The Hardy–Schrödinger operator with interior singularity: the remaining cases

Abstract

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in ${\mathbb {R}}^n$ ($n\ge 3$) having the singularity 0 in its interior. Here $\gamma <\frac{(n-2)^2}{4}$, $0\le s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\le \lambda <\lambda _1(L_\gamma )$, the latter being the first eigenvalue of the Hardy–Schrödinger operator $L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}$. There is a threshold $\lambda ^*(\gamma , \Omega ) \ge 0$ beyond which the minimal energy is achieved, but below which, it is not. It is well known that $\lambda ^*(\Omega )=0$ in higher dimensions, for example if $0\le \gamma \le \frac{(n-2)^2}{4}-1$. Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when $\frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}$, to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of $\Omega$ and $\gamma$. If either $s>0$ or if $\gamma > 0$, i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of $\Omega$, a notion that we introduce herein. On the other hand, and just like the case when $\gamma =s=0$ studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), $n=3$ is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when $s=0$, $\gamma \le 0$.

Appendices

Appendix A: Green’s function for $-\Delta -\gamma |x|^{-2}-h(x)$ on a bounded domain

Theorem 6

Let $\Omega$ be a smooth bounded domain of ${\mathbb {R}}^n$ such that $0\in \Omega$ is an interior point. We fix $\gamma <\frac{(n-2)^2}{4}$. We let $h\in C^{0,\theta }(\overline{\Omega })$ be such that $-\Delta -\gamma |x|^{-2}-h$ is coercive. Then there exists $G: (\Omega {\setminus }\{0\})^2\setminus \{(x,x)/\, x\in \Omega {\setminus }\{0\}\}\rightarrow {\mathbb {R}}$ such that for all $p\in \Omega {\setminus }\{0\}$,

• (i) For any $p\in \Omega {\setminus }\{0\}$, $G_p:=G(p,\cdot )\in H_{1}^2(\Omega {\setminus } B_\delta (p))$ for all $\delta >0$, $G_p\in C^{2,\theta }(\overline{\Omega }{\setminus }\{0,p\})$

• (ii) For all $f\in L^{\frac{2n}{n+2}}(\Omega )\cap L^p_{loc}(\overline{\Omega }-\{0\})$, $p>n/2$, and all $\varphi \in H_0^1(\Omega )$ such that

  $$\begin{aligned} -\Delta \varphi -\left( \frac{\gamma }{|x|^2}+h(x)\right) \varphi =f\hbox { in }\Omega ;\quad \varphi _{|\partial \Omega }=0, \end{aligned}$$

  then we have that

  $$\begin{aligned} \varphi (p)=\int _{\Omega }G(p,x)f(x)\,{ dx} \end{aligned}$$

  (127)

  In addition, $G>0$ is unique and

• (iii) For all $p\in \Omega {\setminus }\{0\}$, there exists $c_0(p)>0$ such that

  $$\begin{aligned} G_p(x)\sim _{x\rightarrow 0} \frac{c_0(p)}{|x|^{\beta _-(\gamma )}}\quad \hbox {and}\quad G_p(x)\sim _{x\rightarrow p}\frac{1}{(n-2)\omega _{n-1}|x-p|^{n-2}} \end{aligned}$$

  (128)

• (iv) There exists $c>0$ such that

  $$\begin{aligned} 0< G_p(x)\le c \left( \frac{\max \{|p|,|x|\}}{\min \{|p|,|x|\}}\right) ^{\beta _-(\gamma )}|x-p|^{2-n}\quad \hbox {for}\quad x\in \Omega -\{0,p\}. \end{aligned}$$

  (129)

• (v) For all $\omega \Subset \Omega$, there exists $c(\omega )>0$ such that

  $$\begin{aligned} c(\omega ) \left( \frac{\max \{|p|,|x|\}}{\min \{|p|,|x|\}}\right) ^{\beta _-(\gamma )}|x-p|^{2-n}\le G_p(x)\quad \hbox {for all}~p,x\in \omega {\setminus }\{0\}. \end{aligned}$$

  (130)

Proof

Fix $\delta _0>0$ such that $B_{\delta _0}(0)\subset \Omega$. We let $\eta _\epsilon (x):={\tilde{\eta }}(\epsilon ^{-1}|x|)$ for all $x\in {\mathbb {R}}^n$ and $\epsilon >0$, where ${\tilde{\eta }}\in C^\infty ({\mathbb {R}})$ is nondecreasing and such that ${\tilde{\eta }}(t)=0$ for $t<1$ and ${\tilde{\eta }}(t)=1$ for $t>1$. Set

$$\begin{aligned} L_\epsilon :=-\Delta -\left( \frac{\gamma \eta _\epsilon }{|x|^2}+h(x)\right) . \end{aligned}$$

It follows from Lemma 1 and the coercivity of $-\Delta -\left( \gamma |x|^{-2}+h\right)$ that there exists $\epsilon _0>0$ and $c>0$ such that such that for all $\varphi \in H_0^1(\Omega )$ and $\epsilon \in (0,\epsilon _0)$,

$$\begin{aligned} \int _\Omega \left( |\nabla \varphi |^2-\left( \frac{\gamma \eta _\epsilon }{|x|^2}+h(x)\right) \varphi ^2\right) \, { dx}\ge c\int _\Omega \varphi ^2\,{ dx}. \end{aligned}$$

As a consequence, there exists $c>0$ such that for all $\varphi \in H_0^1(\Omega )$ and $\epsilon \in (0,\epsilon _0)$,

$$\begin{aligned} \int _\Omega \left( |\nabla \varphi |^2-\left( \frac{\gamma \eta _\epsilon }{|x|^2}+h(x)\right) \varphi ^2\right) \, { dx}\ge c\Vert \varphi \Vert _{D^{1,2}}^2. \end{aligned}$$

(131)

Let $G_\epsilon >0$ be the Green’s function of $-\Delta -\left( \gamma \eta _\epsilon |x|^{-2}+h\right)$ on $\Omega$ with Dirichlet boundary condition. The existence follows from the coercivity and the $C^{0,\theta }$ regularity of the potential for any $\epsilon >0$.

Step 1: Integral bounds for $G_\epsilon$. We claim that for all $\delta >0$ and $1<q<\frac{n}{n-2}$ and $\delta '\in (0,\delta )$, there exists $C(\delta ,q)>0$ and $C(\delta ,\delta ')>0$ such that

$$\begin{aligned} \Vert G_\epsilon (x,\cdot )\Vert _{L^q(\Omega )}\le C(\delta ,q)\quad \hbox {and}\quad \Vert G_\epsilon (x,\cdot )\Vert _{L^{\frac{2n}{n-2}}(\Omega {\setminus }B_{\delta '}(x))}\le C(\delta ,\delta ') \end{aligned}$$

(132)

for all $x\in \Omega$, $|x|>\delta$.

Indeed, fix $f\in C^\infty _c(\Omega )$ and let $\varphi _\epsilon \in C^{2,\theta }(\overline{\Omega })$ be the solution to the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} L_\epsilon \varphi _\epsilon =-\Delta \varphi _\epsilon -\left( \frac{\gamma \eta _\epsilon }{|x|^2}+h(x)\right) \varphi _\epsilon = f &{}\hbox {in}~\Omega \\ \varphi _\epsilon =0&{}\hbox {on}~\partial \Omega \end{array}\right. \end{aligned}$$

(133)

Multiplying the equation by $\varphi _\epsilon$, integrating by parts on $\Omega$, using (131) and Hölder’s inequality, we get that

$$\begin{aligned} \int _\Omega |\nabla \varphi |^2\,{ dx}\le C\Vert f\Vert _{\frac{2n}{n+2}}\Vert \varphi _\epsilon \Vert _{\frac{2n}{n-2}} \end{aligned}$$

where $C>0$ is independent of $\epsilon$, f and $\varphi _\epsilon$. The Sobolev inequality $\Vert \varphi \Vert _{\frac{2n}{n-2}}\le C\Vert \nabla \varphi \Vert _2$ for $\varphi \in H_0^1(\Omega )$ then yields

$$\begin{aligned} \Vert \varphi _\epsilon \Vert _{\frac{2n}{n-2}}\le C\Vert f\Vert _{\frac{2n}{n+2}} \end{aligned}$$

where $C>0$ is independent of $\epsilon$, f and $\varphi _\epsilon$.

Fix $p>n/2$ and $\delta \in (0,\delta _0)$ and $\delta _1,\delta _2>0$ such that $\delta _1+\delta _2<\delta$, and $x\in \Omega$ such that $|x|>\delta$. It follows from standard elliptic theory that

$$\begin{aligned} |\varphi _\epsilon (x)|\le & {} \Vert \varphi \Vert _{C^0(B_{\delta _1}(x))}\\\le & {} C\left( \Vert \varphi _\epsilon \Vert _{L^{2^{\star }}(B_{\delta _1+\delta _2}(x))}+\Vert f\Vert _{L^{p}(B_{\delta _1+\delta _2}(x))}\right) \\\le & {} C\left( \Vert f\Vert _{L^{\frac{2n}{n+2}}(\Omega )}+\Vert f\Vert _{L^{p}(B_{\delta _1+\delta _2}(x))}\right) \end{aligned}$$

where $C>0$ depends on $p,\delta ,\delta _1,\delta _2$, $\gamma$ and $\Vert h\Vert _\infty$. Therefore, Green’s representation formula yields

$$\begin{aligned} \left| \int _\Omega G_\epsilon (x,\cdot )f\,{ dy}\right| \le C\left( \Vert f\Vert _{L^{\frac{2n}{n+2}}(\Omega )}+\Vert f\Vert _{L^{p}(B_{\delta _1+\delta _2}(x))}\right) \end{aligned}$$

(134)

for all $f\in C^\infty _c(\Omega )$. It follows from (134) that

$$\begin{aligned} \left| \int _\Omega G_\epsilon (x,\cdot )f\,{ dy}\right| \le C\cdot \Vert f\Vert _{L^{p}(\Omega )} \end{aligned}$$

for all $f\in C^\infty _c(\Omega )$ where $p>n/2$. It then follows from duality arguments that for any $q\in (1, n/(n-2))$ and any $\delta >0$, there exists $C(\delta ,q)>0$ such that $\Vert G_\epsilon (x,\cdot )\Vert _{L^q(\Omega )}\le C(\delta ,q)$ for all $\epsilon <\epsilon _0$ and $x\in \Omega {\setminus }B_\delta (0)$.

Let $\delta '\in (0,\delta )$ and $\delta _1,\delta _2>0$ such that $\delta _1+\delta _2<\delta '$. We get from (134) that

$$\begin{aligned} \left| \int _\Omega G_\epsilon (x,\cdot )f\,{ dy}\right| \le C\Vert f\Vert _{L^{\frac{2n}{n+2}}(\Omega {\setminus }B_{\delta '}(x))} \end{aligned}$$

(135)

for all $f\in C^\infty _c(\Omega {\setminus }B_{\delta '}(x))$. Here again, a duality argument yields (132), which proves the claim in Step 1.

Step 2: Convergence of $G_\epsilon$. Fix $x\in \Omega {\setminus }\{0\}$. For $0<\epsilon <\epsilon '$, since $G_\epsilon (x,\cdot )$, $G_{\epsilon '}(x,\cdot )$ are $C^2$ outside x, we have

$$\begin{aligned} -\Delta (G_\epsilon (x,\cdot )-G_{\epsilon '}(x,\cdot ))-\left( \frac{\gamma \eta _\epsilon }{|\cdot |^2}+h\right) (G_\epsilon (x,\cdot )-G_{\epsilon '}(x,\cdot ))= \frac{\gamma (\eta _\epsilon -\eta _{\epsilon '})}{|\cdot |^2}G_{\epsilon '}(x,\cdot ) \end{aligned}$$

in the strong sense. The coercivity (131) then yields

$$\begin{aligned} G_\epsilon (x,\cdot )\ge G_{\epsilon '}(x,\cdot )\quad \hbox {for}\quad 0<\epsilon <\epsilon '\quad \hbox {if}\quad \gamma \ge 0, \end{aligned}$$

and the reverse inequality if $\gamma <0$. It then follows from the integral bound (132) and elliptic regularity that there exists $G(x,\cdot )\in C^{2,\theta }(\overline{\Omega }{\setminus }\{0,x\})$ such that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}G_\epsilon (x,\cdot )=G(x,\cdot )\quad \hbox {in}~C^2_{loc}(\overline{\Omega }-\{0,x\}). \end{aligned}$$

In particular, G is symmetric and

$$\begin{aligned} -\Delta G(x,\cdot )-\left( \frac{\gamma }{|\cdot |^2}+h\right) G(x,\cdot )=0\quad \hbox {in}~\overline{\Omega }{\setminus }\{0,x\}. \end{aligned}$$

(136)

Moreover, passing to the limit $\epsilon \rightarrow 0$ in (132) and using elliptic regularity, we get that for all $\delta >0$, $1<q<\frac{n}{n-2}$ and $\delta '\in (0,\delta )$, there exist $C(\delta ,q)>0$ and $C(\delta ,\delta ')>0$ such that for all $x\in \Omega$, $|x|>\delta$,

$$\begin{aligned} \Vert G(x,\cdot )\Vert _{L^q(\Omega )}\le C(\delta ,q)\quad \hbox {and}\quad \Vert G(x,\cdot )\Vert _{L^{\frac{2n}{n-2}}(\Omega {\setminus }B_{\delta '}(x))}\le C(\delta ,\delta '). \end{aligned}$$

(137)

Moreover, for any $f\in L^p(\Omega )$, $p>n/2$, let $\varphi _\epsilon \in C^2(\overline{\Omega })$ be such that (133) holds, and fix $x\in \Omega {\setminus }\{0\}$. Passing to the limit $\epsilon \rightarrow 0$ in the Green identity $\varphi _\epsilon (x)=\int _\Omega G_\epsilon (x,\cdot )f\, dy$ yields

$$\begin{aligned} \varphi (x)=\int _\Omega G(x,\cdot )f\,{ dy}\quad \hbox {for all}~x\in \Omega {\setminus }\{0\} \end{aligned}$$

(138)

where $\varphi \in H_0^1(\Omega )\cap C^{0}(\overline{\Omega }{\setminus }\{0\})$ is the only weak solution to

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta \varphi -\left( \frac{\gamma }{|x|^2}+h(x)\right) \varphi = f &{}\hbox {in}~\Omega \\ \varphi =0&{}\hbox {on}~\partial \Omega \end{array}\right. \end{aligned}$$

In particular, the strong comparision principle yields $G(x,\cdot )>0$ for $x\in \Omega {\setminus }\{0\}$.

Step 3: Upper bound for G(x, y) when one variable is far from 0.

It follows from (136), elliptic theory and (137) that for any $\delta >0$, there exists $C(\delta )>0$ such that

$$\begin{aligned} 0<G(x,y)\le C(\delta )\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|x|>\delta ,\quad |y|>\delta ,\quad |x-y|>\delta . \end{aligned}$$

(139)

We claim that for any $\delta >0$, there exists $C(\delta )>0$ such that

$$\begin{aligned} 0<|x-y|^{n-2}G(x,y)\le C(\delta )\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|x|>\delta \quad \hbox {and}\quad |y|>\delta . \end{aligned}$$

(140)

Indeed, with no loss of generality, we can assume that $\delta \in (0,\delta _0)$. Define now $\Omega _\delta :=\Omega {\setminus }B_{\delta /2}(0)$, and fix $x\in \Omega$ such that $|x|>\delta$. Let $H_x$ be the Green’s function for $-\Delta -\left( \frac{\gamma }{|x|^2}+h(x)\right)$ in $\Omega _\delta$ with Dirichlet boundary condition. Classical estimates (see [22]) yield the existence of $C(\delta )>0$ such that $|x-y|^{n-2}H_x(y)\le C(\delta )$ for all $x,y\in \Omega _\delta$. It is easy to check that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta (G_x-H_x)-\left( \frac{\gamma }{|x|^2}+h\right) (G_x-H_x)= 0 &{}\hbox {weakly in }\Omega _\delta \\ G_x-H_x=0&{}\hbox {on }\partial \Omega \\ G_x-H_x=G_x&{}\hbox {on }\partial B_{\delta /2}(0). \end{array}\right. \end{aligned}$$

Regularity theory then yields that $G_x-H_x\in C^{2,\theta }(\overline{\Omega _\delta })$. It follows from (139) that $G_x$ is bounded by a constant depending only on $\delta$ on $\partial B_{\delta /2}(0)$ for $|x|>\delta$. The comparison principle then yields $|G_x(y)-H_x(y)|\le C(\delta )$ for $y\in \Omega _\delta$ and $|x|>\delta$. The above bound for $H_x$ and (139) then yields (140).

We now claim that for any $0<\delta '<\delta$, there exists $C(\delta ,\delta ')>0$ such that

$$\begin{aligned} 0<|y|^{\beta _-(\gamma )}G(x,y)\le C(\delta ,\delta ')\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|x|>\delta>\delta '>|y|>0.\quad \end{aligned}$$

(141)

Indeed, fix $\delta _1<\delta$ and use (139) to deduce that $G_x(y)\le C(\delta ,\delta _1)$ for all $x\in \Omega {\setminus }B_\delta (0)$ and $y\in \partial B_{\delta _1}(0)$. Since $\delta _1<|x|$, we have that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta G_x-\left( \frac{\gamma }{|x|^2}+h\right) G_x= 0 &{}\hbox {in }H_1^2(B_{\delta _1}(0))\\ 0<G_x\le C(\delta ,\delta ')&{}\hbox {on }\partial B_{\delta _1}(0). \end{array}\right. \end{aligned}$$

It follows from (162) below that for $\delta _1>0$ small enough, there exists $u_{\beta _-}\in H_1^2(B_{\delta _1}(0))$ such that $c_1\le |z|^{\beta _-(\gamma )}u_{\beta _-}(z)<c_2$ for all $z\in B_{\delta _1}(0)$, and

$$\begin{aligned} -\Delta u_{\beta _-}-\left( \frac{\gamma }{|x|^2}+h\right) u_{\beta _-}\ge 0\quad \hbox {in }H_1^2(B_{\delta _1}(0)). \end{aligned}$$

Therefore, there exists $C(\delta ,\delta ')>0$ such that $G_x(z)\le C(\delta ,\delta ')u_{\beta _-}(z)$ for all $z\in \partial B_{\delta _1}(0)$. It then follows from the comparison principle that $G_x(y)\le C(\delta ,\delta ')u_{\beta _-}(y)$ for all $y\in B_{\delta _1}(0){\setminus }\{0\}$. Combining this with (139), we obtain (141).

Note that by symmetry, we also get that for any $0<\delta '<\delta$, there exists $C(\delta ,\delta ')>0$ such that

$$\begin{aligned} |x|^{\beta _-(\gamma )}G(x,y)\le C(\delta ,\delta ')\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|y|>\delta>\delta '>|x|>0. \end{aligned}$$

(142)

Step 4: Upper bound for G(x, y) when both variables approach 0.

We claim first that for all $c_1,c_2,c_3>0$, there exists $C(c_1,c_2,c_3)>0$ such that for $x,y\in \Omega$ such that $c_1|x|<|y|<c_2|x|$ and $|x-y|>c_3|x|$, we have

$$\begin{aligned} |x-y|^{n-2}G(x,y)\le C(c_1,c_2,c_3). \end{aligned}$$

(143)

Indeed, fix $x\in B_{\delta _0/2}(0){\setminus }\{0\}\subset \Omega {\setminus }\{0\}$, and define

$$\begin{aligned} H(z):=G_x(|x|z)\quad \hbox {for}\quad z\in B_{\delta _0/|x|}(0){\Big \backslash }\left\{ 0,\frac{x}{|x|}\right\} , \end{aligned}$$

so that

$$\begin{aligned} -\Delta H-\left( \frac{\gamma }{|z|^2}+|x|^2h(|x|z)\right) H=0\quad \hbox {in }B_{\delta _0/|x|}(0){\Big \backslash }\left\{ 0,\frac{x}{|x|}\right\} . \end{aligned}$$

Since $H>0$, it follows from the Harnack inequality that for all $R>0$ large enough and $\alpha >0$ small enough, there exist $\delta _1>0$ and $C>0$ independent of $|x|<\delta _1$ such that

$$\begin{aligned} H(z)\le C H(z')\quad \hbox {for all}~z,z'\in B_R(0){\setminus }\left( B_\alpha (0)\cup B_\alpha \left( \frac{x}{|x|}\right) \right) , \end{aligned}$$

which rewrites as:

$$\begin{aligned} G_x(y)\le C G_x(y')\quad \hbox {for all}~y,y'\in B_{R|x|}(0){\setminus }\left( B_{\alpha |x|}(0)\cup B_{\alpha |x|}(x)\right) . \end{aligned}$$

(144)

Let $u_{\beta _+}$ be a sub-solution to (163). In particular, for $|x|<\delta _2$ small, there exists $C>0$ such that

$$\begin{aligned} G_x(z)\ge c|x|^{\beta _+(\gamma )}\left( \inf _{\partial B_{R|x|}(0)}G_x\right) u_{\beta _+}(z)\quad \hbox {for all}~z\in \partial B_{R|x|}(0). \end{aligned}$$

Since $-\Delta G_x-(\gamma |\cdot |^{-2}+h)G_x=0$ outside 0, it follows from coercivity and the comparison principle that

$$\begin{aligned} G_x(z)\ge c|x|^{\beta _+(\gamma )}\left( \inf _{\partial B_{R|x|}(0)}G_x\right) u_{\beta _+}(z)\quad \hbox {for all}~z\in \Omega {\setminus }B_{R|x|}(0). \end{aligned}$$

Fix $z_0\in \Omega {\setminus }\{0\}$. Then for $\delta _3$ small enough, it follows from (142) and the Harnack inequality (144) that there exists $C>0$ independent of x such that

$$\begin{aligned} G_x(y)\le C |x|^{-\beta _+(\gamma )-\beta _-(\gamma )}\quad \hbox {for all}~y\in B_{R|x|}(0){\setminus }\left( B_{\alpha |x|}(0)\cup B_{\alpha |x|}(x)\right) \end{aligned}$$

Taking $\alpha >0$ small enough and $R>0$ large enough, we then get (143) for $|x|<\delta _3$. The general case for arbitrary $x\in \Omega {\setminus }\{0\}$ then follows from (140). This prove (143).

Next we claim that for all $c_1,c_2>0$, there exists $C(c_1,c_2)>0$ such that

$$\begin{aligned} |x-y|^{n-2}G(x,y)\le C(c_1,c_2)\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~c_1|x|<|y|<c_2|x|. \end{aligned}$$

(145)

For that, we fix $x\in B_{\delta _0/2}(0){\setminus }\{0\}$ and set

$$\begin{aligned} H(z):=|x|^{n-2}G_x(x+|x|z)\quad \hbox {for all}~z\in B_{1/2}(0){\setminus }\{0\}. \end{aligned}$$

We have that $H\in C^2(\overline{B_{1/2}(0)}{\setminus }\{0\})$ and satisfies

$$\begin{aligned} -\Delta H-\left( \frac{\gamma }{\left| \frac{x}{|x|}+z\right| ^2} +|x|^2h(x+|x|z)\right) H=\delta _0~\hbox {weakly in}~B_{1/2}(0). \end{aligned}$$

We now argue as in the proof of (140). From (143), we have that $|H(z)|\le C$ for all $z\in \partial B_{1/2}(0)$ where C is independent of $x\in B_{\delta _0/2}(0){\setminus }\{0\}$. Let $\Gamma _0$ be the Green’s function of $-\Delta -\left( \frac{\gamma }{\left| \frac{x}{|x|}+z\right| ^2}+|x|^2h(x+|x|z)\right)$ at 0 on $B_{1/2}(0)$ with Dirichlet boundary condition. Therefore, $H-\Gamma _0\in C^2(\overline{B_{1/2}(0)})$ and, via the comparison principle, it is bounded by its supremum on the boundary. Therefore $|z|^{n-2}H(z)\le C$ for all $B_{1/2}(0){\setminus }\{0\}$ where C is independent of $x\in B_{\delta _0/2}(0){\setminus }\{0\}$. Scaling back and using (143), we get (145) for $x\in B_{\delta _0/2}(0){\setminus }\{0\}$. The general case is a consequence of (140). This ends the proof of (145).

We now show that there exists $C>0$ such that

$$\begin{aligned} |y|^{\beta _-(\gamma )}|x|^{\beta _+(\gamma )}G(x,y)\le C\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|y|<\frac{1}{2}|x|. \end{aligned}$$

(146)

Indeed, the proof goes essentially as in (141). Fix $x\in B_{\delta _0/2}(0)$, $x\ne 0$, and set $H(z):=|x|^{n-2}G_x(|x|z)$ for $z\in B_{1/2}(0){\setminus }\{0\}$. We have that

$$\begin{aligned} -\Delta H-\left( \frac{\gamma }{|z|^2}+|x|^2h(|x|z)\right) H=0\quad \hbox {in}~H_1^2(B_{1/2}(0)). \end{aligned}$$

Moreover, it follows from (143) that there exists $C>0$ such that $|H(z)|\le C$ for all $z\in \partial B_{1/2}(0)$. Then, as above, using a super-solution, we get that there exists $C>0$ such that $0<H(z)\le C|z|^{-\beta _-(\gamma )}$ for all $z\in B_{1/2}(0){\setminus }\{0\}$. Scaling back yields (146) when $x\in B_{\delta _0/2}(0)$. The general case follows from (141). This proves (146).

Again, by symmetry, we conclude that there exists $C>0$ such that

$$\begin{aligned} |x|^{\beta _-(\gamma )}|y|^{\beta _+(\gamma )}G(x,y)\le C\quad \hbox {for}\quad x,y\in \Omega ~\hbox {such that}~|x|<\frac{1}{2}|y|. \end{aligned}$$

(147)

Finally, one easily checks that (129) is a direct consequence of (146), (147) and (145). When $f\in C^\infty _c(\Omega )$, identity (127) is a consequence of (138). The general case follows from density and the integral controls on G. The behavior (128) is a consequence of the classification of solutions to harmonic equations and Theorem 9.

To conclude, we shall briefly sketch the proof of the lower bound (130). Indeed, in Steps 3 and 4, we repeatedly used the comparison principle to get the upper bound for G by considering domains on the boundary of which G was bounded from above. As one checks, in the case when x, y are in $\omega \subset \subset \Omega$, G is also bounded from below by some positive constant on the boundary of these domains. This yields the lower bound (130), and completes the proof of Theorem 6. $\square$

Appendix B: Green’s function for $-\Delta -\gamma |x|^{-2}$ on ${\mathbb {R}}^n$

In this section, we prove the following:

Theorem 7

Fix $\gamma <\frac{(n-2)^2}{4}$. For all $p\in {\mathbb {R}}^n{\setminus }\{0\}$, there exists $G:{\mathbb {R}}^n{\setminus }\{0,p\}\rightarrow {\mathbb {R}}$ such that

• (i) $G\in H_{1,loc}^2({\mathbb {R}}^n{\setminus }\{p\})$,

• (ii) For all $\varphi \in C^\infty _c({\mathbb {R}}^n)$, we have that

  $$\begin{aligned} \varphi (p)=\int _{{\mathbb {R}}^n}G(x)\left( -\Delta \varphi -\frac{\gamma }{|x|^2}\varphi \right) \,{ dx}\quad \hbox {for all}~\varphi \in C^\infty _c({\mathbb {R}}^n) \end{aligned}$$

  (148)

  Moreover, if $G,G'$ satisfy (i) and (ii) and are positive, then there exists $C\in {\mathbb {R}}$ such that $G(x)-G'(x)=C|x|^{-\beta _-(\gamma )}$ for all $x\in {\mathbb {R}}^n{\setminus }\{0,p\}$.

  In addition, there exists one and only one function $G:=G_p>0$ such that (i) and (ii) hold and

• (iii) For all $p\in {\mathbb {R}}^n{\setminus }\{0\}$, there exists $c_0(p),c_\infty (p)>0$ such that

  $$\begin{aligned} G_p(x)\sim _{x\rightarrow 0} \frac{c_0(p)}{|x|^{\beta _-(\gamma )}}\quad \hbox {and}\quad G_p(x)\sim _{x\rightarrow \infty } \frac{c_\infty (p)}{|x|^{\beta _+(\gamma )}} \end{aligned}$$

  and

  $$\begin{aligned} G_p(x)\sim _{x\rightarrow p}\frac{1}{(n-2)\omega _{n-1}|x-p|^{n-2}}. \end{aligned}$$

  (149)

• (iv) There exists $c>0$ independent of p such that

  $$\begin{aligned} c^{-1}\left( \frac{\max \{|p|,|x|\}}{\min \{|p|,|x|\}} \right) ^{\beta _-(\gamma )}|x-p|^{2-n}\le G_p(x)\le c \left( \frac{\max \{|p|,|x|\}}{\min \{|p|,|x|\}}\right) ^{\beta _-(\gamma )}|x-p|^{2-n} \end{aligned}$$

  (150)

Remark

Note that when $\gamma =0$, we have $\beta _-(\gamma )=0$, $\beta _+(\gamma )=n-2$ and $G(p,x)=\frac{1}{(n-2)\omega _{n-1}}|x-p|^{2-n}$ for all $x,p\in {\mathbb {R}}^n$, $x\ne p$.

Proof

We shall again proceed with several steps.

Step 1: Construction of a positive kernel at a given point: For a fixed $p_0\in {\mathbb {R}}^n{\setminus }\{0\}$, we show that there exists $G\in C^2({\mathbb {R}}^n{\setminus }\{0,p_0\})$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta G-\frac{\gamma }{|x|^2}G=0&{}\hbox {in}~{\mathbb {R}}^n{\setminus }\{0,p_0\}\\ G>0&{}\\ G\in L^{\frac{2n}{n-2}}(B_\delta (0))&{}\hbox {with}~\delta :=|p_0|/4\\ G\hbox { satisfies (ii)}. \end{array}\right. \end{aligned}$$

(151)

Indeed, let ${\tilde{\eta }}\in C^\infty ({\mathbb {R}})$ be a nondecreasing function such that $0\le {\tilde{\eta }}\le 1$, ${\tilde{\eta }}(t)=0$ for all $t\le 1$ and ${\tilde{\eta }}(t)=1$ for all $t\ge 2$. For $\epsilon >0$, set $\eta _\epsilon (x):={\tilde{\eta }}\left( \frac{|x|}{\epsilon }\right)$ for all $x\in {\mathbb {R}}^n$. For $R>0$, we argue as in the proof of (131) to deduce that the operator $-\Delta -\frac{\gamma \eta _\epsilon }{|x|^2}$ is coercive on $B_R(0)$ and that there exists $c>0$ independent of $R,\epsilon >0$ such that

$$\begin{aligned} \int _{B_R(0)}\left( |\nabla \varphi |^2-\frac{\gamma \eta _\epsilon }{|x|^2}\varphi ^2\right) \,{ dx}\ge c\int _{B_R(0)}|\nabla \varphi |^2\,{ dx}\quad \hbox {for all}~ \varphi \in C^\infty _c(B_R(0)). \end{aligned}$$

Consider $R,\epsilon >0$ such that $R>2|p_0|$ and $\epsilon <\frac{|p_0|}{6}$, and let $G_{R,\epsilon }$ be the Green’s function of $-\Delta -\frac{\gamma \eta _\epsilon }{|x|^2}$ in $B_R(0)$ at the point $p_0$ with Dirichlet boundary condition. We have that $G_{R,\epsilon }>0$ since the operator is coercive.

Fix $R_0>0$ and $q'\in (1,\frac{n}{n-2})$, then by arguing as in the proof of (132), we get that there exists $C=C(\gamma ,p_0, q', R_0)$ such that

$$\begin{aligned} \Vert G_{R,\epsilon }\Vert _{L^{q'}(B_{R_0}(0))}\le C\quad \hbox {for all}~R>R_0\quad \hbox {and}\quad 0<\epsilon <\frac{|p_0|}{6}, \end{aligned}$$

(152)

and

$$\begin{aligned} \Vert G_{R,\epsilon }\Vert _{L^{\frac{2n}{n-2}}(B_{\delta _0}(0))}\le C\quad \hbox {for all}~R>R_0\quad \hbox {and}\quad 0<\epsilon <\frac{|p_0|}{6}, \end{aligned}$$

(153)

where $\delta :=|p_0|/4$. Arguing again as in Step 2 of the proof of Theorem 6, there exists $G\in C^2({\mathbb {R}}^n{\setminus }\{0,p_0\})$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} G_{R,\epsilon }\rightarrow G\ge 0&{}\hbox {in}~C^2_{loc}({\mathbb {R}}^n{\setminus }\{0,p_0\})~\hbox {as}~R\rightarrow +\infty ,\,\epsilon \rightarrow 0\\ -\Delta G-\frac{\gamma }{|x|^2}G=0&{}\hbox {in}~{\mathbb {R}}^n{\setminus }\{0,p_0\}\\ G\in L^{\frac{2n}{n-2}}(B_\delta (0)) \end{array}\right. \end{aligned}$$

(154)

Fix $\varphi \in C^\infty _c({\mathbb {R}}^n)$. For $R>0$ large enough, we have that $\varphi (p_0)=\int _{{\mathbb {R}}^n}G_{R,\epsilon }(-\Delta \varphi -\gamma \eta _\epsilon |x|^{-2}\varphi )\,{ dx}$. With the integral bounds above, we then get that $x\mapsto G(x) |x|^{-2}\in L^1_{{ loc}}({\mathbb {R}}^n)$. Therefore, we get

$$\begin{aligned} \varphi (p_0)=\int _{{\mathbb {R}}^n}G(x)\left( -\Delta \varphi -\frac{\gamma }{|x|^2}\varphi \right) \,{ dx}\quad \hbox {for all}~\varphi \in C^\infty _c({\mathbb {R}}^n). \end{aligned}$$

(155)

As a consequence, $G>0$.

Step 2: Asymptotic behavior at 0 and p for solutions to (151). It follows from Theorem 9 below that either G behaves like $|x|^{-\beta _-(\gamma )}$ or $|x|^{-\beta _+(\gamma )}$ at 0. Since $G\in L^{\frac{2n}{n-2}}(B_\delta (0))$ for some small $\delta >0$ and $\beta _-(\gamma )<\frac{n-2}{2}<\beta _+(\gamma )$, we get that there exists $c>0$ such that

$$\begin{aligned} \lim _{x\rightarrow 0}|x|^{\beta _-(\gamma )}G(x)=c. \end{aligned}$$

(156)

In addition, Theorem 9 yields $G\in H_{1,loc}^2({\mathbb {R}}^n{\setminus }\{p_0\})$. Since G is positive and smooth in a neighborhood of p, it follows from (155) and the classification of solutions to harmonic equations that

$$\begin{aligned} G(x)\sim _{x\rightarrow p_0}\frac{1}{(n-2)\omega _{n-1}|x-p_0|^{n-2}}. \end{aligned}$$

(157)

Step 3: Asymptotic behavior at $\infty$ for solutions to (151): We let

$$\begin{aligned} \tilde{G}(x):=\frac{1}{|x|^{n-2}}G\left( \frac{x}{|x|^2}\right) \quad \hbox {for all}~x\in {\mathbb {R}}^n{\Big \backslash }\left\{ 0, \frac{p_0}{|p_0|^2}\right\} , \end{aligned}$$

be the Kelvin’s transform of G. We have that

$$\begin{aligned} -\Delta \tilde{G}-\frac{\gamma }{|x|^2}\tilde{G}=0\quad \hbox {in}~{\mathbb {R}}^n{\Big \backslash }\left\{ 0, \frac{p_0}{|p_0|^2}\right\} . \end{aligned}$$

Since $\tilde{G}>0$, it follows from Theorem 9 that there exists $c_1>0$ such that

$$\begin{aligned} \hbox {either}~\tilde{G}(x)\sim _{x\rightarrow 0}\frac{c_1}{|x|^{\beta _-(\gamma )}}~\hbox {or}~\tilde{G}(x)\sim _{x\rightarrow 0}\frac{c_1}{|x|^{\beta _+(\gamma )}}. \end{aligned}$$

Coming back to G, we get that

$$\begin{aligned} \hbox {either}~G(x)\sim _{x\rightarrow \infty }\frac{c_1}{|x|^{\beta _+(\gamma )}}~\hbox {or}~G(x)\sim _{|x|\rightarrow \infty }\frac{c_1}{|x|^{\beta _-(\gamma )}}. \end{aligned}$$

Assuming we are in the second case, for any $c\le c_1$, we define

$$\begin{aligned} \bar{G}_c(x):=G(x)-\frac{c}{|x|^{\beta _-(\gamma )}}~\hbox {in}~{\mathbb {R}}^n{\setminus }\{0, p_0\}, \end{aligned}$$

which satisfy $-\Delta \bar{G}-\frac{\gamma }{|x|^2}\bar{G}=0$ in ${\mathbb {R}}^n{\setminus }\{0, p_0\}$. It follows from (156) and (157) that for $c<c_1$, $\bar{G}_c>0$ around $p_0$ and $\infty$. It then follows from the coercivity of $-\Delta -\gamma |x|^{-2}$ that $\bar{G}_c>0$ in ${\mathbb {R}}^n{\setminus }\{0, p\}$ for $c<c_1$. Letting $c\rightarrow c_1$ yields $\bar{G}_{c_1}\ge 0$, and then $\bar{G}_{c_1}> 0$ since it is positive around $p_0$. Since $\bar{G}_{c_1}(x)=o(|x|^{-\beta _-(\gamma )})$ as $|x|\rightarrow \infty$, performing again a Kelvin transform and using Theorem 9, we get that $|x|^{\beta _+(\gamma )}\bar{G}_{c_1}(x)\rightarrow c_2>0$ as $|x|\rightarrow \infty$. Then there exists $c_3>0$ such that

$$\begin{aligned} \lim _{x\rightarrow 0}|x|^{\beta _-(\gamma )}\bar{G}_{c_1}(x)=c_3>0\quad \hbox {and}\quad \lim _{x\rightarrow \infty }|x|^{\beta _+(\gamma )}\bar{G}_{c_1}(x)=c_2. \end{aligned}$$

Since $x\mapsto |x|^{-\beta _-(\gamma )}\in H_{1,loc}^2({\mathbb {R}}^n)$, we get that $\varphi (p)=\int _{{\mathbb {R}}^n}\bar{G}_{c_1}(x)\left( -\Delta \varphi -\frac{\gamma }{|x|^2}\varphi \right) \,{ dx}$ for all $\varphi \in C^\infty _c({\mathbb {R}}^n)$.

Step 4: Uniqueness: Let $G_1,G_2>0$ be 2 functions such that (i), (ii) hold for $p:=p_0$, and set $H:=G_1-G_2$. It follows from Steps 2 and 3 that there exists $c\in {\mathbb {R}}$ such that $H'(x):=H(x)-c|x|^{-\beta _-(\gamma )}$ satisfies

$$\begin{aligned} H'(x)=_{x\rightarrow 0}O\left( |x|^{-\beta _-(\gamma )}\right) \quad \hbox {and}\quad H'(x)=_{|x|\rightarrow \infty }O\left( |x|^{-\beta _+(\gamma )}\right) . \end{aligned}$$

(158)

We then have that $H\in H_{1,{ loc}}^2({\mathbb {R}}^n{\setminus }\{p_0\})$ is such that

$$\begin{aligned} \int _{{\mathbb {R}}^n}H'(x)\left( -\Delta \varphi -\frac{\gamma }{|x|^2}\varphi \right) \, { dx}=0 \quad \hbox {for all}~\varphi \in C^\infty _c({\mathbb {R}}^n). \end{aligned}$$

The ellipticity of the Laplacian then yields that $H'\in C^\infty ({\mathbb {R}}^n{\setminus }\{0\})$. The pointwise bounds (158) yield that $H'\in D^{1,2}({\mathbb {R}}^n)$. Multiplying $-\Delta H'-\frac{\gamma }{|x|^2}H'=0$ by $H'$, integrating by parts and using the coercivity yields that $H'\equiv 0$, and therefore, $G_1-G_2=c|\cdot |^{-\beta _-(\gamma )}$. This proves uniqueness.

Step 5: Existence. It follows from Step 3 that, up to substracting a multiple of $|\cdot |^{-\beta _-(\gamma )}$, there exists $G_{p_0}>0$ satisfying (i), (ii) and the pointwise controls (iii) at $p_0$. It is a consequence of (iii) that there exists $c>0$ such that

$$\begin{aligned} c^{-1} \left( \frac{\max \{1,|x|\}}{\min \{1,|x|\}}\right) ^{\beta _-(\gamma )}|x-p_0|^{2-n}\le G_{p_0}(x)\le c \left( \frac{\max \{1,|x|\}}{\min \{1,|x|\}}\right) ^{\beta _-(\gamma )}|x-p_0|^{2-n} \end{aligned}$$

for all $x\in {\mathbb {R}}^n{\setminus }\{0,p_0\}$, c depending on $p_0$. For $p\in {\mathbb {R}}^n{\setminus }\{0\}$, consider $\rho _p: {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n$ a linear isometry such that $\rho _p(\frac{p_0}{|p_0|})=\frac{p}{|p|}$, and define

$$\begin{aligned} G_p(x):=\left( \frac{|p_0|}{|p|}\right) ^{n-2}G_{p_0} \left( \left( \rho _p^{-1}\left( \frac{|p_0|}{|p|}x\right) \right) \right) \quad \hbox {for all}~x\in {\mathbb {R}}^n{\setminus }\{0,p\}. \end{aligned}$$

It is easy to check that $G_p>0$ and that it satisfies (i)–(iv). $\square$

Appendix C: Singular solutions to $-\Delta u-c(x) |x|^{-2}u=0$

We collect here a few results that should be classical, but quite difficult to find in the literature. These results and their proofs are closely related to the work of the authors in [15], to which we shall frequently refer for details.

Theorem 8

(Optimal regularity and Generalized Hopf’s Lemma) Fix $\gamma <\frac{(n-2)^2}{4}$ and let $f: \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a Caratheodory function such that

$$\begin{aligned} |f(x, v)|\le C|v| \left( 1+\frac{|v|^{2^\star (s)-2}}{|x|^s}\right) \quad \hbox { for all}~x\in \Omega \quad \hbox {and}\quad v\in {\mathbb {R}}. \end{aligned}$$

Let $u\in H_1^2(B_1(0))$ be a weak solution of

$$\begin{aligned} -\Delta u-\frac{\gamma +O(|x|^\theta )}{|x|^2}u=f(x,u)\quad \hbox {in}~H_1^2(B_{1/2}(0)) \end{aligned}$$

(159)

for some $\theta >0$. Then, there exists $K\in {\mathbb {R}}$ such that

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u(x)}{|x|^{-\beta _-(\gamma )}}=K. \end{aligned}$$

(160)

Moreover, if $u\ge 0$ and $u\not \equiv 0$, we have that $K>0$.

Theorem 9

Let $u\in C^2(B_1(0){\setminus }\{0\})$ be a positive solution to

$$\begin{aligned} -\Delta u-\frac{c(x)}{|x|^2}u=0\quad \hbox {in}~B_1(0){\setminus }\{0\} \end{aligned}$$

where $c(x)=\gamma +O(|x|^\theta )$ as $x\rightarrow 0$ with $\gamma <(n-2)^2/4$ and $\theta \in (0,1)$. Then there exists $\alpha >0$ such that

$$\begin{aligned} \hbox {either}~u(x)\sim _{x\rightarrow 0}\frac{\alpha }{|x|^{\beta _-(\gamma )}}~\hbox {or}~u(x)\sim _{x\rightarrow 0}\frac{\alpha }{|x|^{\beta _+(\gamma )}}. \end{aligned}$$

In particular, $u\in H_1^2(B_{1/2}(0))$ if and only if the first case holds.

Proposition 11

Let $u\in C^2({\mathbb {R}}^n{\setminus }\{0\})$ be a nonnegative function such that

$$\begin{aligned} -\Delta u-\frac{\gamma }{|x|^2}u=0\quad \hbox {in}~{\mathbb {R}}^n{\setminus }\{0\}. \end{aligned}$$

(161)

Then there exist $\lambda _-,\lambda _+\ge 0$ such that

$$\begin{aligned} u(x)=\lambda _- |x|^{-\beta _-(\gamma )}+\lambda _+ |x|^{-\beta _+(\gamma )}\quad \hbox {for all}~x\in {\mathbb {R}}^n{\setminus }\{0\}. \end{aligned}$$

Proofs: The proofs of these results follow closely the proofs of Theorems 6.1 and 7.1 and Proposition 7.4 of [15]. Here are the ingredients to adapt:

Sub- and super-solutions: The first step is the following result:

Proposition 12

Fix $\gamma <(n-2)^2/4$ and $\theta \in (0,1)$ and let $c:B_1(0)\rightarrow {\mathbb {R}}$ be such that $c(x)=\gamma +O(|x|^\theta )$ as $x\rightarrow 0$. We choose $\beta \in \{\beta _-(\gamma ),\beta _+(\gamma )\}$. Then there exists $u_\beta ^{(+)}, u_\beta ^{(-)}\in C^2(B_1(0))$ such that for $\delta >0$ small enough,

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_{\beta }^{(+)}-\frac{c(x)}{|x|^2}u_{\beta }^{(+)}>0 &{}\hbox {in}~B_\delta (0)\\ u_{\beta }^{(+)}(x)\sim |x|^{-\beta }&{}\hbox {as}~x\rightarrow 0 \end{array}\right. ;\quad \left\{ \begin{array}{ll} -\Delta u_{\beta }^{(-)}-\frac{c(x)}{|x|^2}u_{\beta }^{(-)}<0 &{}\hbox {in}~B_\delta (0)\\ u_{\beta }^{(-)}(x)\sim |x|^{-\beta }&{}\hbox {as}~x\rightarrow 0 \end{array}\right. \end{aligned}$$

(162)

The proof is as follows. For $\beta \in \{\beta _-(\gamma ),\beta _+(\gamma )\}$, we define $u_{\beta }:\, x\mapsto |x|^{-\beta }+\lambda |x|^{-\beta '}$. A straightforward computation yields

$$\begin{aligned} -\Delta u_{\beta }-\frac{c(x)}{|x|^2}u_{\beta }=|x|^{-\beta '-2} \left( \lambda (\beta '(n-2-\beta ')-\gamma )+O(|x|^\theta ) +O(|x|^{\theta -(\beta -\beta ')}\right) \end{aligned}$$

as $x\rightarrow 0$. Then, choosing $\beta '\in {\mathbb {R}}$ such that $0<\beta -\beta '<\theta$ and $\beta '(n-2-\beta ')-\gamma \ne 0$, we get either a sub- or a supersolution taking $\lambda$ positive or negative. This proves the proposition.

Sub-solution with Dirichlet boundary condition: We let $u_{\beta _+(\gamma )}$ as above be a super-solution on $B_\delta (0){\setminus }\{0\}$. Take $\eta \in C^\infty ({\mathbb {R}}^n)$ such that $\eta (x)=0$ for $x\in B_{\delta /4}(0)$ and $\eta (x)=1$ for $x\in {\mathbb {R}}^n{\setminus }B_{\delta /3}(0)$. Define on $B_\delta (0)$ the function

$$\begin{aligned} f(x):=\left( -\Delta -\frac{c(x)}{|x|^2}\right) (\eta u_{\beta _+(\gamma )}), \end{aligned}$$

Note that f vanishes around 0 and that it is in $C^\infty (\overline{B_\delta (0)})$. Let $v\in D^{1,2}(B_\delta (0))$ be such that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta v-\frac{c(x)}{|x|^2}v=f&{}\hbox {in}~B_\delta (0)\\ v=0&{}\hbox {on}~\partial B_\delta (0). \end{array}\right. \end{aligned}$$

Note that for $\delta >0$ small enough, $-\Delta -(\gamma +O(|x|^\theta ))|x|^{-2}$ is coercive on $B_\delta (0)$, and therefore, the existence of v is ensured for small $\delta$. Define

$$\begin{aligned} u_{\beta _+(\gamma )}^{(d)}:=u_{\beta _+(\gamma )}-\eta u_{\beta _+(\gamma )}+v. \end{aligned}$$

The definition of $\eta$ and v yields

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_{\beta _+(\gamma )}^{(d)}-\frac{c(x)}{|x|^2}u_{\beta _+(\gamma )}^{(d)}>0&{} \hbox {in}~B_\delta (0)-\{0\}\\ u_{\beta _+(\gamma )}^{(d)}=0&{}\hbox {in}~\partial B_\delta (0) \end{array}\right. \end{aligned}$$

(163)

Moreover, since $-\Delta v-c(x)|x|^{-2}v=0$ around 0 and $v\in D^{1,2}(B_\delta (0))$, it follows from Theorem 8 that there exists $C>0$ such that $|v(x)|\le C |x|^{-\beta _-(\gamma )}$ for all $x\in B_\delta (0)$. Then it follows from the expression of $u_{\beta _+(\gamma )}$ that

$$\begin{aligned} u_{\beta _+(\gamma )}(x)\sim _{x\rightarrow 0}|x|^{-\beta _+(\gamma )}. \end{aligned}$$

We then get a supersolution satisfying (163) with the above behavior at 0. This is similar for a subsolution.

In the proof above, it is important that the operator $-\Delta -c(x)|x|^{-2}$ is coercive on $B_\delta (0)$ for $\delta >0$ small enough. Now let $\Omega$ be a smooth bounded domain of ${\mathbb {R}}^n$ and let $\gamma <(n-2)^2/4$ and $h\in C^{0,\theta (\overline{\Omega })}$ be such that $-\Delta -(\gamma |x|^{-2}+h)$ is coercive on $\Omega$. Arguing as above, we get that there exists $u_{\beta _+(\gamma )}^{(d,\Omega )}\in C^{2,\theta }(\overline{\Omega }{\setminus }\{0\})$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_{\beta _+(\gamma )}^{(d,\Omega )}-\left( \frac{\gamma }{|x|^2}+h\right) u_{\beta _+(\gamma )}^{(d)}>0&{}\hbox {in}~\Omega {\setminus }\{0\}\\ u_{\beta _+(\gamma )}^{(d,\Omega )}>0 &{}\hbox {in}~\Omega {\setminus }\{0\}\\ u_{\beta _+(\gamma )}^{(d,\Omega )}=0 &{}\hbox {in}~\partial \Omega \end{array}\right. \end{aligned}$$

(164)

and

$$\begin{aligned} u_{\beta _+(\gamma )}^{(d,\Omega )}(x)\sim _{x\rightarrow 0}|x|^{-\beta _+(\gamma )}. \end{aligned}$$

Similarly, we get a subsolution.

These points are enough to adapt the proofs of the above-mentioned results of [15] to our context. $\square$