Isolated singularities for the n-Liouville equation

Abstract

In dimension n isolated singularities—at a finite point or at infinity—for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in Esposito (Anal Non Linéaire 35(3):781–801, 2018) and establish a quantization result for entire solutions of the singular n-Liouville equation.

1 Introduction

The behavior near an isolated singularity has been discussed by Serrin in [19, 20] for a very general class of second-order quasi-linear equations. The simplest example is given by the prototypical equation $-\Delta _n u=f$, where $\Delta _n(\cdot ) =\hbox {div} (|\nabla (\cdot ) |^{n-2}\nabla (\cdot ) )$, $n \ge 2$, is the n-Laplace operator. In dimension n, the case $f \in L^1$ is very delicate as it represents a limiting situation where Serrin’s results do not apply. We will be interested in the n-Liouville equation, where f is taken as an exponential function of u according to Liouville’s seminal paper [16], and the singularity might be at a finite point or at infinity.

To this aim, it is enough to consider the generalized n-Liouville equation

$$\begin{aligned} -\Delta _n u=|x|^{n \alpha } e^u \text{ in } \Omega {\setminus } \{0\}, \, \int _{\Omega } |x|^{n \alpha } e^u <+\infty \end{aligned}$$

(1.1)

on an open set $\Omega \subset {\mathbb {R}}^n$ with $0 \in \Omega$, and we will be concerned with describing the behavior of u at 0. A solution u of (1.1) stands for a function $u \in C^{1,\eta }_{loc}(\Omega {\setminus } \{0\})$ which satisfies

$$\begin{aligned} \int _{\Omega } |\nabla u|^{n-2}\langle \nabla u, \nabla \varphi \rangle =\int _{\Omega } |x|^{n \alpha } e^u \varphi \qquad \forall \ \varphi \in C_0^1(\Omega {\setminus } \{0\}). \end{aligned}$$

The regularity assumption on u is not restrictive since a solution in $W^{1,n}_{\mathrm {loc}}(\Omega {\setminus } \{0\})$ is automa-tically in $C^{1,\eta }_{loc}(\Omega {\setminus } \{0\})$, for some $\eta \in (0,1)$, thanks to [9, 19, 22], see Theorem 2.3 in [11].

Concerning the behavior near an isolated singularity, our main result is

Theorem 1.1

Let u be a solution of (1.1). Then there exists $\gamma >- n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$, $\omega _n=|B_1(0)|$, so that

$$\begin{aligned} -\Delta _n u=|x|^{n \alpha } e^u -\gamma \delta _0 \hbox { in } \Omega \end{aligned}$$

(1.2)

with

$$\begin{aligned} u-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |x|\in L^\infty _{\mathrm {loc}} (\Omega ) \end{aligned}$$

(1.3)

and

$$\begin{aligned} \lim _{x \rightarrow 0} \left[ |x|\nabla u(x)-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \frac{x}{|x|} \right] =0. \end{aligned}$$

(1.4)

The case $\alpha =-2$ is relevant for the asymptotic behavior at infinity for solutions u of

$$\begin{aligned} -\Delta _n u=e^u \text{ in } \Omega , \, \int _\Omega e^u<+\infty , \end{aligned}$$

(1.5)

where $\Omega$ is an unbounded open set so that $B_R(0)^c \subset \Omega$ for some $R>0$. Indeed, let us recall that $\Delta _n$ is invariant under Kelvin transform: if u solves (1.1), then ${{\hat{u}}}(x)=u(\frac{x}{|x|^2})$ does satisfy

$$\begin{aligned} -\Delta _n {{\hat{u}}}=|x|^{-2n} (-\Delta _n u) \left( \frac{x}{|x|^2}\right) =|x|^{-n(\alpha +2)} e^{{{\hat{u}}}} \hbox { in }{{\hat{\Omega }}}=\left\{ x \not =0 :\ \frac{x}{|x|^2} \in \Omega \right\} . \end{aligned}$$

(1.6)

By Theorem 1.1 applied with $\alpha =-2$ to ${{\hat{u}}}$ at 0 we find:

Corollary 1.2

Let u be a solution of (1.5) on an unbounded open set $\Omega$ with $B_R(0)^c \subset \Omega$ for some $R>0$. Then there holds

$$\begin{aligned} u=-\left( \frac{\gamma _\infty }{n\omega _n} \right) ^{\frac{1}{n-1}} \log |x|+O(1) \end{aligned}$$

(1.7)

as $|x| \rightarrow \infty$ for some $\gamma _\infty >n^n \omega _n$. In particular, when $\Omega ={\mathbb {R}}^n$ there holds

$$\begin{aligned} u= - \left( \frac{1}{n \omega _n}\int _{{\mathbb {R}}^n} e^u \right) ^{\frac{1}{n-1}}\log |x|+O(1) \end{aligned}$$

(1.8)

as $|x| \rightarrow \infty$.

When $n=2$ the asymptotic expansion (1.8) is a well known property established in [6] by means of the Green representation formula—unfortunately not available in the quasi-linear case—and of the growth properties of entire harmonic functions. Notice that

$$\begin{aligned} \gamma _\infty =\int _{{\mathbb {R}}^n} |x|^{-2n} e^{{{\hat{u}}}} =\int _{{\mathbb {R}}^n} e^u \end{aligned}$$

follows by integrating (1.2) written for ${{\hat{u}}}$ on ${\mathbb {R}}^n$. Property (1.8) has been already proved in [11] under the assumption $\gamma _\infty >n^n \omega _n$ and the present full generality allows to simplify the classification argument in [11]: a Pohozaev identity leads for $\gamma _\infty$ to the quantization property

$$\begin{aligned} \int _{{\mathbb {R}}^n} e^u=n \left( \frac{n^{2}}{n-1}\right) ^{n-1} \omega _n \end{aligned}$$

(1.9)

and an isoperimetric argument concludes the classification result thanks to (1.9).

In the punctured plane $\Omega ={\mathbb {R}}^n {\setminus } \{0\}$ the isoperimetric argument fails and in general the classification result is no longer available. The two-dimensional case $n=2$ has been treated via complex analysis in [7, 17]: solutions u to

$$\begin{aligned} -\Delta u=e^u -\gamma \delta _0 \text{ in } {\mathbb {R}}^2,\, \int _{{\mathbb {R}}^2} e^u<+\infty , \end{aligned}$$

(1.10)

have been classified for $\gamma >-4\pi$ of the form

$$\begin{aligned} u(x)=\log \frac{8 (\alpha +1)^2 \lambda ^2 |x|^{2\alpha }}{(1+\lambda ^2 |x^{\alpha +1}+c|^2)^2},\, \alpha =\frac{\gamma }{4\pi }, \end{aligned}$$

with $\lambda >0$ and a complex number $c =0$ if $\alpha \notin {\mathbb {N}}$, and in particular satisfy

$$\begin{aligned} \int _{{\mathbb {R}}^n} e^u=8\pi (\alpha +1). \end{aligned}$$

(1.11)

The structure of entire solutions u to (1.10) changes drastically passing from radial solutions when $\alpha \notin {\mathbb {N}}$ to non-radial solutions when $\alpha \in {\mathbb {N}}$ (and $c \not =0$). Unfortunately a PDE approach is not available for $n=2$ and a classification result is completely out of reach when $n \ge 3$. However, quantization properties are still in order as it follows by Theorem 1.1 and the Pohozaev identities:

Theorem 1.3

Let u be a solution of

$$\begin{aligned} -\Delta _n u=e^u -\gamma \delta _0 \text{ in } {\mathbb {R}}^n,\, \int _{{\mathbb {R}}^n} e^u<+\infty . \end{aligned}$$

(1.12)

Then $\gamma >-n^n \omega _n$ and

$$\begin{aligned} \int _{{\mathbb {R}}^n} e^u=\gamma +\gamma _\infty \end{aligned}$$

(1.13)

with $\gamma _\infty$ the unique solution in $(n^n \omega _n,+\infty )$ of

$$\begin{aligned} \frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} \gamma _\infty ^{\frac{n}{n-1}} -n \gamma _\infty =n\gamma + \frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} |\gamma |^{\frac{n}{n-1}} . \end{aligned}$$

(1.14)

When $n=2$ notice that for $\gamma > -4\pi$ the unique solution $\gamma _\infty >4\pi$ of (1.14) is given explicitly as $\gamma _\infty =\gamma +8\pi$ and then $\int _{{\mathbb {R}}^n} e^u=2\gamma +8\pi =8\pi (\alpha +1)$ in accordance with (1.11). To have Theorem 1.3 meaningful, in Sect. 4 we will show the existence of a family of radial solutions u to (1.12) but we don’t know whether other solutions might exist or not, depending on the value of $\gamma$. Notice that (1.10) is equivalent to

$$\begin{aligned} -\Delta v=|x|^{2\alpha } e^v \text{ in } {\mathbb {R}}^2,\, \int _{{\mathbb {R}}^2} |x|^{2\alpha } e^v<+\infty , \end{aligned}$$

in terms of $v=u-2\alpha \log |x|$. For $n\ge 3$ such equivalence breaks down and the problem

$$\begin{aligned} -\Delta _n v=|x|^{n \alpha } e^v \text{ in } {\mathbb {R}}^n,\, \int _{{\mathbb {R}}^n} |x|^{n \alpha } e^v<+\infty , \end{aligned}$$

(1.15)

has its own interest, independently of (1.12). As in Theorem 1.3, for (1.15) we have the following quantization result:

Theorem 1.4

Let v be a solution of (1.15). Then $\alpha > -1$ and

$$\begin{aligned} \int _{{\mathbb {R}}^n} |x|^{n \alpha }e^v=n \left( \frac{n^{2}}{n-1}\right) ^{n-1} (\alpha +1)^{n-1} \omega _n. \end{aligned}$$

Radial solutions v of (1.15) can be easily obtained as $v=n \log (\alpha +1)+u (|x|^{\alpha +1})$ in terms of a radial entire solution u to (1.5). Thanks to the classification result in [11], for (1.15) we can therefore exhibit the following family of radial solutions:

$$\begin{aligned} v_\lambda =\log \frac{c_n (\alpha +1)^n \lambda ^n}{\left( 1+\lambda ^{\frac{n}{n-1}} |x|^{\frac{n(\alpha +1)}{n-1}}\right) ^n}, \, c_n=n (\frac{n^{2}}{n-1})^{n-1}. \end{aligned}$$

Problems with exponential nonlinearities on a bounded domain with given singular sources can exhibit non-compact solution-sequences, whose shape near a blow-up point is asymptotically described by the limiting problem (1.12). In the regular case (i.e. in absence of singular sources) a concentration-compactness principle has been established [5] for $n=2$ and [1] for $n \ge 2$. In the non-compact situation the exponential nonlinearity concentrates at the blow-up points as a sum of Dirac measures. Theorem 1.3 gives information on the concentration mass of such Dirac measures at a singular blow-up point, which is expected bo te a super-position of several masses $c_n\omega _n$ carried by multiple sharp collapsing peaks governed by (1.12)$_{\gamma =0}$ and possibly the mass (1.13) of a sharp peak described by (1.12). In the regular case such quantization property on the concentration masses has been proved [14] for $n=2$ and extended [12] to $n \ge 2$ by requiring an additional boundary assumption, while the singular case has been addressed in [2, 21] for $n=2$. For Theorem 1.4 a similar comment is in order.

Let us briefly explain the main ideas behind Theorem 1.1. We can re-adapt the argument in [11] to show that $u \in \displaystyle \bigcap \nolimits _{1\le q <n} W^{1,q}_{\mathrm {loc}}(\Omega )$ and then u satisfies (1.2) for some $\gamma \in {\mathbb {R}}$. On a radial ball $B \subset \subset \Omega$ decompose u as $u=u_0+h$, where h is given by

$$\begin{aligned} \Delta _n h= \gamma \delta _0 \hbox { in } B,\quad h=u \hbox { on }\partial B \end{aligned}$$

and satisfies (1.3) thanks to [13, 19, 20]. The key property stems from a simple observation: $|x|^{n\alpha }e^h \in L^1$ near 0 implies $|x|^{n\alpha }e^h \in L^p$ near 0 for some $p>1$ whenever h has a logarithmic singularity at 0. Back to [19, 20] thanks to such improved integrability, one aims to show that $u_0 \in L^\infty (B)$ and then u has the same logarithmic behavior (1.3) as h. In order to develop a regularity theory for the solution $u_0$ of

$$\begin{aligned} -\Delta _n (u_0+h)+\Delta _n h=|x|^{n \alpha } e^{u_0+h} \text{ in } B, \, u_0=0 \hbox { on } \partial B, \end{aligned}$$

(1.16)

the crucial point is to establish several integral inequalities involving $u_0$ paralleling the estimates available for entropy solutions in [1, 3] and for $W^{1,n}$-solutions in [19]. To this aim, we make use of the deep uniqueness result [10] to show that u can be regarded as a Solution Obtained as Limit of Approximations (the so-called SOLA, see for example [4]).

The paper is organized as follows. In Sect. 2 we develop the above argument to prove Theorem 1.1. Section 3 is devoted to establish Theorems 1.3–1.4 via Pohozaev identities: going back to an idea of Y.Y. Li and N. Wolanski for $n=2$, the Pohozaev identities have revealed to be a fundamental tool to derive information on the mass of a singularity (see for example [2, 12, 18]). In Sect. 4 a family of radial solutions u to (1.12) is constructed.

2 Proof of Theorem 1.1

Assume $B_1(0) \subset \subset \Omega$. Let us first establish the following property on u.

Proposition 2.1

Let u be a solution of (1.1). There exists $C>0$ so that

$$\begin{aligned} u(x) \le C- n(\alpha +1) \log |x| \qquad \hbox {in } B_1(0) {\setminus } \{ 0 \}. \end{aligned}$$

(2.1)

Proof

Letting $U_r(y)={{\hat{u}}} (\frac{y}{r})+n(\alpha +1) \log r=u(\frac{ry}{|y|^2})+n(\alpha +1) \log r$ for $0<r\le \frac{1}{2}$, we have that $U_r$ solves

$$\begin{aligned}&-\Delta _n U_r=|y|^{-n(\alpha +2)} e^{U_r} \qquad \hbox {in } {\mathbb {R}}^n {\setminus } B_{\frac{1}{2}}(0), \nonumber \\&\quad \int _{{\mathbb {R}}^n {\setminus } B_{\frac{1}{2}}(0)} |y|^{-n(\alpha +2)} e^{U_r}= \int _{B_{2r}(0)} |x|^{n\alpha } e^u \end{aligned}$$

(2.2)

in view of (1.6). Given a ball $B_{\frac{1}{2}}(x_0)$ for $x_0 \in {\mathbb {S}}^{n-1}$, let us consider the n-harmonic function $H_r$ in $B_{\frac{1}{2}}(x_0)$ so that $H_r=U_r$ on $\partial B_{\frac{1}{2}}(x_0)$. By the weak maximum principle we deduce that $H_r \le U_r$ in $B_{\frac{1}{2}}(x_0)$ and then

$$\begin{aligned} \int _{B_{\frac{1}{2}}(x_0)} (H_r)_+^n\le & {} \int _{B_{\frac{1}{2}}(x_0)} (U_r)_+^n \le n! \int _{B_{\frac{1}{2}}(x_0)} e^{U_r} \nonumber \\\le & {} C \int _{{\mathbb {R}}^n {\setminus } B_{\frac{1}{2}}(0)} |y|^{-n(\alpha +2)} e^{U_r} \le C \int _{\Omega } |x|^{n\alpha } e^u<+\infty \end{aligned}$$

(2.3)

for all $0<r\le \frac{1}{2}$. By the estimates in [19] we have that there exists $C>0$ so that

$$\begin{aligned} \Vert H_r\Vert _{\infty , B_{\frac{1}{4}} (x_0)} \le C \end{aligned}$$

(2.4)

for all $0<r\le \frac{1}{2}$. At the same time, by the exponential estimate in [1] we have that there exist $0<r_0 \le \frac{1}{2}$ and $C>0$ so that

$$\begin{aligned} \int _{B_{\frac{1}{2}}(x_0)} e^{2 |U_r-H_r|} \le C \end{aligned}$$

(2.5)

for all $0<r \le r_0$ in view of $\displaystyle \lim _{ r \rightarrow 0^+} \int _{B_{\frac{1}{2}}(x_0)} |y|^{-n(\alpha +2)} e^{U_r}= 0$ thanks to (1.1) and (2.2). Since $|y|^{-n(\alpha +2)} e^{U_r} \le C e^{|U_r-H_r|}$ on $B_{\frac{1}{4}} (x_0)$ for all $0<r\le \frac{1}{2}$ in view of (2.4), we deduce that $|y|^{-n(\alpha +2)} e^{U_r}$ and $(U_r)_+^{\frac{n}{2}}$ are uniformly bounded in $L^2( B_{\frac{1}{4}} (x_0))$ for all $0<r \le r_0$ in view of (2.3) and (2.5). Thanks again to the estimates in [19], we finally deduce that

$$\begin{aligned} \Vert U_r^+\Vert _{\infty , B_{\frac{1}{8}} (x_0)} \le C \end{aligned}$$

(2.6)

for all $0<r \le r_0$. Since ${\mathbb {S}}^{n-1}$ can be covered by a finite number of balls $B_{\frac{1}{8}} (x_0)$, $x_0 \in {\mathbb {S}}^n$, going back to u from (2.6) one deduces that

$$\begin{aligned} u(x) \le C-n(\alpha +1) \log |x| \end{aligned}$$

for all $|x|=r \le r_0$. Since this estimate does hold in $B_1(0) {\setminus } B_{r_0}(0)$ too, we have established the validity of (2.1). $\square$

From now on, set $B=B_r(0)$ for $0<r\le 1$. We are now ready to establish the starting point for the argument we will develop in the sequel. There holds

Proposition 2.2

Let u be a solution of (1.1). Then

$$\begin{aligned} u \in \bigcap _{1\le q<n} W^{1,q}(\Omega ). \end{aligned}$$

(2.7)

Proof

Let us go through the argument in [11] to obtain $W^{1,q}$-estimates on u. For $0<\epsilon<r<1$ let us introduce $h_{\epsilon ,r} \in W^{1,n}(A_{\epsilon ,r})$, $A_{\epsilon ,r}:=B {\setminus } \overline{B_\epsilon (0)}$, as the solution of

$$\begin{aligned} \Delta _n h_{\epsilon ,r}=0 \hbox { in }A_{\epsilon ,r} ,\, h_{\epsilon ,r}= u \hbox { on }\partial A_{\epsilon ,r}. \end{aligned}$$

Regularity issues for quasi-linear PDEs involving $\Delta _n$ are well established since the works of DiBenedetto, Evans, Lewis, Serrin, Tolksdorf, Uhlenbeck, Uraltseva. For example, by [9, 15, 19, 22] we deduce that $h_{\epsilon ,r},\, u_{\epsilon ,r}=u-h_{\epsilon ,r} \in C^{1,\eta }(\overline{A_{\epsilon ,r}})$ and $u_{\epsilon ,r}$ satisfies

$$\begin{aligned} -\Delta _n (u_{\epsilon ,r}+h_{\epsilon ,r})+\Delta _n h_{\epsilon ,r}= |x|^{n\alpha }e^ u \hbox { in }A_{\epsilon ,r},\, u_{\epsilon ,r}=0 \hbox { on }\partial A_{\epsilon ,r}. \end{aligned}$$

(2.8)

By the techniques in [1, 3, 4] we have the following estimates, see Proposition 2.1 in [11]: for all $1\le q<n$ and all $p\ge 1$ there exist $0<r_0<1$ and $C>0$ so that

$$\begin{aligned} \int _{A_{\epsilon ,r}} |\nabla u_{\epsilon ,r}|^q +\int _{A_{\epsilon ,r}} e^{p u_{\epsilon ,r}} \le C \end{aligned}$$

(2.9)

for all $0<\epsilon <r \le r_0$ thanks to (2.8) and $\displaystyle \lim _{r \rightarrow 0^+} \int _{B} |x|^{n \alpha } e^u =0.$ Since by the Sobolev embedding $W^{1,\frac{n}{2}}_0(B_1(0)) \hookrightarrow L^n(B_1(0))$ there holds $\int _{A_{\epsilon ,r}} |u_{\epsilon ,r}|^n \le C$ for all $0<\epsilon <r \le r_0$ in view of (2.9) with $q=\frac{n}{2}$ and $A_{\epsilon ,r} \subset B_1(0)$, we have that

$$\begin{aligned} \Vert h_{\epsilon ,r} \Vert _{L^n(A)} \le C(A) \qquad \forall \ A \subset \subset {\overline{B}} {\setminus } \{0\} , \, \forall \ 0<\epsilon <r \le r_0 \end{aligned}$$

in view of $u \in C^{1,\eta }_{loc}(B_1(0) {\setminus } \{0\})$ and then

$$\begin{aligned} \Vert h_{\epsilon ,r}\Vert _{C^{1,\eta }(A)} \le C(A) \qquad \forall \ A \subset \subset {\overline{B}} {\setminus } \{0\} , \, \forall \ 0<\epsilon <r \le r_0 \end{aligned}$$

thanks to [9, 15, 19, 22]. By the Ascoli–Arzelá’s Theorem and a diagonal process we can find a sequence $\epsilon \rightarrow 0$ so that $h_{\epsilon ,r} \rightarrow h_r$ and $u_{\epsilon ,r} \rightarrow u_r:=u-h_r$ in $C^1_{\mathrm {loc}} ({\overline{B}}{\setminus } \{0\})$ as $\epsilon \rightarrow 0$, where $h_r \le u$ is a n-harmonic function in $B {\setminus } \{0\}$ and $u_r$ satisfies

$$\begin{aligned} u_r \in W^{1,q}_0(B) ,\qquad e^{u_r} \in L^p(B) \end{aligned}$$

(2.10)

for all $1\le q<n$ and all $p\ge 1$ if r is sufficiently small in view of (2.9). Since

$$\begin{aligned} h_r(x) \le C- n(\alpha +1) \log |x| \qquad \hbox {in }B \end{aligned}$$

in view of $h_r \le u$ and (2.1), we have that $H^\lambda (y)=- \frac{h_r(\lambda y)}{\log \lambda }$ is a n-harmonic function in $B_{\frac{r}{\lambda }}(0)$ so that $H^\lambda \le n(\alpha +1)+1$ in $B_2(0){\setminus } B_{\frac{1}{2}}(0)$ for all $0<\lambda \le \lambda _0$, where $\lambda _0 \in (0,\frac{r}{2}]$ is a suitable small number. By the Harnack inequality in Theorem 7- [19] applied to $n(\alpha +1)+1-H^\lambda \ge 0$ we deduce that

$$\begin{aligned} \max _{|y|=1} H^\lambda \le C\left[ \frac{n|\alpha +1|+1}{C}+\min _{|y|=1} H^\lambda \right] \end{aligned}$$

(2.11)

for all $0<\lambda \le \lambda _0$, for a suitable $C\in (0,1)$. There are two possibilities:

• either $\displaystyle \min _{|y|=1} H^\lambda \ge -\frac{n|\alpha +1|+1}{C}$ for all $0<\lambda \le \lambda _1$ and some $\lambda _1\in (0, \lambda _0]$, which implies $\displaystyle \max _{|y|=1} |H^\lambda | \le \frac{n|\alpha +1|+1}{C}$ for all $0<\lambda \le \lambda _1$ and in particular

  $$\begin{aligned} |h_r| \le -C_0 \log |x| \qquad \hbox {in }B_{\lambda _1}(0) \end{aligned}$$

  (2.12)

  for some $C_0>0$;

• or $\displaystyle \min _{|y|=1} H^{\lambda _n} \le -\frac{n|\alpha +1|+1}{C}$ for a sequence $\lambda _n \downarrow 0$, which implies $\displaystyle \max _{|y|=1} H^{\lambda _n} \le 0$ in view of (2.11) and in turn $h_r \le 0$ on $|x|=\lambda _n$ for all $n \in {\mathbb {N}}$, leading to

  $$\begin{aligned} h_r \le 0\qquad \hbox {in }B_{\lambda _1}(0) \end{aligned}$$

  (2.13)

  by the weak maximum principle.

Notice that (2.13) implies the validity of (2.12) for some $C_0>0$ in view of Theorem 12- [19]. Thanks to (2.12) one can apply Theorem 1.1- [13] to show that

$$\begin{aligned} h_r \in W^{1,q}(B) \end{aligned}$$

(2.14)

for all $1\le q <n$ and there exists $\gamma _r \in {\mathbb {R}}$ so that

$$\begin{aligned} h_r-\gamma _r (n\omega _n |\gamma _r|^{n-2})^{-\frac{1}{n-1}} \log |x|\in L^\infty (B), \qquad \Delta _n h_r=\gamma _r \delta _0 \hbox { in } B. \end{aligned}$$

(2.15)

In particular, $u \in \displaystyle \bigcap _{1\le q<n} W^{1,q}(\Omega )$ in view of (2.10) and (2.14), and (2.7) is established. $\square$

Even if $\gamma _r>-n^n|\alpha +1|^{n-2}(\alpha +1)\omega _n$, at this stage we cannot exclude that $\displaystyle \lim _{r \rightarrow 0}\gamma _r=-n^n|\alpha +1|^{n-2}(\alpha +1)\omega _n$. Therefore, we are not able to use (2.10) and (2.15) for improving the exponential integrability on u to reach $|x|^{n\alpha } e^u=|x|^{n\alpha } e^{h_r} e^{u_r} \in L^p$ near 0 for some $p>1$ and r sufficiently small, as it would be necessary to prove $L^\infty$-bounds on $u_r$ via (2.8) on $u_{\epsilon ,r}$.

We need to argue in a different way. Since $u \in W^{1,n-1}(\Omega )$ in view of (2.7), we can extend (1.1) at 0 as

$$\begin{aligned} -\Delta _n u= |x|^{n\alpha } e^u-\gamma \delta _0 \quad \hbox {in }\Omega . \end{aligned}$$

(2.16)

To see it, let $\varphi \in C_0^1(\Omega )$ and consider a function $\chi _\epsilon \in C^\infty (\Omega )$ with $0\le \chi _\epsilon \le 1$, $\chi _\epsilon =0$ in $B_{\frac{\epsilon }{2}}(0)$, $\chi _\epsilon =1$ in $\Omega {\setminus } B_\epsilon (0)$ and $\epsilon |\nabla \chi _\epsilon | \le C$. Taking $\chi _\epsilon \varphi \in C_0^1(\Omega {\setminus } \{0\})$ as a test function in (1.1) we have that

$$\begin{aligned} \int _\Omega |\nabla u|^{n-2}\langle \nabla u, \varphi \nabla \chi _\epsilon +\chi _\epsilon \nabla \varphi \rangle =\int _\Omega \chi _\epsilon |x|^{n\alpha } e^u \varphi . \end{aligned}$$

(2.17)

Since $u \in W^{1,n-1}(\Omega )$ and $|x|^{n\alpha } e^u \in L^1(\Omega )$ it is easily seen that

$$\begin{aligned}&\int _\Omega \chi _\epsilon |\nabla u|^{n-2}\langle \nabla u, \nabla \varphi \rangle \rightarrow \int _\Omega |\nabla u|^{n-2}\langle \nabla u, \nabla \varphi \rangle ,\nonumber \\&\quad \int _\Omega \chi _\epsilon |x|^{n\alpha } e^u \varphi \rightarrow \int _\Omega |x|^{n\alpha } e^u \varphi \end{aligned}$$

(2.18)

as $\epsilon \rightarrow 0$. Since

$$\begin{aligned} \int _\Omega |\nabla u|^{n-1}|\varphi - \varphi (0)| |\nabla \chi _\epsilon | \le C \int _{B_\epsilon (0) {\setminus } B_{\frac{\epsilon }{2}}(0)} |\nabla u|^{n-1} \rightarrow 0 \end{aligned}$$

as $\epsilon \rightarrow 0$ in view of $|\varphi - \varphi (0)|\le C\epsilon$ in $B_\epsilon (0) {\setminus } B_{\frac{\epsilon }{2}}(0)$ and $u \in W^{1,n-1}(\Omega )$, the remaining term in (2.17) can be re-written as follows:

$$\begin{aligned} \int _\Omega |\nabla u|^{n-2} \varphi \langle \nabla u, \nabla \chi _\epsilon \rangle = \varphi (0) \int _\Omega |\nabla u|^{n-2} \langle \nabla u,\nabla \chi _\epsilon \rangle +o(1) \end{aligned}$$

(2.19)

as $\epsilon \rightarrow 0$. By inserting (2.18)–(2.19) into (2.17) we deduce the existence of

$$\begin{aligned} \gamma =\lim _{\epsilon \rightarrow 0} \int _\Omega |\nabla u|^{n-2} \langle \nabla u,\nabla \chi _\epsilon \rangle \end{aligned}$$

and the validity of (2.16) for u. Moreover, if we assume $u \in C^1({\overline{\Omega }} {\setminus } \{0\})$, we can interpret $\gamma$ as

$$\begin{aligned} \gamma =\lim _{\epsilon \rightarrow 0} [\int _\Omega |x|^{n\alpha } e^u \chi _\epsilon + \int _{\partial \Omega } |\nabla u|^{n-2}\partial _n u]=\int _\Omega |x|^{n\alpha } e^u + \int _{\partial \Omega } |\nabla u|^{n-2}\partial _n u. \end{aligned}$$

Since $\gamma _r \ge \gamma +o(1)$ as $r \rightarrow 0$ according to (4.16)- [11], we find that $h_r$ is possibly much lower than u and then needs to be compensated by an unbounded function $u_r \ge 0$ in order to keep the validity of $u=u_r+h_r$. Instead, thanks to Theorem 2.1- [13] introduce $h \in \displaystyle \bigcap _{1\le q<n} W^{1,q}(B)$ as the solution of

$$\begin{aligned} \Delta _n h= \gamma \delta _0 \hbox { in } B,\quad h=u \hbox { on }\partial B \end{aligned}$$

so that

$$\begin{aligned} h-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |x|\in L^\infty (B). \end{aligned}$$

(2.20)

Decomposing u as $u=u_0+h$, the solution $u_0$ of (1.16) on B is very likely a bounded function, as we will prove below.

In order to establish some crucial integral inequalities involving $u_0$, let us introduce the following approximation scheme. By convolution with mollifiers consider sequences $f_j, g_j \in C_0^\infty (B)$ so that $f_j \rightharpoonup |x|^{n \alpha } e^u-\gamma \delta _0$ weakly in the sense of measures and $0\le f_j-g_j \rightarrow |x|^{n \alpha } e^u$ in $L^1(B)$ as $j \rightarrow +\infty$. Since $u \in C^{1,\eta }(\partial B)$, let $\varphi \in C^{1,\eta } (B)$ be the n-harmonic extension of $u \mid _{\partial B}$ in B. Let $v_j,w_j \in W_0^{1,n}(B)$ be the weak solutions of $-\hbox {div}\ \mathbf{a}(x,\nabla v_j)=f_j$ and $-\hbox {div}\ \mathbf{a}(x,\nabla w_j)=g_j$ in B, where $\mathbf{a}(x,p)=|p+\nabla \varphi |^{n-2} (p+\nabla \varphi )-|\nabla \varphi |^{n-2} \nabla \varphi$. In this way, $u_j=v_j+\varphi$ and $h_j=w_j+\varphi$ do solve

$$\begin{aligned} -\Delta _n u_j=f_j \hbox { and } -\Delta _n h_j=g_j \hbox { in }B,\, u_j=h_j=u \hbox { on }\partial B. \end{aligned}$$

Since $f_j,g_j$ are uniformly bounded in $L^1(B)$, by (21) in [4] we can assume that $v_j \rightarrow v$ and $w_j \rightarrow w$ in $W^{1,q}_0(B)$ for all $1\le q<n$ as $j \rightarrow +\infty$, where v and w do satisfy

$$\begin{aligned} -\hbox {div}\ \mathbf{a}(x,\nabla v)=|x|^{n \alpha } e^u-\gamma \delta _0 \hbox { and }-\hbox {div}\ \mathbf{a}(x,\nabla w)=-\gamma \delta _0 \hbox { in }B \end{aligned}$$

(2.21)

in view of $g_j \rightharpoonup -\gamma \delta _0$ weakly in the sense of measures as $j \rightarrow +\infty$. Since $u-\varphi ,h-\varphi \in \displaystyle \bigcap _{1\le q<n} W^{1,q}_0(B)$ do solve the first and the second equation in (2.21), respectively, by the uniqueness result in [10] (see Theorems 1.2 and 4.2 in [10]) we have that $v=u-\varphi$ and $w=h-\varphi$, i.e.

$$\begin{aligned} u_j \rightarrow u \hbox { and } h_j \rightarrow h \hbox { in } W^{1,q}(B) \hbox { for all }1\le q<n \hbox { as }j \rightarrow +\infty . \end{aligned}$$

Thanks to the approximation given by the $u_j$’s and $h_j$’s, we can now derive some crucial integral inequalities on $u_0$.

Proposition 2.3

Let $u_0$ be a solution of (1.16). Then $u_0 \ge 0$ and we have:

$$\begin{aligned} \int _{\{k<|u_0| < k+a\} } |\nabla u_0|^n \le \frac{a}{d} \int _B |x|^{n\alpha } e^u \qquad \forall \, k,a>0 \end{aligned}$$

(2.22)

and, if $|x|^{n\alpha } e^u \in L^p(B)$ for some $p>1$,

$$\begin{aligned} \left( \int _B u_0^{2mnq} \right) ^{\frac{1}{2m}} \le \frac{C q^{n-1}}{d} |B|^{\frac{n-1}{mnq}} \left( \int _B |x|^{n p \alpha }e^{pu} \right) ^{\frac{1}{p}} \left( \int _B u_0^{mnq}\right) ^{\frac{n(q-1)+1}{mnq}}, \end{aligned}$$

(2.23)

where $m=\frac{p}{p-1}$ and

$$\begin{aligned} d= \inf _{X \not = Y} \frac{\langle |X|^{n-2}X-|Y|^{n-2}Y,X-Y \rangle }{|X-Y|^n}>0. \end{aligned}$$

(2.24)

Proof

First use $-(v_j-w_j)_- \in W^{1,n}_0(B)$ as a test function for $-\hbox {div}\ \mathbf{a}(x,\nabla v_j)+\hbox {div}\ \mathbf{a}(x,\nabla w_j)$ to get

$$\begin{aligned} d \int _{\{ v_j-w_j<0\} } |\nabla (v_j-w_j)|^n\le & {} - \int _B \langle \mathbf{a}(x,\nabla v_j)-\mathbf{a}(x,\nabla w_j), \nabla (v_j-w_j)_- \rangle \\= & {} -\int _B (f_j-g_j) (v_j-w_j)_- \le 0 \end{aligned}$$

in view of (2.24) and $f_j-g_j\ge 0$. Hence, $v_j- w_j \ge 0$ and then $u_0\ge 0$ in view of $v_j-w_j \rightarrow u-h=u_0$ in $W^{1,q}_0(B)$ for all $1\le q<n$ as $j \rightarrow +\infty$. Now, introduce the truncature operator $T_{k,a}$, for $k,a>0$, as

$$\begin{aligned} T_{k,a}(s)=\left\{ \begin{array}{cl} s- k \hbox { sign}(s) &{}\hbox {if }k< |s|< k+a, \\ a \hbox { sign}(s) &{}\hbox {if }|s|\ge k+a,\\ 0 &{}\hbox {if }|s|\le k, \end{array} \right. \end{aligned}$$

and use $T_{k,a}(v_j-w_j) \in W^{1,n}_0(B)$ as a test function for $-\hbox {div}\ \mathbf{a}(x,\nabla v_j)+\hbox {div}\ \mathbf{a}(x,\nabla w_j)$ to get

$$\begin{aligned}&d \int _{\{ k<|v_j-w_j| < k+a\} } |\nabla (v_j-w_j)|^n \le \int _B \langle \mathbf{a}(x,\nabla v_j)-\mathbf{a}(x,\nabla w_j), \nabla T_{k,a}(v_j-w_j) \rangle \nonumber \\&\quad =\int _B (f_j-g_j) T_{k,a}(v_j-w_j) \end{aligned}$$

(2.25)

in view of (2.24). Since $v_j-w_j \rightarrow u_0$ in $W^{1,q}_0(B)$ for all $1\le q<n$ and $f_j-g_j \rightarrow |x|^{n \alpha } e^u$ in $L^1(B)$ as $j \rightarrow +\infty$, we can let $j \rightarrow +\infty$ in (2.25) and get by Fatou’s Lemma that

$$\begin{aligned} d \int _{\{ k<|u_0| < k+a\} } |\nabla u_0|^n \le \int _B |x|^{n \alpha } e^u T_{k,a}(u_0) \le a \int _B |x|^{n \alpha } e^u \end{aligned}$$

yielding the validity of (2.22). Finally, if $|x|^{n\alpha } e^u \in L^p(B)$ for some $p>1$, we can assume that $f_j-g_j \rightarrow |x|^{n \alpha } e^u$ in $L^p(B)$ as $j \rightarrow +\infty$ and use $T_a[|v_j-w_j|^{n(q-1)}(v_j-w_j)] \in W^{1,n}_0(B)$, where $T_a=T_{0,a}$ and $a>0,q \ge 1$, as a test function for $-\hbox {div}\ \mathbf{a}(x,\nabla v_j)+\hbox {div}\ \mathbf{a}(x,\nabla w_j)$ to get by Hölder’s inequality

$$\begin{aligned}&d \frac{n(q-1)+1}{q^n} \int _{\{ |v_j-w_j|^{n(q-1)+1} < a\} } |\nabla |v_j-w_j|^q|^n \\&\quad \le \int _B |f_j-g_j| |v_j-w_j|^{n(q-1)+1}\\&\quad \le |B|^{\frac{n-1}{mnq}} \Vert f_j-g_j\Vert _p \left( \int _B |v_j-w_j|^{mnq}\right) ^{\frac{n(q-1)+1}{mnq}} \end{aligned}$$

in view of $|T_a(s)|\le |s|$ and (2.24). We have used that $v_j-w_j \in W^{1,n}_0(B) \subset \displaystyle \bigcap _{q \ge 1} L^q(B)$ by the Sobolev embedding Theorem. Letting $a \rightarrow +\infty$, by Fatou’s Lemma we get that

$$\begin{aligned} \int _B |\nabla |v_j-w_j|^q|^n \le \frac{q^n}{d [n(q-1)+1]} |B|^{\frac{n-1}{mnq}} \Vert f_j-g_j\Vert _p \left( \int _B |v_j-w_j|^{mnq}\right) ^{\frac{n(q-1)+1}{mnq}}. \end{aligned}$$

In particular, $|v_j-w_j|^q \in W^{1,n}_0(B)$ and by the Sobolev embedding $W^{1,n}_0(B) \subset L^{2mn}(B)$ we have that

$$\begin{aligned} \left( \int _B |v_j-w_j|^{2mnq} \right) ^{\frac{1}{2m}} \le \frac{C q^n}{d [n(q-1)+1]} |B|^{\frac{n-1}{mnq}} \Vert f_j-g_j\Vert _p \left( \int _B |v_j-w_j|^{mnq}\right) ^{\frac{n(q-1)+1}{mnq}}. \end{aligned}$$

Letting $j \rightarrow +\infty$, we finally deduce the validity of (2.23):

$$\begin{aligned} \left( \int _B u_0^{2mnq} \right) ^{\frac{1}{2m}} \le \frac{C q^{n-1}}{d} |B|^{\frac{n-1}{mnq}} \left( \int _B |x|^{n p \alpha }e^{pu} \right) ^{\frac{1}{p}} \left( \int _B u_0^{mnq}\right) ^{\frac{n(q-1)+1}{mnq}} \end{aligned}$$

in view of $v_j-w_j \rightarrow u_0$ in $W^{1,q}_0(B)$ for all $1\le q<n$ and $f_j-g_j \rightarrow |x|^{n \alpha } e^u$ in $L^p(B)$ as $j \rightarrow +\infty$. $\square$

We are now ready to complete the proof of Theorem 1.1.

Proof (of Theorem 1.1)

Since $u_0 \ge 0$ by Proposition 2.3, we have that $h \le u$. By (1.1) and (2.20) we have that

$$\begin{aligned} \int _{B} |x|^{n\alpha +\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}}} \le C \int _{B} |x|^{n\alpha }e^h \le C \int _\Omega |x|^{n\alpha }e^u<+\infty , \end{aligned}$$

which implies

$$\begin{aligned} n\alpha +\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}}>-n \end{aligned}$$

(2.26)

or equivalently

$$\begin{aligned} \gamma >-n^n|\alpha +1|^{n-2}(\alpha +1)\omega _n. \end{aligned}$$

(2.27)

Since $|x|^{n\alpha }e^h \in L^1$ near 0 and h has a logarithmic singularity at 0, then, as already observed in the Introduction, a stronger integrability follows:

$$\begin{aligned} |x|^{n\alpha } e^h \in L^p(B) \end{aligned}$$

(2.28)

for some $p>1$. Inequality (2.22) is used in [1] to deduce exponential estimates on $u_0$ like

$$\begin{aligned} \int _B e^{\frac{\delta u_0}{\Vert f\Vert _1}} \le C_r \end{aligned}$$

(2.29)

for some $\delta >0$ where $f=|x|^{n\alpha } e^u$. Since $\displaystyle \lim _{r \rightarrow 0} \int _{B} |x|^{n \alpha } e^u=0$, by (2.29) we deduce that $e^{u_0} \in L^p(B)$ for all $p\ge 1$ if r is sufficiently small and then by (2.28)

$$\begin{aligned} |x|^{n\alpha } e^u=|x|^{n\alpha } e^h e^{u_0} \in L^p(B) \end{aligned}$$

for some $p>1$. Inequality (2.23) is used in Proposition 4.1-[11] (compare with (4.4) in [11]) to get $u_0 \in L^\infty (B)$ and then (2.20) does hold for u too, yielding the validity of (1.3). In order to prove (1.4), set $H=u-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |x|$ and introduce the function

$$\begin{aligned} U_r(y)=u(ry)-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log r= \gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |y|+H(ry) \end{aligned}$$

for a given sequence $r \rightarrow 0$. Since

$$\begin{aligned} -\Delta _n U_r= r^{n(1+\alpha )} |y|^{n\alpha } e^{u(ry)}= r^{n(1+\alpha ) +\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}}} |y|^{n\alpha +\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}}} e^{H(ry)}, \end{aligned}$$

by (1.3) and (2.26)–(2.27) we have that $U_r$ and $\Delta _n U_r$ are bounded in $L^\infty _{\mathrm {loc}}({\mathbb {R}}^n {\setminus } \{0\})$, uniformly in r. By [9, 19, 22] we deduce that $U_r$ is bounded in $C^{1,\eta }_{\mathrm {loc}}({\mathbb {R}}^n {\setminus } \{0\})$, uniformly in r. By the Ascoli–Arzelá’s Theorem and a diagonal process, up to a sub-sequence we have that $U_r \rightarrow U_0$ in $C^1_{\mathrm {loc}}({\mathbb {R}}^n {\setminus } \{0\})$, where $U_0$ is a n-harmonic function in ${\mathbb {R}}^n {\setminus } \{0\}$. Setting $H_r(y)=H(ry)$, we deduce that $H_r \rightarrow H_0$ in $C^1_{\mathrm {loc}}({\mathbb {R}}^n {\setminus } \{0\})$, where $H_0 \in L^\infty ({\mathbb {R}}^n)$ in view of (1.3). Since $U_0=\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |y|+H_0$ with $H_0 \in L^\infty ({\mathbb {R}}^n)\cap C^1({\mathbb {R}}^n {\setminus } \{0\})$, it is well known that $H_0$ is a constant function, as shown in Corollary 2.2-[13] (see also [11] for a direct proof). In particular we get that

$$\begin{aligned} \sup _{|x|=r} |x| \Big | \nabla [ u-\gamma (n\omega _n |\gamma |^{n-2})^{-\frac{1}{n-1}} \log |x|]\Big | =\sup _{|y|=1} |\nabla H_r| \rightarrow \sup _{|y|=1} |\nabla H_0| =0. \end{aligned}$$

Since this is true for any sequence $r \rightarrow 0$ up to extracting a sub-sequence, we have established the validity of (1.4). The proof of Theorem 1.1 is concluded. $\square$

3 Quantization results

In this section we will make crucial use of the following integral identity: for any solution u of

$$\begin{aligned} -\Delta _n u=|x|^{n\alpha } e^u \hbox { in } {\mathbb {R}}^n {\setminus } \{0\} \end{aligned}$$

(3.1)

there holds

$$\begin{aligned} n (\alpha +1) \int _A |x|^{n\alpha } e^u=\int _{\partial A} \left[ |x|^{n\alpha } e^u \langle x,\nu \rangle +|\nabla u|^{n-2} \partial _{\nu }u \ \langle \nabla u,x \rangle -\frac{|\nabla u|^n}{n}\langle x,\nu \rangle \right] ,\nonumber \\ \end{aligned}$$

(3.2)

where A is the annulus $A=B_R(0) {\setminus } B_\epsilon (0)$, $0<\epsilon<R<+\infty$, and $\nu$ is the unit outward normal vector at $\partial A$. Notice that (3.2) is simply a special case of the well-known Pohozaev identities associated to (3.1). Even though the classical Pohozaev identities require more regularity than simply $u \in C^{1,\eta }({\mathbb {R}}^n {\setminus } \{0\})$, (3.2) is still valid in the quasilinear case and we refer to [8] for a justification. Thanks to (3.2) we are able to show the following general result.

Proposition 3.1

Let u be a solution of (3.1) so that (1.3)–(1.4) do hold at 0 and $\infty$ with $\gamma$ and $-\gamma _\infty$, respectively, so that $\gamma > - n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$ and $\gamma _\infty > n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$. Then $\int _{{\mathbb {R}}^n} |x|^{n \alpha } e^u =\gamma +\gamma _\infty$ satisfies

$$\begin{aligned} n (\alpha +1) (\gamma +\gamma _\infty )= \frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} \left[ |\gamma _\infty |^{\frac{n}{n-1}} -|\gamma |^{\frac{n}{n-1}} \right] . \end{aligned}$$

(3.3)

Proof

By (1.3)–(1.4) at 0 with $\gamma > - n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$ we deduce that

$$\begin{aligned}&|\nabla u| = \frac{1}{|x|} \left[ (\frac{|\gamma |}{n\omega _n})^{\frac{1}{n-1}} + o(1)\right] ,\quad \langle \nabla u, x \rangle =\gamma \left( n\omega _n |\gamma |^{n-2}\right) ^{-\frac{1}{n-1}}+o(1),\nonumber \\&|x|^{n\alpha }e^u = o(\frac{1}{|x|^n}) \end{aligned}$$

(3.4)

as $x \rightarrow 0$ thanks to the equivalence between (2.26) and (2.27). By (3.4) we have that

$$\begin{aligned} \int _{\partial B_\epsilon (0)} |x| \left[ |x|^{n\alpha } e^u +|\nabla u|^{n-2} \langle \nabla u, \frac{x}{|x|} \rangle ^2-\frac{|\nabla u|^n}{n} \right] \rightarrow \frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} |\gamma |^{\frac{n}{n-1}}\nonumber \\ \end{aligned}$$

(3.5)

as $\epsilon \rightarrow 0^+$ in view of $\hbox {Area}({\mathbb {S}}^{n-1})= n\omega _n$. Similarly, by (1.3)–(1.4) at $\infty$ with $-\gamma _\infty$ so that $\gamma _\infty > n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$ we deduce that

$$\begin{aligned}&|\nabla u|=\frac{1}{|x|} \left[ (\frac{|\gamma _\infty |}{n\omega _n})^{\frac{1}{n-1}} + o(1)\right] ,\quad \langle \nabla u, x \rangle =-\gamma _\infty (n\omega _n |\gamma _\infty |^{n-2})^{-\frac{1}{n-1}}+o(1),\\&|x|^{n\alpha }e^u=o\left( \frac{1}{|x|^n}\right) \end{aligned}$$

as $|x| \rightarrow \infty$ and then

$$\begin{aligned} \int _{\partial B_R(0)} |x| \left[ |x|^{n\alpha } e^u +|\nabla u|^{n-2} \langle \nabla u,\frac{x}{|x|} \rangle ^2-\frac{|\nabla u|^n}{n} \right] \rightarrow \frac{n-1}{n}(n \omega _n)^{-\frac{1}{n-1}} |\gamma _\infty |^{\frac{n}{n-1}}\nonumber \\ \end{aligned}$$

(3.6)

as $R \rightarrow +\infty$. In view of (1.4) at 0 and $\infty$ we easily get that

$$\begin{aligned} -\Delta _n u=|x|^{n \alpha } e^u -\gamma \delta _0 -\gamma _\infty \delta _{\infty } \hbox { in } {\mathbb {R}}^n \end{aligned}$$

in the sense

$$\begin{aligned} \int _{{\mathbb {R}}^n} |\nabla u|^{n-2} \langle \nabla u,\nabla \varphi \rangle =\int _{{\mathbb {R}}^n} |x|^{n \alpha } e^u \varphi -\gamma \varphi (0) -\gamma _\infty \varphi (\infty ) \end{aligned}$$

for all $\varphi \in C^1({\mathbb {R}}^n)$ so that $\varphi (\infty ):=\displaystyle \lim _{|x| \rightarrow \infty } \varphi (x)$ does exist. Choosing $\varphi =1$ we deduce that

$$\begin{aligned} \int _{{\mathbb {R}}^n} |x|^{n \alpha } e^u =\gamma +\gamma _\infty . \end{aligned}$$

(3.7)

By inserting (3.5)–(3.7) into (3.2) and letting $\epsilon \rightarrow 0^+,\, R\rightarrow +\infty$ we deduce the validity of (3.3).

$\square$

Let us now apply Proposition 3.1 to problems (1.12) and (1.15).

Proof (of Theorem 1.3)

Let u be a solution of (1.12). By Theorem 1.1 (1.3)–(1.4) do hold for u at 0 with $\gamma > - n^n \omega _n$. By (1.6) the Kelvin transform ${{\hat{u}}}$ satisfies

$$\begin{aligned} -\Delta _n {{\hat{u}}}=|x|^{-2n} e^{{{\hat{u}}}} \hbox { in } {\mathbb {R}}^n {\setminus } \{0\}. \end{aligned}$$

Let us apply Theorem 1.1 to deduce the validity of (1.3)–(1.4) for ${{\hat{u}}}$ at 0 with $\gamma _\infty > n^n \omega _n$. Back to u, (1.3)–(1.4) do hold for u at $\infty$ with $-\gamma _\infty$ so that $\gamma _\infty > n^n \omega _n$. Let us apply Proposition 3.1 with $\alpha =0$ to get $\int _{{\mathbb {R}}^n} e^u =\gamma +\gamma _\infty$ with $\gamma _\infty$ satisfying (1.14). Notice that the function $f(s)=n s+\frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} |s|^{\frac{n}{n-1}}$ is increasing in $(-n^n \omega _n,+\infty )$ and then $f(s) >f(-n^n \omega _n)=-n^n \omega _n$ for all $s \in (-n^n \omega _n,+\infty )$. At the same time the function $g(s)=\frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} s^{\frac{n}{n-1}} -n s$ is increasing in $(n^n \omega _n,+\infty )$ and then $g(s)>g(n^n \omega _n)=-n^n \omega _n$ for all $s \in (n^n\omega _n,+\infty )$. Therefore, for any $\gamma >- n^n \omega _n$ Equation (1.14) has a unique solution $\gamma _\infty >n^n \omega _n$. The proof of Theorem 1.3 is concluded. $\square$

Remark 3.2

Concerning Corollary 1.2, observe that in the argument above we have esta-blished (1.7) for problem (1.5) on $\Omega ={\mathbb {R}}^n$ and a similar proof is in order for a general unbounded open set $\Omega$. Since $\gamma =0$, we deduce the validity of (1.8) in view of (1.13).

Proof (of Theorem 1.4)

Let v be a solution of (1.15). Applying Theorem 1.1 to the Kelvin transform ${{\hat{v}}}$, solution of

$$\begin{aligned} -\Delta _n {{\hat{v}}}=|x|^{-n(\alpha +2)} e^{{{\hat{v}}}} \hbox { in } {\mathbb {R}}^n {\setminus } \{0\}, \end{aligned}$$

we deduce the validity of (1.3)–(1.4) for v at $\infty$ with $-\gamma _\infty$ so that $\gamma _\infty > n^n |\alpha +1|^{n-2}(\alpha +1) \omega _n$. By Proposition 3.1 with $\gamma =0$ we deduce that $\gamma _\infty =\int _{{\mathbb {R}}^n} |x|^{n\alpha } e^v$ satisfies

$$\begin{aligned} n (\alpha +1) \gamma _\infty = \frac{n-1}{n} (n\omega _n)^{-\frac{1}{n-1}} \gamma _\infty ^{\frac{n}{n-1}} . \end{aligned}$$

Therefore, $\alpha > -1$ and

$$\begin{aligned} \int _{{\mathbb {R}}^n} |x|^{n \alpha } e^v=n\left( \frac{n^2}{n-1}\right) ^{n-1} (\alpha +1)^{n-1} \omega _n , \end{aligned}$$

concluding the proof of Theorem 1.4. $\square$

4 Radial solutions for (1.12)

Fix $M>1$ and assume that

$$\begin{aligned} \frac{1}{M}\le r_0 \le M, \quad \alpha _0 \le M, \quad \frac{1}{M}\le |\alpha _1| \le M. \end{aligned}$$

(4.1)

Let us first discuss the local existence theory for the following Cauchy problem:

$$\begin{aligned} \left\{ \begin{array}{l} -\frac{1}{r^{n-1}}(r^{n-1} |U'|^{n-2} U')'=e^U \\ \, U(r_0)=\alpha _0,\,\,\, U'(r_0)=\alpha _1. \end{array} \right. \end{aligned}$$

(4.2)

Given $0<\delta <\frac{1}{2M}$, define $I=[r_0-\delta ,r_0+\delta ]$ and $E=\{U \in C(I,[\alpha _0-1,\alpha _0+1]):\ U(r_0)=\alpha _0 \}$, which is a Banach space endowed with $\Vert \cdot \Vert _\infty$ as a norm. We can re-formulate (4.2) as $U=TU$, where

$$\begin{aligned} TU(r)= & {} \alpha _0+\int _{r_0}^r \frac{ds}{s} \Big | r_0^{n-1}|\alpha _1|^{n-2}\alpha _1\\&-\int _{r_0}^s t^{n-1} e^{U(t)} dt \Big |^{-\frac{n-2}{n-1}} \left( r_0^{n-1} |\alpha _1|^{n-2}\alpha _1-\int _{r_0}^s t^{n-1} e^{U(t)} dt\right) . \end{aligned}$$

In view of

$$\begin{aligned} |s^n-r_0^n| \le n (M+1)^{n-1} \delta \qquad \forall \, s \in I \end{aligned}$$

(4.3)

we have that $\displaystyle \max _I U \le M+1$ and $\displaystyle \max _I |\int _{r_0}^s t^{n-1} e^{U(t)} dt| \le e^{M+1} (M+1)^{n-1} \delta$ for all $U \in E$, and then for $0<\delta < \frac{e^{-M-1}}{2 (M+1)^{3n-3}}$ we have that

$$\begin{aligned} r_0^{n-1} |\alpha _1|^{n-2}\alpha _1-\int _{r_0}^s t^{n-1} e^{U(t)} dt \hbox { has the same sign as } \alpha _1 \,\, \forall s \in I \end{aligned}$$

(4.4)

and

$$\begin{aligned} \frac{1}{2 M^{2n-2}}\le & {} \frac{1}{2} r_0^{n-1} |\alpha _1|^{n-1} \le |r_0^{n-1} |\alpha _1|^{n-2}\alpha _1\nonumber \\&-\int _{r_0}^s t^{n-1} e^{U(t)} dt| \le \frac{3}{2} r_0^{n-1} |\alpha _1|^{n-1} \le \frac{3}{2} M^{2n-2} \end{aligned}$$

(4.5)

for all $U \in E$. Since $\log \frac{r_0+\delta }{r_0} \le \log \frac{r_0}{r_0-\delta } \le \frac{\delta }{r_0-\delta } \le 2 M \delta$ in view of $\delta <\frac{r_0}{2}$ and

$$\begin{aligned} ||x|^{-\frac{n-2}{n-1}}x-|y|^{-\frac{n-2}{n-1}}y| \le C_M |x-y| \quad \forall \, x,y \in {\mathbb {R}}: \, xy\ge 0, \, \min \{|x|,|y|\}\ge \frac{1}{2 M^{2n-2}} \end{aligned}$$

(for example, take $C_M=(1+\frac{n-2}{n-1})(4M^{2n-2})^{\frac{n-2}{n-1}}$), by (4.4)–(4.5) we have that

$$\begin{aligned} \Vert TU-\alpha _0 \Vert _{\infty ,I}\le & {} \sup _{r \in I} |\int _{r_0}^r \frac{ds}{s} \Big | r_0^{n-1}|\alpha _1|^{n-2}\alpha _1 -\int _{r_0}^s t^{n-1} e^{U(t)} dt \Big |^{\frac{1}{n-1}}| \\\le & {} 2 (\frac{3}{2})^{\frac{1}{n-1}} M^3 \delta \le 3 M^3 \delta \end{aligned}$$

and

$$\begin{aligned} \Vert TU-TV\Vert _{\infty ,I}\le & {} C_M \sup _{r \in I} |\int _{r_0}^r \frac{ds}{s} |\int _{r_0}^s t^{n-1} [e^{U(t)}-e^{V(t)}] dt| |\\\le & {} 2 C_M (M+1)^n e^{M+1} \delta \Vert U-V\Vert _{\infty ,I} \end{aligned}$$

for all $U,V \in C^1 (I)$ in view of $\delta <1$ and (4.3). In conclusion, if

$$\begin{aligned} 0<\delta <\min \left\{ \frac{1}{3M^3}, \frac{e^{-M-1}}{2 (M+1)^{3n-3}}, \frac{e^{-M-1}}{2C_M (M+1)^n} \right\} , \end{aligned}$$

(4.6)

then T is a contraction map from E into itself and a unique fixed point $U \in E$ of T is found by the Contraction Mapping Theorem, providing a solution U of (4.2) in $I=[r_0-\delta ,r_0+\delta ]$.

Once a local existence result has been established for (4.2), we can turn the attention to global issues. Given $r_0>0$, $\alpha _0$ and $\alpha _1 \not =0$, let $I=(r_1,r_2)$, $0\le r_1<r_0<r_2\le +\infty$, be the maximal interval of existence for the solution U of (4.2). We claim that $r_1=0$ when $\alpha _1>0$ and $r_2=+\infty$ when $\alpha _1<0$.

Consider first the case $\alpha _1>0$ and assume by contradiction $r_1>0$. Since

$$\begin{aligned} U'(r) =\frac{1}{r} \left( r_0^{n-1}\alpha _1^{n-1}+\int _r^{r_0} t^{n-1} e^{U(t)} dt \right) ^{\frac{1}{n-1}} \ge \frac{r_0 \alpha _1}{r}>0 \end{aligned}$$

(4.7)

for all $r \in (r_1,r_0]$, one would have that

$$\begin{aligned} U(r) \le \alpha _0,\quad \alpha _1 \le U'(r) \le \frac{1}{r_1} [r_0^{n-1} \alpha _1^{n-1}+\frac{r_0^n}{n} e^{\alpha _0}]^{\frac{1}{n-1}} \end{aligned}$$

for all $r\in (r_1,r_0]$ and then (4.1) would hold for initial conditions $\alpha _0'=U(r_0')$, $\alpha _1'=U'(r_0')$ in (4.2) at $r_0'$ approaching $r_1$ from the right. Since this would allow to continue the solution U on the left of $r_1$ in view of the estimate (4.6) on the time for local existence, we would reach a contradiction and then the property $r_1=0$ has been established.

In the case $\alpha _1<0$ assume by contradiction $r_2<+\infty$. Since

$$\begin{aligned} U'(r) =- \frac{1}{r} \left( r_0^{n-1}|\alpha _1|^{n-1}+\int _{r_0}^r t^{n-1} e^{U(t)} dt \right) ^{\frac{1}{n-1}} \le - \frac{r_0 |\alpha _1|}{r}<0 \end{aligned}$$

(4.8)

for all $r \in [r_0,r_2)$, one would have that

$$\begin{aligned} U(r) \le \alpha _0,\quad -\frac{1}{r_0} [r_0^{n-1} |\alpha _1|^{n-1}+\frac{r_2^n}{n} e^{\alpha _0}]^{\frac{1}{n-1}} \le U'(r) \le - \frac{r_0 |\alpha _1|}{r_2} \end{aligned}$$

for all $r\in [r_0,r_2)$ and then (4.1) would hold for initial conditions $\alpha _0'=U(r_0')$, $\alpha _1'=U'(r_0')$ in (4.2) at $r_0'$ approaching $r_2$ from the left. Since one could continue the solution U past $r_2$ thanks to (4.6), a contradiction would arise. Then, we have shown that $r_2=+\infty$.

Given $\epsilon >0$, let now $U_\epsilon ^\pm$ be the maximal solution of

$$\begin{aligned} \left\{ \begin{array}{l} -\frac{1}{r^{n-1}}(r^{n-1} |U'|^{n-2} U')'=e^U \\ \, U(1)=\alpha _0,\,\,\, U'(1)=\pm \epsilon . \end{array} \right. \end{aligned}$$

By the discussion above we have that $U_\epsilon ^+$ and $U_\epsilon ^-$ are well defined in (0, 1] and $[1,+\infty )$, respectively. According to (4.7)–(4.8) one has

$$\begin{aligned} (U_\epsilon ^+)'= & {} \frac{1}{r} \left( \epsilon ^{n-1}+\int _r^1 t^{n-1} e^{U_\epsilon ^+(t)} dt \right) ^{\frac{1}{n-1}} \hbox { in }(0,1],\nonumber \\ (U_\epsilon ^-)'= & {} - \frac{1}{r} \left( \epsilon ^{n-1}+\int _1^r t^{n-1} e^{U_\epsilon ^-(t)} dt \right) ^{\frac{1}{n-1}} \hbox { in }[1,+\infty ) \end{aligned}$$

(4.9)

and then $U_\epsilon ^+$, $U_\epsilon ^-$ are uniformly bounded in $C^{1,\gamma }_{loc} (0,1]$, $C^{1,\gamma }_{loc} [1,+\infty )$, respectively, in view of $U_\epsilon ^+,U_\epsilon ^- \le \alpha _0$. Up to a subsequence and a diagonal argument, we can assume that $U_\epsilon ^+ \rightarrow U^+$ in $C^1_{loc} (0,1]$ and $U_\epsilon ^- \rightarrow U^-$ in $C^1_{loc} [1,+\infty )$ as $\epsilon \rightarrow 0^+$, where

$$\begin{aligned} (U^+)'= & {} \frac{1}{r} \left( \int _r^1 t^{n-1} e^{U^+(t)} dt \right) ^{\frac{1}{n-1}} \hbox { in }(0,1],\nonumber \\ (U^-)'= & {} - \frac{1}{r} \left( \int _1^r t^{n-1} e^{U_-(t)} dt \right) ^{\frac{1}{n-1}} \hbox { in }[1,+\infty ) \end{aligned}$$

(4.10)

thanks to (4.9). Since $U^+(1)=U^-(1)=\alpha _0$ and $(U^+)'(1)=(U^-)'(1)=0$ in view of (4.10), we have that

$$\begin{aligned} U=\left\{ \begin{array}{ll} U^+ &{} \hbox {in }(0,1]\\ U^- &{} \hbox {in }[1,+\infty ) \end{array}\right. \end{aligned}$$

is in $C^1(0,+\infty )$ with $U\le U(1)=\alpha _0$, $U'(1)=0$ and

$$\begin{aligned} U'(r) =\frac{1}{r} \Big | \int _r^1 t^{n-1} e^{U(t)} dt \Big |^{-\frac{n-2}{n-1}} \int _r^1 t^{n-1} e^{U(t)} dt \hbox { in }(0,+\infty ). \end{aligned}$$

(4.11)

It is not difficult to check that U satisfies $-\Delta _n U=e^U$ in ${\mathbb {R}}^n {\setminus } \{0 \}$ and

$$\begin{aligned} \lim _{r \rightarrow 0} \frac{U(r)}{\log r}=\lim _{ r \rightarrow 0} rU'(r)=( \int _0^1 t^{n-1} e^{U(t)} dt )^{\frac{1}{n-1}}= \left( \frac{1}{n \omega _n} \int _{B_1(0)} e^U \right) ^{\frac{1}{n-1}} \end{aligned}$$

(4.12)

in view of (4.11). By Theorem 1.1 and (4.12) we deduce that U is a radial solution of

$$\begin{aligned} -\Delta _n U=e^U -\gamma \delta _0 \hbox { in } {\mathbb {R}}^n ,\quad U \le U(1)=\alpha _0, \end{aligned}$$

with $\gamma =\int _{B_1(0)} e^{U}$ depending on the choice of $\alpha _0$. By the Pohozaev identity (3.2) on $A=B_1(0) {\setminus } B_\epsilon (0)$, $\epsilon \in (0,1)$, we have that

$$\begin{aligned} \omega _n[e^{\alpha _0}- \epsilon ^n e^{U(\epsilon )}]=\int _{B_1(0){\setminus } B_\epsilon (0)} e^U +\frac{n-1}{n} \omega _n [\epsilon U'(\epsilon )]^n \end{aligned}$$

in view of $U(1)=\alpha _0$ and $U'(1)=0$, and letting $\epsilon \rightarrow 0^+$ one deduces that

$$\begin{aligned} \omega _n e^{\alpha _0}= \gamma +\frac{n-1}{n} \omega _n \left( \frac{\gamma }{n \omega _n} \right) ^{\frac{n}{n-1}} \end{aligned}$$

in view of (4.12). Since $\gamma \in (0,+\infty ) \rightarrow \gamma +\frac{n-1}{n} \omega _n \left( \frac{\gamma }{n \omega _n} \right) ^{\frac{n}{n-1}} \in (0,+\infty )$ is a bijection, for any given $\gamma >0$ let $\alpha _0= \log [\frac{\gamma }{\omega _n}+\frac{n-1}{n} ( \frac{\gamma }{n \omega _n})^{\frac{n}{n-1}}]$ and the corresponding U is the solution of (1.12) we were searching for. Notice that $\int _{{\mathbb {R}}^n} e^U<+\infty$ in view of $\int _1^\infty t^{n-1} e^{U(t)} dt<+\infty$, as it can be deduced by

$$\begin{aligned} \lim _{r \rightarrow +\infty } \frac{U(r)}{\log r}=\lim _{ r \rightarrow +\infty } rU'(r)=-\left( \int _1^\infty t^{n-1} e^{U(t)} dt \right) ^{\frac{1}{n-1}} \end{aligned}$$

due to (4.11). We have established the following result:

Theorem 4.1

For any $\gamma >0$ there exists a 1-parameter family of distinct solutions $U_\lambda$, $\lambda >0$, to (1.12) given by $U_\lambda (x)=U(\lambda x)+n \log \lambda$ such that $U_\lambda$ takes its unique absolute maximum point at $\frac{1}{\lambda }$.