Existence of positive weak solutions for a nonlocal singular elliptic system

1.   Introduction and statement of the main results

Singular elliptic problems of the form

with $g$ such that $\lim_{s\rightarrow0^{+}}g\left(x, s\right) = \infty,$ have been extensively studied in the literature. Starting with the pioneering works ^[9,10,17], a vast amount of works was devoted to these problems, see for instance, ^{[2,5,8,12,13,14,16,19,20,25,26,27,30]}, and ^[37].

In particular, ^[26] gives an existence result for classical solutions to problem (1.1), in the case when $g\left(., u\right) = ad_{\Omega}^{-\gamma}u^{-\beta},$ with $0\leq a\in C^{\sigma }\left(\overline{\Omega}\right)$ for some $\sigma\in\left(0, 1\right),$ $\beta > 0$ and $\gamma < 2;$ and ^[13] gives an existence result for very weak solutions of the same problem. Notice that, in this case, $g\left(x, s\right)$ becomes singular at $s = 0$ and also at $x\in\partial\Omega.$ Let us mention also that existence and uniqueness results for singular problems involving the $p$-laplacian operator on exterior domains were recently obtained in ^[6].

The existence of positive solutions of singular elliptic systems is addressed (in the local case), in ^[22], ^[29], and ^[1]. In ^[22] and ^[29] the results are obtained via the sub-supersolutions method, while in ^[1] (where appear also multiplicity results), the methods are variational and topological.

A systematic study of local singular elliptic problems, as well as additional references, can be found in ^[21,33]. For a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators, we refer the reader to the reference ^[28].

Concerning nonlocal elliptic problems, let us mention that in ^[32], existence and multiplicity results were obtained for some singular elliptic problems driven by fractional powers of the p-Laplacian operator. In ^[11], global bifurcation problems for the fractional p-Laplacian were studied and, in ^[3], existence and multiplicity results were obtained for singular bifurcation problems of the form $\left(-\Delta\right) ^{s}u = f\left(x\right) u^{-\beta}+\lambda u^{p}$ in $\Omega,$ $u = 0$ in $\mathbb{R}^{n}\setminus\Omega,$ $u > 0$ in $\Omega,$ in the case where $\Omega$ is a bounded and regular enough domain in $\mathbb{R}^{n},$ $s\in\left(0, 1\right),$ $n > 2s,$ $\beta > 0,$ $p > 1,$ $\lambda > 0,$ and $f$ is a nonnegative function belonging to a suitable Lebesgue space. There, it was proved the existence of at least two solutions for this problem when $\lambda$ is positive and small enough. In ^[23], a more precise existence and multiplicity result was obtained for the same problem in the case when $f\equiv1$ and the nonlinearity has critical growth at infinity, (i.e., when $p = 2_{s}^{\ast}-1,$ with $2_{s}^{\ast} = \frac{2n}{n-2s}$). In fact, in ^[23], it was proved that, under these assumptions, there exists $\Lambda > 0$ such that:

ⅰ) There exist exactly two positive solutions when $0 < \lambda < \Lambda,$

ⅱ) There exists at least one positive solution when $\lambda = \Lambda,$

ⅲ) No solution exists when $\lambda > \Lambda.$

Also, in ^[24], it was investigated the existence of positive weak solutions to problems like $\left(-\Delta\right) ^{s}u = -au^{-\beta }+\lambda h$ in $\Omega,$ $u = 0$ in $\mathbb{R}^{n}\setminus\Omega,$ $u > 0$ in $\Omega,$ in the case where $s\in\left(0, 1\right),$ $n > 2s,$ $\beta \in\left(0, 1\right),$ $\lambda > 0,$ and where $a$ and $h$ are nonnegative bounded functions with $h\not \equiv 0$.

Our aim in this work is to obtain sufficient conditions on $\beta_{1},$ $\beta_{2},$ $\gamma_{1}$ and $\gamma_{2}$ for the existence of positive weak solutions to the following problem

Here, and from now on, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{1, 1}$ boundary, $s\in\left(0, 1\right),$ $d_{\Omega}: = dist\left(., \partial\Omega\right),$ $\beta_{1}\in\left(0, 1\right),$ $\beta_{2} \in\left(0, 1\right),$ $\gamma_{1} < 2s,$ $\gamma_{2} < 2s,$ $a$ and $b$ belong to $L^{\infty}\left(\Omega\right),$ and satisfy $\inf_{\Omega}a > 0$ and $\inf_{\Omega}b > 0.$

Before stating our main results, let us recall the definition of the fractional Sobolev space $H^{s}\left(\mathbb{R}^{n}\right)$ and some well known facts related to this space. For $s\in\left(0, 1\right)$ and $n\in\mathbb{N},$ let

and for $u\in H^{s}\left(\mathbb{R}^{n}\right),$ let

and let

and for $u\in X_{0}^{s}\left(\Omega\right),$ let

With these norms, $H^{s}\left(\mathbb{R}^{n}\right)$ and $X_{0}^{s}\left(\Omega\right)$ are Hilbert spaces (see ^[36], Lemma 7), $C_{c}^{\infty}\left(\Omega\right)$ is dense in $X_{0}^{s}\left(\Omega\right)$ (see ^[18], Theorem 6). Also, $X_{0}^{s}\left(\Omega\right)$ is a closed subspace of $H^{s}\left(\mathbb{R}^{n}\right)$, and from the fractional Poincarè inequality (as stated e.g., in ^[15], Theorem 6.5; see also Remark 2.1 below), if $n > 2s$ then $\left\Vert.\right\Vert _{X_{0}^{s}\left(\Omega\right) }$ and $\left\Vert.\right\Vert _{H^{s}\left(\mathbb{R}^{n}\right) }$ are equivalent norms on $X_{0} ^{s}\left(\Omega\right).$

For $f\in L_{loc}^{1}\left(\Omega\right)$ we will write $f\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ to mean that exists a positive constant $c$ such that $\left\vert \int_{\Omega}f\varphi\right\vert \leq c\left\Vert u\right\Vert _{X_{0}^{s}\left(\Omega\right) }$ for any $\varphi\in X_{0}^{s}\left(\Omega\right).$ For $f\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ we will write $\left(\left(-\Delta\right) ^{s}\right) ^{-1}f$ for the unique weak solution $u$ (given by the Riesz theorem) of the problem

The notion of weak solution that we use in this work is the given by the following definition:

Definition 1.1. Let $s\in\left(0, 1\right),$ let $f:\Omega\rightarrow\mathbb{R}$ be a Lebesgue measurable function such that $f\varphi\in L^{1}\left(\Omega\right)$ for any $\varphi\in X_{0} ^{s}\left(\Omega\right).$ We say that $u:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a weak solution to the problem

if $u\in X_{0}^{s}\left(\Omega\right),$ $u = 0$ in $\mathbb{R}^{n} \setminus\Omega$ and, for any $\varphi\in$ $X_{0}^{s}\left(\Omega\right),$

For $u\in X_{0}^{s}\left(\Omega\right)$ and $f\in L_{loc} ^{1}\left(\Omega\right),$ we will write $\left(-\Delta\right) ^{s}u\leq f$ in $\Omega$ (respectively $\left(-\Delta\right) ^{s}u\geq f$ in $\Omega$) to mean that, for any nonnegative $\varphi\in H_{0}^{s}\left(\Omega\right),$ it hold that $f\varphi\in L^{1}\left(\Omega\right)$ and

For $u, v\in X_{0}^{s}\left(\Omega\right),$ we will write $\left(-\Delta\right) ^{s}u\leq\left(-\Delta\right) ^{s}v$ in $\Omega$ (respectively $\left(-\Delta\right) ^{s}u\geq\left(-\Delta\right) ^{s}v$ in $\Omega$)$,$ to mean that $\left(-\Delta\right) ^{s}\left(u-v\right) \leq0$ in $\Omega$ (resp. $\left(-\Delta\right) ^{s}\left(u-v\right) \geq0$ in $\Omega$).

If $f$ and $g$ are measurable real valued functions defined on $\Omega,$ we will write $f\approx g$ to mean that there exists a positive constant $c,$ such that $c^{1}f\leq g\leq cf$ $a.e.$ in $\Omega.$ We will write $f\lessapprox g$ (respectively $f\gtrapprox g$ in $\Omega$) to mean that, for some positive constant $c,$ $f\leq cg$ $a.e.$ in $\Omega$ (resp. $f\geq cg$ $a.e.$ in $\Omega$).

Also, we set $\omega_{0}: = 2diam\left(\Omega\right).$ With these notations, our main results read as follow:

Theorem 1.2. Let $\beta_{1}\in\left(0, 1\right),$ $\beta_{2}\in\left(0, 1\right),$ let $\gamma_{1} < 2s,$ $\gamma_{2} < 2s,$ and let $a$ and $b$ be functions in $L^{\infty}\left(\Omega\right)$ such that $a\approx1,$ $b\approx1.$ Assume that one of the following three conditions i) - iii) holds:

i) $\gamma_{1}+s\beta_{1} < s$ and $\gamma_{2}+s\beta_{2} < s,$

ii) $\gamma_{1}+s\beta_{1} < s$ and $\gamma_{2}+s\beta_{2} = s,$

iii) $\gamma_{1}+s\beta_{1} = s$ and $\gamma_{2}+s\beta_{2} < s.$

Then problem has a weak solution $\left(u, v\right) \in X_{0}^{s}\left(\Omega\right) \times X_{0}^{s}\left(\Omega\right)$ such that $u\approx\vartheta_{1}$ and $v\approx\vartheta_{2}$ in $\Omega,$ where

Theorem 1.3. Let $\beta_{1}\in\left(0, 1\right),$ $\beta_{2}\in\left(0, 1\right),$ let $\gamma_{1} < 2s,$ $\gamma_{2} < 2s,$ and let $a$ and $b$ be functions in $L^{\infty}\left(\Omega\right)$ such that $a\approx1,$ $b\approx1.$ Assume that $\gamma_{1}+s\beta_{1} = s$ and $\gamma_{2}+s\beta _{2} = s.$ Then problem (1.2) has a weak solution $\left(u, v\right) \in X_{0}^{s}\left(\Omega\right) \times X_{0}^{s}\left(\Omega\right)$ such that $d_{\Omega}^{s}\lessapprox u\lessapprox d_{\Omega }^{s}\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right)$ and $d_{\Omega} ^{s}\lessapprox v\lessapprox d_{\Omega}^{s}\ln\left(\frac{\omega_{0} }{d_{\Omega}}\right)$ in $\Omega.$

Theorem 1.4. Let $\beta_{1}\in\left(0, 1\right),$ $\beta_{2}\in\left(0, 1\right),$ let $\gamma_{1} < 2s,$ $\gamma_{2} < 2s,$ and let $a$ and $b$ be functions in $L^{\infty}\left(\Omega\right)$ such that $a\approx1,$ $b\approx1.$ Assume that one of the following two conditions holds:

i) $\gamma_{1}+s\beta_{1} < s$ and $s < \gamma_{2}+s\beta_{2} < \min\left\{ 2s, \frac{1}{2}+s\right\},$

ii) $s < \gamma_{1}+s\beta_{1} < \min\left\{ 2s, \frac{1}{2}+s\right\}$ and $\gamma_{2}+\beta_{2}s < s.$

Then problem (1.2) has a weak solution $\left(u, v\right) \in X_{0}^{s}\left(\Omega\right) \times X_{0}^{s}\left(\Omega\right)$ such that $u\approx\vartheta_{1}$ and $v\approx\vartheta_{2}$ in $\Omega,$ where

The article is organized as follows: In Section 2, we quote some known facts and state some preliminary results. Lemma 2.2 quotes a result from ^[7], which gives accurate two side estimates for the values of the Green operator on negative powers of the distance function $d_{\Omega}$ (where the Green operator is the associated to the fractional laplacian with homogeneous Dirichlet condition on $\mathbb{R}^{n}\setminus\Omega$). Using this result and some of its consequences, Lemmas 2.4 and 2.5 states that, if the assumptions of Theorem 1.2 (respectively of Theorem 1.4) are assumed, and if $\vartheta_{1}$ and $\vartheta_{2}$ are as given in the statement of the respective Theorem, then $d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}$ and $d_{\Omega}^{-\gamma_{2}}\vartheta_{1}^{-\beta_{2}}$ belong to $\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime},$ $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma_{1}} \vartheta_{2}^{-\beta_{1}}\right) \approx\vartheta_{1},$ and $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma_{2}} \vartheta_{1}^{-\beta_{2}}\right) \approx\vartheta_{2}$ in $\Omega.$ Similarly, using again Lemma 2.2, Lemma 2.6 states that if $\gamma+\beta s = s$ and $\vartheta: = d_{\Omega}^{s}\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right),$ then $d_{\Omega}^{-\gamma}\vartheta^{-\beta}$ belongs to $\left(X_{0} ^{s}\left(\Omega\right) \right) ^{\prime}$ and $d_{\Omega}^{s} \lessapprox\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma}\vartheta^{-\beta}\right) \lessapprox\vartheta.$

In Section 3, Lemmas 3.1 and 3.2 adapt, to our setting, the ideas of the sub-supersolution method developed, for (local) elliptic systems, in (^[29], Theorem 3.2).

In Lemma 3.1 we consider, for $\varepsilon > 0$ and under the hypothesis of either Theorem 1.2 or Theorem 1.4, the set $\mathcal{C}_{\varepsilon}: = \left\{ \left(\zeta_{1}, \zeta_{2}\right) \in L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right) :\varepsilon\vartheta_{i}\leq\zeta_{i}\leq\varepsilon^{-1}\vartheta_{i}\text{ for }i = 1, 2\right\},$ and the operator $T:\mathcal{C}_{\varepsilon }\rightarrow$ $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ defined by

and we show that $T$ is a continuous and compact map and that, for $\varepsilon$ small enough, $T\left(\mathcal{C}_{\varepsilon}\right) \subset\mathcal{C}_{\varepsilon}.$ Lemma 3.2 says that the same conclusions hold if the hypothesis of Theorem 1.3 are assumed and $\mathcal{C}_{\varepsilon}$ is defined by

Finally, Theorems 1.2, 1.3, and 1.4 are proved using the Schauder fixed point theorem combined with Lemmas 3.1 and 3.2.

2.   Preliminaries

Remark 2.1. (ⅰ) (see e.g., ^[34], Proposition 4.1 and Corollary 4.2) The following comparison principle holds: If $u, v\in X_{0} ^{s}\left(\Omega\right)$ and $\left(-\Delta\right) ^{s}u\geq\left(-\Delta\right) ^{s}v$ in $\Omega,$ then $u\geq v$ in $\Omega.$ In particular, if $v\in X_{0}^{s}\left(\Omega\right),$ $\left(-\Delta\right) ^{s} v\geq0$ in $\Omega,$ and $v\geq0$ in $\mathbb{R}^{n}\setminus\Omega,$ then $v\geq0$ in $\Omega.$

(ⅱ) (see e.g., ^[34], Lemma 7.3) Let $f:\Omega\rightarrow\mathbb{R}$ be a nonnegative and nonidentically zero measurable function such that $f\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime},$ and let $u$ be the weak solution of problem (1.3). Then $u$ satisfies, for some positive constant $c,$

(ⅲ) (see e.g., ^[35], Proposition 1.1) If $f\in L^{\infty}\left(\Omega\right)$ then the weak solution $u$ of problem (1.3) belongs to $C^{s}\left(\mathbb{R}^{n}\right).$ In particular, there exists a positive constant $c$ such that

For additional regularity resuls see, for instance, ^[4] and ^[28].

(ⅳ) (Poincarè inequality, see ^[15], Theorem 6.5) Let $s\in\left(0, 1\right),$ let $n > 2s,$ and let $2_{s}^{\ast}: = \frac{2n} {n-2s}.$ Then there exists a positive constant $C = C\left(n, s\right)$ such that, for any measurable and compactly supported function $f:\mathbb{R} ^{n}\rightarrow\mathbb{R}$,

(ⅴ) From the Hölder's inequality and the Poincarè inequality it follows that $v\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ for any $v\in L^{\left(2_{s}^{\ast}\right) ^{\prime}}\left(\Omega\right).$

(ⅵ) (Hardy inequality, see ^[32], Theorem 2.1) There exists a positive constant $c$ such that, for any $\varphi\in X_{0}^{s}\left(\Omega\right),$

(ⅶ) Let $G:\Omega\times\Omega\rightarrow\mathbb{R\cup}\left\{ \infty\right\}$ be the Green function for $\left(-\Delta\right) ^{s}$ in $\Omega,$ with homogeneous Dirichlet boundary condition on $\mathbb{R}^{n}\setminus\Omega.$ Then, for any $f\in C\left(\overline{\Omega}\right),$ the weak solution $u$ of problem (1.3) is given by $u\left(x\right) = \int_{\Omega}G\left(x, y\right) f\left(y\right) dy$ for $x\in\Omega$ and by $u\left(x\right) = 0$ for $x\in\mathbb{R}^{n}\setminus\Omega.$

Let us recall the following result of ^[7]:

Lemma 2.2. (See ^[7], Lemma 2) Let $G$ be the Green function for $\left(-\Delta\right) ^{s}$ in $\Omega,$ with homogeneous Dirichlet boundary condition on $\mathbb{R}^{n}\setminus\Omega.$ Then

As a consequence of Lemma 2.2, we have the following

Lemma 2.3. Let $\rho\in\left[0, s+\frac{1}{2}\right).$ Then $d_{\Omega}^{-\rho}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ and

Proof. Let $\varphi\in X_{0}^{s}\left(\Omega\right).$ Since $d_{\Omega}^{s-\rho }\in L^{2}\left(\Omega\right),$ the Holder and the Hardy inequalities give $\int_{\Omega}\left\vert d_{\Omega}^{-\rho}\varphi\right\vert \leq\int _{\Omega}d_{\Omega}^{s-\rho}\left\vert \frac{\varphi}{d_{\Omega}^{s} }\right\vert \leq c\left\Vert d_{\Omega}^{s-\rho}\right\Vert _{2}\left\Vert \varphi\right\Vert _{X_{0}^{s}\left(\Omega\right) }\leq c^{\prime }\left\Vert \varphi\right\Vert _{X_{0}^{s}\left(\Omega\right) }$ with $c$ and $c^{\prime}$ positive constants independent of $\varphi.$ Thus $d_{\Omega }^{-\rho}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}.$

Let $G$ be the Green function for $\left(-\Delta\right) ^{s}$ in $\Omega,$ with homogeneous Dirichlet boundary condition on $\mathbb{R} ^{n}\setminus\Omega.$ To prove (2.4) it is enough (thanks to Lemma 2.2) to show that $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\rho}\right) = \int_{\Omega}G\left(., y\right) d_{\Omega}^{-\rho}\left(y\right) dy.$ Let $\left\{ \varepsilon_{j}\right\} _{j\in N}\subset\left(0, 1\right)$ be a decreasing sequence such that $\lim_{j\rightarrow\infty}\varepsilon _{j} = 0,$ and for $j\in\mathbb{N},$ let $u_{\varepsilon_{j}}\in X_{0} ^{s}\left(\Omega\right)$ be the weak solution of the problem

Thus $u_{\varepsilon_{j}} = \int_{\Omega}G\left(., y\right) \left(d_{\Omega }\left(y\right) +\varepsilon_{j}\right) ^{-\rho}dy$ in $\Omega$ and, by Lemma 2.2, there exists a positive constant $c,$ independent of $j,$ such that $u_{\varepsilon_{j}}\leq cd_{\Omega}^{s}$ if $\rho < s,$ $u_{\varepsilon_{j}}\leq cd_{\Omega}^{s}\ln\left(\frac{\omega_{0} }{d_{\Omega}}\right)$ if $\rho = s,$ and $\ u_{\varepsilon_{j}}\leq cd_{\Omega}^{2s-\rho}$ if $s < \rho < \frac{1}{2}+s.$ In particular, there exists a positive constant $c^{\prime}$ such that $\int_{\Omega}u_{\varepsilon_{j} }d_{\Omega}^{-\rho}\leq c^{\prime}$ for all $j\in\mathbb{N}.$ Let $u\left(x\right) : = \lim_{j\rightarrow\infty}u_{\varepsilon_{j}}\left(x\right).$ By the monotone convergence theorem, $u\left(x\right) = \int_{\Omega }G\left(x, y\right) d_{\Omega}^{-\beta}\left(y\right) dy.$ Taking $u_{\varepsilon_{j}}$ as a test function in (2.5) we get

with $c^{\prime}$ independent of $j.$ For $j\in\mathbb{N},$ let $U_{\varepsilon_{j}}$ and $U$ be the functions, defined on $\mathbb{R} ^{n}\times\mathbb{R}^{n},$ by

Then $\left\{ U_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}}$ is bounded in $\mathcal{H} = L^{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n}, \frac {1}{\left\vert x-y\right\vert ^{n+2s}}dxdy\right).$ Thus, after pass to a subsequence if necessary, we can assume that $\left\{ U_{\varepsilon_{j} }\right\} _{j\in\mathbb{N}}$ is weakly convergent in $\mathcal{H}$ to some $V\in\mathcal{H}.$ Since $\left\{ U_{\varepsilon_{j}}\right\} _{j\in \mathbb{N}}$ converges pointwise to $U$ on $\mathbb{R}^{n}\times\mathbb{R} ^{n},$ we conclude that $U\in\mathcal{H}$ and that $\left\{ U_{\varepsilon _{j}}\right\} _{j\in\mathbb{N}}$ converges weakly to $U$ in $\mathcal{H}.$ Thus $u\in X_{0}^{s}\left(\Omega\right)$ and, for any $\varphi\in X_{0}^{s}\left(\Omega\right),$

Therefore $u = \left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\rho}\right)$ and so $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\rho}\right) = \int_{\Omega}G\left(x, y\right) d_{\Omega}^{-\beta}\left(y\right) dy.$

Lemma 2.4. Assume the hypothesis of Theorem 1.2 and let $\vartheta_{1}$ and $\vartheta_{2}$ be as given there. Then, in each one of the cases i) and ii) of Theorem 1.2, $d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime},$ $d_{\Omega}^{-\gamma_{2}}\vartheta _{1}^{-\beta_{2}}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime },$ $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega }^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}\right) \approx\vartheta_{1},$ and $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega} ^{-\gamma_{2}}\vartheta_{1}^{-\beta_{2}}\right) \approx\vartheta_{2}$ in $\Omega.$

Proof. When the condition i) of Theorem 1.2 holds we have $\vartheta_{1} = \vartheta_{2} = d_{\Omega}^{s},$ and the lemma follows directly from Lemma 2.3. If the condition ii) holds, then $\gamma_{1}+s\beta_{1} < s,$ $\gamma_{2}+s\beta_{2} = s,$ $\vartheta _{1} = d_{\Omega}^{s}$ and $\vartheta_{2} = d_{\Omega}^{s}\ln\left(\frac {\omega_{0}}{d_{\Omega}}\right).$ Since $\left(\ln\left(\frac{\omega _{0}}{d_{\Omega}}\right) \right) ^{-\beta_{1}}\in L^{\infty}\left(\Omega\right)$ we have $d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1} } = d_{\Omega}^{-\gamma_{1}-s\beta_{1}}\left(\ln\left(\frac{\omega_{0} }{d_{\Omega}}\right) \right) ^{-\beta_{1}}\lessapprox d_{\Omega} ^{-\gamma_{1}-s\beta_{1}}$ and so, by Lemma 2.3, $d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ and $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1} }\right) \lessapprox d_{\Omega}^{s} = \vartheta_{1}$ in $\Omega.$ Also, for $\delta > 0$ we have $\inf_{\Omega}d_{\Omega}^{-\delta}\left(\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right) \right) \theta^{-\beta_{1}} > 0,$ and so

Then, by the comparison principle of Remark 2.1 ⅰ), and by Lemma 2.3,

On the other hand, $d_{\Omega}^{-\gamma_{2}}\vartheta_{1}^{-\beta_{2} } = d_{\Omega}^{-\gamma_{2}-s\beta_{2}} = d_{\Omega}^{-s},$ and so, again by Lemma 2.3, $d_{\Omega}^{-\gamma_{2}}\vartheta_{1} ^{-\beta_{2}}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ and $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega }^{-\gamma_{2}}\vartheta_{1}^{-\beta_{2}}\right) \approx d_{\Omega}^{s} \ln\left(\frac{\omega_{0}}{d_{\Omega}}\right) = \vartheta_{2}$ in $\Omega.$

By replacing $\beta_{1},$ $\gamma_{1},$ $\vartheta_{1}$ and $\vartheta_{2}$ by $\beta_{2},$ $\gamma_{2},$ $\vartheta_{2}$ and $\vartheta_{1}$ respectively, the same argument proves the lemma in the case iii).

Lemma 2.5. Assume the hypothesis of Theorem 1.4 and let $\vartheta_{1}$ and $\vartheta_{2}$ be as given there. Then the conclusions of Lemma 2.4 remain true for $\vartheta_{1}$ and $\vartheta_{2}.$

Proof. Consider the case when the condition i) of Theorem 1.4 holds, i.e., the case when $\gamma_{1}+s\beta_{1} < s,$ $s < \gamma_{2}+s\beta _{2} < \min\left\{ 2s, \frac{1}{2}+s\right\},$ $\vartheta_{1} = d_{\Omega}^{s}$ and $\vartheta_{2} = d_{\Omega}^{2s-\gamma_{2}-s\beta_{2}}.$ Then $d_{\Omega }^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}} = d_{\Omega}^{-\gamma_{1}-\beta _{1}\left(2s-\gamma_{2}-s\beta_{2}\right) }.$ Since $0 < \gamma_{1}+\beta _{1}\left(2s-\gamma_{2}-s\beta_{2}\right) < \gamma_{1}+s\beta_{1} < s,$ Lemma 2.3 gives that $d_{\Omega}^{-\gamma_{1}}\vartheta _{2}^{-\beta_{1}}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime }$ and that $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}\right) \approx d_{\Omega }^{s} = \vartheta_{1}$ in $\Omega.$ On the other hand, $d_{\Omega} ^{-\gamma_{2}}\vartheta_{1}^{-\beta_{2}} = d_{\Omega}^{-\gamma_{2}-s\beta_{2}}$ and $s < \gamma_{2}+s\beta_{2} < \min\left\{ 2s, \frac{1}{2}+s\right\},$ and so, by Lemma 2.3,

The proof when the condition ii) holds is similar.

Lemma 2.6. Let $\vartheta: = d_{\Omega}^{s}\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right).$ If $\gamma+s\beta = s$ and $\beta > 0,$ then $d_{\Omega}^{-\gamma}\vartheta^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ and $d_{\Omega}^{s}\lessapprox\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma }\vartheta^{-\beta}\right) \lessapprox\vartheta$ in $\Omega.$

Proof. Since $\left(\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right) \right) ^{-\beta}\in L^{\infty}\left(\Omega\right),$ we have $d_{\Omega}^{-\gamma }\vartheta^{-\beta} = d_{\Omega}^{-s}\left(\ln\left(\frac{\omega_{0} }{d_{\Omega}}\right) \right) ^{-\beta}\lessapprox d_{\Omega}^{-s}.$ Then, by Lemma 2.3 and the comparison principle, $d_{\Omega }^{-\gamma}\vartheta^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}$ and $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma}\vartheta^{-\beta}\right) \lessapprox\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-s}\right) \approx d_{\Omega}^{s}\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right) = \vartheta$ in $\Omega.$ On the other hand, since $\inf_{\Omega}d_{\Omega }^{-\delta}\left(\ln\left(\frac{\omega_{0}}{d_{\Omega}}\right) \right) ^{-\beta_{1}} > 0$ for any $\delta > 0,$ we have

and then so, by Lemma 2.3 and the comparison principle, $\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\gamma}\vartheta^{-\beta}\right) \gtrapprox\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{-\left(\gamma +s\beta-\delta\right) }\right) \approx d_{\Omega}.$

3.   Proof of the main results

Lemma 3.1. Assume the hypothesis of Theorem 1.2 (respectively of Theorem 1.4), and let $\vartheta_{1}$ and $\vartheta_{2}$ be as defined there. For $\varepsilon > 0,$ let

and let $T:\mathcal{C}_{\varepsilon}\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ be defined by

Then:

1) $\mathcal{C}_{\varepsilon}$ is a closed convex set in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$

2) $T\left(\mathcal{C}_{\varepsilon}\right) \subset\mathcal{C}_{\varepsilon}$ for any $\varepsilon$ positive and small enough.

3) $T:\mathcal{C} _{\varepsilon}\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ is continuous

4) $T:\mathcal{C}_{\varepsilon }\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ is a compact map.

Proof. Clearly $\mathcal{C}_{\varepsilon}$ is a closed convex set in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ To see 2), note that, for any $\left(\zeta_{1}, \zeta_{2}\right) \in\mathcal{C}_{\varepsilon}$, $ad_{\Omega}^{-\gamma_{1}}\zeta_{2}^{-\beta_{1}}\approx d_{\Omega} ^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}$ and $bd_{\Omega}^{-\gamma_{2}} \zeta_{1}^{-\beta_{2}}\approx d_{\Omega}^{-\gamma_{2}}\vartheta_{1} ^{-\beta_{2}}$ and then, when the hypothesis of Theorem 1.2 hold (respectively of Theorem 1.4 hold), Lemma 2.5 (resp. Lemma 2.4) gives that $T$ is well defined on $\mathcal{C}_{\varepsilon}$ and that $T\left(\mathcal{C}_{\varepsilon}\right) \subset X_{0}^{s}\left(\Omega\right) \times X_{0}^{s}\left(\Omega\right) \subset L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$

To see 2) observe that, for any $\left(\zeta_{1}, \zeta_{2}\right) \in\mathcal{C}_{\varepsilon},$

and then, by the comparison principle and by Lemmas 2.5 and 2.4, there exist positive constants $c_{1}$ and $c_{2},$ both independent of $\varepsilon,$ $\zeta_{1}$ and $\zeta_{2},$ such that

and, similarly,

Since $0 < \beta_{1} < 1$ and $0 < \beta_{2} < 1,$ for $\varepsilon$ small enough we have

Thus, for such a $\varepsilon,$ $T\left(\mathcal{C}_{\varepsilon}\right) \subset\mathcal{C}_{\varepsilon}.$

To prove that $T:\mathcal{C} _{\varepsilon}\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ is continuous, consider an arbitrary $\left(\zeta_{1}, \zeta_{2}\right) \in\mathcal{C}_{\varepsilon},$ and a sequence $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}} \subset\mathcal{C}_{\varepsilon}$ that converges to $\left(\zeta_{1}, \zeta_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ After pass to a subsequence we can assume that $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $\left(\zeta_{1}, \zeta_{2}\right)$ $a.e.$ in $\Omega.$ Since $0\leq ad_{\Omega}^{–\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\leq\sup_{\Omega}\left(a\right) \varepsilon^{-\beta_{1}}d_{\Omega}^{-\gamma_{1}}\vartheta _{2}^{-\beta_{1}}$ and since, by Lemmas 2.5 and 2.4, $d_{\Omega}^{-\gamma_{1}}\vartheta _{2}^{-\beta_{1}}\in\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime },$ it follows that $\left\{ ad_{\Omega}^{-\gamma_{1}}\zeta_{2, j}^{-\beta _{1}}\right\} _{j\in\mathbb{N}}$ is bounded in $\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime}.$ Similarly, $\left\{ ad_{\Omega} ^{-\gamma_{2}}\zeta_{1, j}^{-\beta_{2}}\right\} _{j\in\mathbb{N}}$ is bounded in $\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime}.$ Let $\left(\xi_{1, j}, \xi_{2, j}\right) : = T\left(\zeta_{1, j}, \zeta_{2, j}\right).$ Then $\left\{ \left(\xi_{1, j}, \xi_{2, j}\right) \right\} _{j\in\mathbb{N}}$ is bounded in $X_{0}^{1}\left(\Omega\right) \times X_{0}^{1}\left(\Omega\right).$ After pass to a further subsequence if necessary, we can assume that there exists $\left(\xi_{1}, \xi_{2}\right) \in X_{0}^{1}\left(\Omega\right) \times X_{0}^{1}\left(\Omega\right)$ such that $\left\{ \left(\xi_{1, j}, \xi_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $\left(\xi_{1}, \xi_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right),$ $\left\{ \left(\xi_{1, j}, \xi_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges $\left(\xi_{1}, \xi_{2}\right)$ $a.e.$ in $\Omega,$ and $\left\{ \xi_{1, j}, \xi_{2, j}\right\} _{j\in \mathbb{N}}$ converges weakly to $\left(\xi_{1}, \xi_{2}\right)$ in $X_{0}^{1}\left(\Omega\right) \times X_{0}^{1}\left(\Omega\right).$ Let $\varphi\in X_{0}^{1}\left(\Omega\right).$ We have, for each $j,$

Now, $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}\subset\mathcal{C}_{\varepsilon}$ and so $\left\vert ad_{\Omega}^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\varphi\right\vert \leq\varepsilon^{-\beta_{1}}\left\Vert a\right\Vert _{\infty}\left\vert d_{\Omega}^{-\gamma_{1}}\vartheta_{2}^{-\beta_{1}}\varphi\right\vert \in L^{1}\left(\Omega\right).$ Therefore, by the Lebesgue dominated convergence theorem,

Similarly,

Then

and so $\left(\xi_{1}, \xi_{2}\right) = T\left(\zeta_{1}, \zeta_{2}\right).$ Then $\left\{ T\left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $T\left(\zeta_{1}, \zeta_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ Thus, for any sequence $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}\subset\mathcal{C}_{\varepsilon}$ that converges to $\left(\zeta_{1}, \zeta_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right),$ we have found a subsequence $\left\{ \left(\zeta_{1, j_{k}}, \zeta_{2, j_{k}}\right) \right\} _{k\in\mathbb{N}}$ such that $\left\{ T\left(\zeta_{1, j_{k}}, \zeta_{2, j_{k}}\right) \right\} _{k\in\mathbb{N}}$ converges to $T\left(\zeta_{1}, \zeta_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ Therefore $T$ is continuous.

To see that $T:\mathcal{C}_{\varepsilon}\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ is a compact map, consider a bounded sequence $\left\{ \left(\zeta_{1, j}, \zeta _{2, j}\right) \right\} _{j\in\mathbb{N}}\subset\mathcal{C}_{\varepsilon}.$ Then $0\leq ad_{\Omega}^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\leq\sup_{\Omega }\left(a\right) \varepsilon^{-\beta_{1}}d_{\Omega}^{-\gamma_{1}} \vartheta_{2}^{-\beta_{1}}$ and so, as above, $\left\{ ad_{\Omega} ^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\right\} _{j\in\mathbb{N}}$ is bounded in $\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime}.$ Then $\left\{ \left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(ad_{\Omega}^{-\gamma_{1}} \zeta_{2, j}^{-\beta_{1}}\right) \right\} _{j\in\mathbb{N}}$ is bounded in $X_{0}^{1}\left(\Omega\right).$ Thus there exists a subsequence $\left\{ \left(\zeta_{1, j_{k}}, \zeta_{2, j_{k}}\right) \right\} _{k\in\mathbb{N}}$ such that $\left(\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(ad_{\Omega} ^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\right) \right\} _{j\in\mathbb{N}}$ converges in $L^{2}\left(\Omega\right).$ Since $0\leq ad_{\Omega} ^{-\gamma_{2}}\zeta_{1, j_{k}}^{-\beta_{2}}\leq\sup_{\Omega}\left(b\right) \varepsilon^{-\beta_{2}}d_{\Omega}^{-\gamma_{1}}\vartheta_{1}^{-\beta_{2}}$ we can repeat the above argument to obtain (after pass to a further subsequence if necessary) that $\left\{ \left(\left(-\Delta\right)^{s}\right) ^{-1}\left(ad_{\Omega }^{-\gamma_{2}}\zeta_{1, j_{k}}^{-\beta_{2}}\right) \right\} _{k\in \mathbb{N}}$ converges in $L^{2}\left(\Omega\right).$ Therefore $\left\{ T\left(\zeta_{1, j_{k}}, \zeta_{2, j_{k}}\right) \right\} _{j\in\mathbb{N}}$ converges in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$

Lemma 3.2. Assume the hypothesis of Theorem 1.3, and let $\vartheta$ be as given in Lemma 2.6. For $\varepsilon > 0,$ let

and let $T:\mathcal{C}_{\varepsilon}\rightarrow L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ be defined by (3.1). Then, for $\varepsilon$ positive and small enough, the conclusions 1)-4) of Lemma 3.1 hold for $\mathcal{C}_{\varepsilon}$ and $T.$

Proof. The proof of the lemma is similar to the proof of Lemma 3.1. Clearly 1) holds. To prove 2), consider an arbitrary $\left(\zeta_{1}, \zeta_{2}\right) \in\mathcal{C}_{\varepsilon}.$ Since $0\leq ad_{\Omega}^{-\gamma_{1}}\zeta_{2}^{-\beta_{1}}\leq \varepsilon^{-\beta_{1}}\sup_{\Omega}\left(a\right) d_{\Omega}^{-s}$ and $0\leq bd_{\Omega}^{-\gamma_{2}}\zeta_{1}^{-\beta_{2}}\leq\varepsilon ^{-\beta_{1}}\sup_{\Omega}\left(b\right) d_{\Omega}^{-s}$ $a.e.$ in $\Omega,$ we have that $ad_{\Omega}^{-\gamma_{1}}\zeta_{2}^{-\beta_{1}}$ and $bd_{\Omega}^{-\gamma_{2}}\zeta_{1}^{-\beta_{2}}$ belong to $\left(X_{0} ^{s}\left(\Omega\right) \right) ^{\prime}.$ Then $T\left(\zeta_{1}, \zeta_{2}\right)$ is well defined and belongs to $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ Also, $\varepsilon ^{\beta_{1}}\inf_{\Omega}\left(a\right) d_{\Omega}^{-\gamma_{1}} \vartheta^{-\beta_{1}}\leq ad_{\Omega}^{-\gamma_{1}}\zeta_{2}^{-\beta_{1}} \leq\varepsilon^{-\beta_{1}}\sup_{\Omega}\left(a\right) d_{\Omega}^{-s}$ in $\Omega,$ and $\varepsilon^{\beta_{2}}\inf_{\Omega}\left(b\right) d_{\Omega}^{-\gamma_{2}}\vartheta^{-\beta_{2}}\leq ad_{\Omega}^{-\gamma_{2} }\zeta_{1}^{-\beta_{2}}\leq\varepsilon^{-\beta_{2}}\sup_{\Omega}\left(b\right) d_{\Omega}^{-s}$ in $\Omega.$ Then, by the comparison principle and Lemma 2.6, there exist positive constants $c_{1}$ and $c_{2},$ both independent of $\varepsilon,$ $\zeta_{1},$ and $\zeta_{2},$ such that

and so, as in Lemma 3.1, (3.2) holds for $\varepsilon$ small enough. Then, for such a $\varepsilon,$ $T\left(\mathcal{C}_{\varepsilon}\right) \subset\mathcal{C}_{\varepsilon}.$

To prove 3), consider an arbitrary $\left(\zeta_{1}, \zeta _{2}\right) \in\mathcal{C}_{\varepsilon},$ and a sequence $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}\subset \mathcal{C}_{\varepsilon}$ that converges to $\left(\zeta_{1}, \zeta _{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ After pass to a subsequence we can assume that $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $\left(\zeta_{1}, \zeta_{2}\right)$ $a.e.$ in $\Omega.$ Since $0\leq ad_{\Omega}^{–\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\leq\sup_{\Omega}\left(a\right) \varepsilon^{-\beta_{1}}d_{\Omega}^{-s},$ and $0\leq bd_{\Omega }^{–\gamma_{2}}\zeta_{1, j}^{-\beta_{2}}\leq\sup_{\Omega}\left(b\right) \varepsilon^{-\beta_{1}}d_{\Omega}^{-s},$ and taking into account that $d_{\Omega}^{-s}\in\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime },$ it follows that $\left\{ ad_{\Omega}^{-\gamma_{1}}\zeta_{2, j}^{-\beta _{1}}\right\} _{j\in\mathbb{N}}$ and $\left\{ bd_{\Omega}^{-\gamma_{2}} \zeta_{1, j}^{-\beta_{2}}\right\} _{j\in\mathbb{N}}$ are bounded in $\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime}.$ Let $\left(\xi _{1, j}, \xi_{2, j}\right) : = T\left(\zeta_{1, j}, \zeta_{2, j}\right).$ Then $\left\{ \left(\xi_{1, j}, \xi_{2, j}\right) \right\} _{j\in\mathbb{N}}$ is bounded in $X_{0}^{1}\left(\Omega\right) \times X_{0}^{1}\left(\Omega\right).$ Therefore, after pass to a further subsequence if necessary, we can assume that, for some $\left(\xi_{1}, \xi_{2}\right) \in X_{0} ^{1}\left(\Omega\right) \times X_{0}^{1}\left(\Omega\right),$ $\left\{ \left(\xi_{1, j}, \xi_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $\left(\xi_{1}, \xi_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right)$ and $a.e.$ in $\Omega;$ and that $\left\{ \xi_{1, j}, \xi_{2, j}\right\} _{j\in\mathbb{N}}$ converges weakly to $\left(\xi_{1}, \xi_{2}\right)$ in $X_{0}^{1}\left(\Omega\right) \times X_{0} ^{1}\left(\Omega\right).$ Let $\varphi\in X_{0}^{1}\left(\Omega\right).$ Since $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}\subset\mathcal{C}_{\varepsilon}$ and $\gamma_{1}+s\beta _{1} = s,$ we have $\left\vert ad_{\Omega}^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1} }\varphi\right\vert \leq\varepsilon^{-\beta_{1}}\left\Vert a\right\Vert _{\infty}\left\vert d_{\Omega}^{-s}\varphi\right\vert$ and by the Hardy inequality, $\left\vert d_{\Omega}^{-s}\varphi\right\vert \in L^{1}\left(\Omega\right).$ Then, from (3.3) and (3.4), the Lebesgue dominated convergence theorem gives (3.5). (3.6) is obtained similarly. Then (3.7) and (3.8) hold. Thus $\left(\xi _{1}, \xi_{2}\right) = T\left(\zeta_{1}, \zeta_{2}\right)$ and so $\left\{ T\left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}$ converges to $T\left(\zeta_{1}, \zeta_{2}\right)$ in $L^{2}\left(\Omega\right) \times L^{2}\left(\Omega\right).$ Then, as in the proof of Lemma 3.1, the conclusion that $T$ is continuous is reached.

To see 4), consider a bounded sequence $\left\{ \left(\zeta_{1, j}, \zeta_{2, j}\right) \right\} _{j\in\mathbb{N}}\subset \mathcal{C}_{\varepsilon}.$ We have $0\leq ad_{\Omega}^{-\gamma_{1}} \zeta_{2, j}^{-\beta_{1}}\leq\sup_{\Omega}\left(a\right) \varepsilon ^{-\beta_{1}}d_{\Omega}^{-s}$ and $0\leq bd_{\Omega}^{-\gamma_{2}} \zeta_{1, j_{k}}^{-\beta_{2}}\leq\sup_{\Omega}\left(b\right) \varepsilon ^{-\beta_{2}}d_{\Omega}^{-s}$ in $\Omega,$ and so $\left\{ ad_{\Omega }^{-\gamma_{1}}\zeta_{2, j}^{-\beta_{1}}\right\} _{j\in\mathbb{N}}$ and $\left\{ bd_{\Omega}^{-\gamma_{2}}\zeta_{1, j}^{-\beta_{2}}\right\} _{j\in\mathbb{N}}$ are bounded in $\left(X_{0}^{1}\left(\Omega\right) \right) ^{\prime}.$ Now 4) follows as in the proof of Lemma 3.1

Proof of Theorems 1.2, 1.3, and 1.4. Theorems 1.2, 1.3 and 1.4 follow from the Schauder fixed point theorem (as stated e.g., in ^[31], Theorem 3.2.20), combined with Lemma 3.1 in the case of Theorems 1.2 and 1.4; and with Lemma 3.2 in the case of Theorem 1.3.

Conflict of interest

The author declare no conflicts of interest in this paper.