\alpha$ is large and the second by using the lower-boundary behaviour of the approximating solution when $\alpha$ is small.

  Type 1: When $1 \leq r < {N}/{2}$, $\gamma \geq 0$, we show that

  \[ u^{\alpha} \in H_0^1(\Omega) \ \text{for any}\ \alpha \in \left[\frac{\gamma+1}{2}, \frac{\mathfrak{S}_r+1}{2}\right] \ \text{where}\ \mathfrak{S}_r:=\frac{N(r-1) + \gamma r(N-2)}{N-2r}. \]
  We notice that as $r \to ({N}/{2})^-$, $\mathfrak {S}_r \to \infty$ and so it is natural to expect the exponential-type Sobolev regularity when $r={N}/{2}.$ In this regard, for $r= {N}/{2}$, we prove the exponential-type Sobolev regularity in the sense mentioned above. The first step in establishing such regularity results is to find an appropriate test function in the energy space to handle the singular nonlinearities, and the second step is to derive uniform a priori estimates for the approximating sequence with the same power-type Sobolev regularity and then pass to the limits. This type of Sobolev regularity result is even new for the classical Laplacian and fractional Laplacian singular and nonsingular problems.

  Type 2: In this case, to handle the singular nonlinearities, we exploit the boundary behaviour of the weak solution in deriving uniform a priori estimates. Precisely, we show that for any $r>1$ and $\gamma >0$:

  \[ u^\alpha \in H_0^1(\Omega)\ \text{for any}\ \alpha \in \left\{\begin{array}{@{}ll} \displaystyle\left(\dfrac{1}{2},\dfrac{\gamma+1}{2}\right] & \text{if}\ \gamma+\dfrac{1}{r} < 1,\\ \left(\dfrac{r\gamma+ 1}{2r},\dfrac{\gamma+1}{2}\right] & \text{if}\ \gamma+\dfrac{1}{r} \geq 1. \end{array}\right. \]
  In addition to that, we also have shown that if $\alpha \leq {1}/{2}$, then $u^\alpha \not \in H_0^1(\Omega )$ if $\gamma +{1}/{r} < 1$, which highlight the optimality of this result.

• • Continuity with respect to datum: As an application of the type 1 power-type Sobolev regularity results, we show that for any $1 \leq r < {N}/{2}$ and $\mathfrak {S} \in [\gamma, \mathfrak {S}_r]$ and for minimal weak solution $u$ and $v$ with respect to datum $f$ and $g$ respectively, the following inequality holds:

  \[ \|\nabla |u-v|^{{(\mathfrak{S}+1)}/{2}}\|^2_{L^2(\Omega)} \leq C \|f-g\|_{L^r(\Omega)}^{{(r(N-2))}/{(N-2r)}}. \]
  The above inequality also implies the continuity of the solution with respect to the given datum and comparison estimates.

For the second class of weight function $f \in \mathcal {A}_\zeta (\Omega )$ for $\zeta \geq 0$, we show:

• • Existence results: For this, we have followed the same classical method of regularizing the singular problem, but here to prove the uniform a priori estimates of the approximating sequence and handle singular nonlinearities, we cannot use the same approach of exploiting Lebesgue summability of the datum $f$, since for $\zeta \geq 1$, $f \not \in L^r(\Omega )$ for any $r \geq 1$. To resolve this issue, we prove new boundary estimates of the approximating sequence by using the lower- and upper-bound estimates of Green's kernel associated with the mixed operator. By considering the interplay of both singular exponents $\gamma >0$ and $\zeta \in [0,2)$, we obtain the following existence results with different Sobolev regularity:

  \[ u \in\left\{\begin{array}{@{}ll} H_0^{1}(\Omega) & \text{ if } \ \gamma+ \zeta \leq 1,\\ H^1_{loc}(\Omega) & \text{ if } \ \gamma+ \zeta > 1. \end{array}\right. \]

• • Optimal boundary behaviour: To prove this, we study the action of Green's operator on the inverse of the distance function perturbed with logarithmic nonlinearity. Using the lower- and upper-bound estimates of Green's kernel [Reference Chen, Kim, Song and Vondraček26] and borrowing some techniques from [Reference Abatangelo, Gómez-Castro and Vázquez2], we show:

  \[ \left\{\begin{array}{@{}ll} u \asymp \delta & \text{if}\ \zeta+\gamma <1,\\ \displaystyle u \asymp \delta \ln^{{1}/{(2-\zeta)}}\left(\dfrac{\mathcal{D}_\Omega}{\delta}\right) & \text{if}\ \zeta+\gamma=1 ,\\ u \asymp \delta^{{(2-\zeta)}/{(\gamma+1)}} & \text{if}\ \zeta+\gamma >1. \end{array}\right. \]

• • Optimal power-type Sobolev regularity and nonexistence results: As an application of the above optimal-boundary behaviour, we prove:

  \[ u^{{(\mathfrak{L}+1)}/{2}}\ \text{belongs to}\ H_0^1(\Omega) \quad \text{if and only if} \quad \mathfrak{L} > \left\{ \begin{array}{@{}ll} 0 & \text{ if } \zeta+ \gamma \leq 1, \\ \mathfrak{L}^* & \text{ if }\ \zeta+ \gamma > 1. \end{array}\right. \]
  and for $\zeta \geq 2$ and $\gamma >0$, nonexistence results is established. These results for mixed-type operator in this class complement the results for fractional Laplacian in [Reference Adimurthi, Giacomoni and Santra4, Reference Arora, Giacomoni, Goel and Sreenadh7].

The article is organized as follows: In § 2, we provide a detailed statement of the main results of this work. In § 3, we prove some preliminary results for the approximated problem, which will be used in the rest of the work. Section 4 is devoted to deriving uniform a priori estimates, Green's function estimates and boundary behaviour of the solution of the approximated problem. In § 5, we provide the proof of our main results. At the end, we provide a short appendix.