1 Introduction
In this paper, we consider the following family of degenerate elliptic equations with a parameter
$h> 1$
:
$$ \begin{align} \begin{cases} \Delta_{\infty}^h u=u^q &{\mathrm{in}}\; \Omega, \\ u=\infty &\mathrm{on} \; \partial\Omega, \end{cases} \end{align} $$
where q is a given constant and
$$ \begin{align*} \Delta_{\infty}^h u:= |Du|^{h-3}\langle D^2uDu,Du \rangle =|Du|^{h-3}\sum_{i,j=1}^{n} u_{x_i}u_{x_j}u_{x_ix_j} \end{align*} $$
denotes the h-homogeneous nonlinear operator. Due to the high degeneracy of the operator
$\Delta _{\infty }^h u$
, the associated problems do not have smooth solutions in general. Therefore, solutions are understood in the viscosity sense (see Section 2 for the precise definition). Notice that the h stands for the homogeneous degree of the operator. Throughout this paper, we denote
$\Omega $
to be a bounded domain in the Euclidean space
$\mathbb {R}^n (n\geq 2) $
and
$\partial \Omega $
to be its boundary.
The boundary condition in (1-1) is understood in the following sense:
$$ \begin{align*} \lim\limits_{x\rightarrow z}u(x)=\infty, \quad z\in \partial\Omega, \end{align*} $$
and the solution of (1-1) is called the ‘boundary blow-up solution’, ‘large solution’, or ‘explosive solution’ due to the explosive boundary condition. The motivation for the name is as follows: if
$ U$
is a boundary blow-up solution, the comparison principle (Theorem 2.3) implies that any solution V to
$\Delta _{\infty }^h u=u^q$
in
$\Omega $
with bounded boundary data satisfies
$V(x)\leq U(x)$
,
$x\in \Omega $
. Hence, the boundary blow-up solution provides local uniform bounds for all other solutions in the domain
$\Omega $
, regardless of the boundary data.
The boundary blow-up problem (1-1) has been studied in two special cases,
${h = 1}$
[Reference Juutinen and Rossi19] and
$h = 3$
[Reference Mohammed and Mohammed29]. For
$h = 1$
,
$\Delta _{\infty }^h u$
is the
$1$
-homogeneous normalized infinity Laplacian operator,
$$ \begin{align*} \Delta_{\infty}^N u:= |Du|^{-2}\langle D^2uDu,Du \rangle. \end{align*} $$
For
$h = 3$
,
$\Delta _{\infty }^h u$
is the
$3$
-homogeneous infinity Laplacian operator,
$$ \begin{align*} \Delta_{\infty} u:= |Du|^{ 2}\Delta_{\infty}^N u. \end{align*} $$
And for other h, we have
$$ \begin{align*} \Delta_{\infty}^h u= |Du|^{ h-3}\Delta_{\infty} u=|Du|^{ h-1}\Delta_{\infty}^N u. \end{align*} $$
The infinity Laplacian
$\Delta _{\infty } $
was first introduced by Aronsson [Reference Aronsson2] in the 1960s in connection with the geometric problem of finding the so-called absolutely minimizing Lipschitz extension. For more properties of the infinity harmonic functions (the viscosity solution to
$\Delta _{\infty } u=0$
), one can see the works of Crandall [Reference Crandall10], Crandall et al. [Reference Crandall, Evans and Gariepy11], Aronsson et al. [Reference Aronsson, Crandall and Juutinen3], and the references therein.
For the inhomogeneous equation
$$ \begin{align*} \Delta_{\infty} u=f(x), \end{align*} $$
Lu and Wang [Reference Lu and Wang26] proved the existence and uniqueness of a viscosity solution of the Dirichlet problem when the inhomogeneous term
$ f$
does not change its sign. In [Reference Bhattacharya and Mohammed4, Reference Bhattacharya and Mohammed5], Bhattacharya and Mohammed studied the existence or nonexistence of viscosity solutions to the Dirichlet problem
$$ \begin{align*} \begin{cases} \Delta_{\infty} u=f(x,u) \quad &\mathrm{in}\; \Omega, \\ u=g \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$
under some conditions for f and
$g.$
The normalized version
$\Delta _{\infty }^N u$
was first introduced by Peres, Schramm, Sheffield, and Wilson from the point of randomized theory named Tug-of-War [Reference Peres, Schramm, Sheffield and Wilson33]. Now let us briefly recall the random-turn,
$\varepsilon $
-tug-of-war game. This is a zero sum game with two players in which the earnings of one of them are the losses of the other. Given a step size
$\varepsilon>0, $
let
$f\in C(\Omega )$
be a running payoff function and
$g\in C(\partial \Omega )$
be a final payoff function. The starting position is
$x_0\in \Omega $
. At the k th step, a fair coin is tossed, and the player who wins the toss may move the token from
$x_{k-1}$
to any
$x_k$
with
$|x_k - x_{k-1}| < \varepsilon .$
The game ends when
$x_m \in \partial \Omega ,$
and player II pays to player I the amount
$$ \begin{align*} \text{Payoff} = g(x_m)+\sum\limits_{i=1}^m f(x_{i-1}). \end{align*} $$
If the token never reaches
$\partial \Omega $
and the game thus fails to terminate, each of the players must pay a fine of
$+\infty .$
The value function
$V_I (x_0)$
for Player I is, roughly, the minimum that Player I can expect to win by playing optimally, while the value function
$V_{II} (x_0) $
is the maximum that Player II can expect to be required to pay, by playing optimally.
According to the dynamic programming principle, one gets
$V_I (x)=V_{II} (x) $
and
$V_I$
satisfies
$$ \begin{align*} \begin{cases} \tfrac{1}{2}(u(x)-\min\limits_{B(x,\varepsilon)\cap \Omega}u(y))- \tfrac{1}{2}(\max\limits_{B(x,\varepsilon)\cap \Omega}u(y)-u(x))=f(x) &\mathrm{in} \;\Omega,\\ u=g &\mathrm{on} \;\partial\Omega. \end{cases} \end{align*} $$
Furthermore, the corresponding expectation value function
$u^{\varepsilon }$
of this game exists and converges (as
$\varepsilon \rightarrow 0$
) to a function u which is the unique solution of the problem
$$ \begin{align*} \begin{cases} \Delta_{\infty}^N u=f(x) &\mathrm{in}\; \Omega, \\ u=g &\mathrm{on} \;\partial\,\Omega. \end{cases} \end{align*} $$
For more stochastic games related to the normalized infinity Laplacian, one can consult the papers [Reference Liu21, Reference Liu and Schikorra23, Reference López-Soriano, Navarro-Climent and Rossi24, Reference Peres, Pete and Somersille32, Reference Rossi35], and so on. The normalized infinity Laplacian equation was also well studied by Lu and Wang based on partial differential equation methods in [Reference Lu and Wang27]. Notice that
$\Delta _{\infty }^N u$
is not only degenerate but also singular when the gradient of u vanishes. In the last three decades, the infinity Laplacian has received much attention because it is not only highly degenerate but also has many applications in image processing [Reference Abderrahim, Xavier, Zakaria and Olivier1, Reference Caselles, Morel and Sbert6, Reference Elmoataz, Toutain and Tenbrinck13] and optimal mass transportation problems [Reference Evans and Gangbo14].
In [Reference Liu and Yang22], Liu and Yang established the existence of the viscosity solutions for the Dirichlet problem of the inhomogeneous equation
$$ \begin{align*} \Delta_{\infty}^h u(x)=f(x). \end{align*} $$
In [Reference Portilheiro and Vázquez34], Portilheiro and Vázquez studied the parabolic version of the operator
$\Delta _{\infty }^h.$
They proved the existence and uniqueness of viscosity solutions for the initial-Dirichlet boundary problem
$$ \begin{align*} \begin{cases} u_t- \Delta_{\infty}^h u=0 \quad&{\mathrm{in}}\; Q=:\Omega\times(0,T), \\ u=g \quad &{\mathrm{on}} \;\partial_p Q. \end{cases} \end{align*} $$
They also established the asymptotic behavior of the viscosity solution for the problem posed in the whole space.
The boundary blow-up solutions to the elliptic equations have many applications in stochastic differential processes [Reference Lasry and Lions20], population dynamics [Reference Fraile, Medina, López-Gómez and Merino15, Reference Ouyang31], and the equilibrium state of charged gas in a container [Reference García-Melián, Gómez-Reñasco, López-Gómez and Sabinade16], and so on. Specifically, Lasry and Lions [Reference Lasry and Lions20] considered a stochastic control problem
$$ \begin{align*} dX_t=a(X_t)\,dt+dB_t,\quad X_0=x\in \Omega,\quad t>0, \end{align*} $$
where the state of the controlled system is a diffusion process,
$B_t$
is a standard Brownian motion, a is the control process,
$X_t$
is the state process, and
$P(X_t\in \partial \Omega )>0.$
To get the state constraints (that is, Brownian motion in a bounded domain), one needs to use the unbounded drifts a. In other words, one will have to choose feedback controls that push back the state process inside
$\Omega $
when it gets near
$\partial \Omega $
. And the intensity of the state blows up at the boundary of
$\Omega $
. Therefore, they defined the class
$\mathcal {A}$
of such feedback controls. And for each
$a\in \mathcal {A}$
, they considered the cost function
$$ \begin{align*} J(x,a)=E\bigg\lbrace\int_{0}^{\infty}\bigg[ f(X_t)+\frac{1}{q}|a(X_t)|^q\bigg] e^{-\lambda t}dt\bigg\rbrace\quad \text{for all } x\in\Omega, \end{align*} $$
where E denotes the expectation,
$q> 1$
, f is a given function on
$\Omega $
which is bounded from below and the positive parameter
$\lambda $
denotes the so-called discount factor. Then, by the dynamic programming principle and probability analysis, they showed that the minimum u (called the value function) of the function J
$$ \begin{align*} u(x)=\underset{a\in\mathcal{A}}{\inf} J(x,a) \quad \text{for all } x\in\Omega, \end{align*} $$
satisfies the following boundary blow-up problem
$$ \begin{align*} \begin{cases} -\dfrac{1}{2}\Delta u +\dfrac{1}{p}|D u|^p +\lambda u =f(x), \quad & x \in \Omega, \\ u(x)=\infty, \quad & x \in\partial\,\Omega, \end{cases} \end{align*} $$
where
$p={q}/({q-1})$
and the optimal feedback control
$a(\cdot )=-|D u|^{p-2} D u(\cdot ).$
Note that for
$h=1,$
the operator
$\Delta _{\infty }^h$
shares the same structure with the standard Laplacian for 1-dimension. Hence, it is meaningful to consider the boundary blow-up problem of
$\Delta _{\infty }^h.$
In the present work, we are interested in the boundary blow-up problem (1-1). The h-degree homogeneous infinity Laplacian equation (1-1) is of intrinsic interest, because it is not only strongly degenerate, but also has no variational structure and divergence form.
Our main results are summarized as follows.
Theorem 1.1. Let
$ \Omega \subset \mathbb {R}^n$
be a bounded domain. If
$q>h,$
then there exists a positive viscosity solution to the boundary blow-up problem (1-1).
When the domain possesses
$C^1$
regularity, we can establish the following asymptotic estimate near the boundary and uniqueness of large solutions to (1-1).
Theorem 1.2. Let
$ \Omega \subset \mathbb {R}^n$
be a bounded domain. Assume that there exists a neighborhood
$\mathcal {N}$
of
$ \partial \Omega $
such that
${\mathrm {dist}}(x,\partial \Omega )\in C^1(\mathcal {N}\cap \Omega )$
and
$q>h,$
then the viscosity solution u to (1-1) satisfies the precise boundary behavior
$$ \begin{align} u(x)\sim \bigg( \bigg( \frac{h+1}{q-h}\bigg) ^h \frac{q+1}{q-h}\bigg)^{{1}/({q-h})} {\mathrm{dist}}( x,\partial\,\Omega)^{-({h+1})/({q-h})} \quad \ \mathrm{as } \; {\mathrm{dist}}( x,\partial\,\Omega)\rightarrow 0. \end{align} $$
Furthermore, the viscosity solution of (1-1) is unique.
Theorem 1.3. Let
$ \Omega \subset \mathbb {R}^n$
be a bounded domain. If
$q\leq h,$
then the boundary blow-up problem (1-1) has no positive solution.
In [Reference Juutinen and Rossi19], Juutinen and Rossi established the existence, uniqueness, and asymptotic behavior for the large solutions of the normalized infinity Laplacian
$$ \begin{align*} \begin{cases} \Delta^N_{\infty} u=u^q \quad &{\mathrm{in}}\; \Omega, \\ u=\infty \quad &\mathrm{on} \; \partial\,\Omega, \end{cases} \end{align*} $$
for
$q>1$
. In [Reference Mohammed and Mohammed29, Reference Mohammed and Mohammed30], Mohammed and Mohammed studied the boundary blow-up solutions of the infinity Laplacian
$$ \begin{align} \begin{cases} \Delta_{\infty} u=b(x)f(u) \quad &{\mathrm{in}}\; \Omega, \\ u=\infty \quad &\mathrm{on} \; \partial\,\Omega, \end{cases} \end{align} $$
where
$ b\in C(\overline {\Omega })$
is nonnegative,
$ f\in C[0,\infty )\cap C^1(0,\infty )$
,
$f(0)=0$
,
$f(s)>0$
,
$ s>0$
, and
$f(s)$
is nondecreasing on
$(0,\infty ].$
They proved that the boundary blow-up problem of (1-3) has a nonnegative viscosity solution if the following Keller–Osserman-type condition holds:
$$ \begin{align*} \varPsi(r):=\int_r^{\infty}\frac{d\tau}{{4F(\tau)}^{{1}/{4}}}<\infty\quad \text{for all } r>0, \end{align*} $$
where
$F(\tau )=\int _0^{\tau } f(\nu )\,d\nu .$
Most recently, Wang et al. [Reference Wang, Gong and Zheng36] studied the second-order asymptotic behavior of boundary blow-up viscosity solutions (1-3) based on Karamata regular variation theory which was first introduced by Cîrstea and Rǎdulescu in a stochastic process to study the boundary behavior and uniqueness of solutions to boundary blow-up elliptic problems. A series of rich and significant information about the boundary behavior of solutions was obtained based on such theory [Reference Cîrstea and Rǎdulescu7–Reference Cîrstea and Rǎdulescu9]. In [Reference Zhang37], under appropriate structure conditions on the nonlinear term
$f,$
Zhang established the following the boundary estimate of large solutions to problem (1-3):
$$ \begin{align*} \lim\limits_{d(x)\rightarrow 0}\frac{u(x)}{\psi(K(d(x)))}=1, \end{align*} $$
where
$k\in C^1$
is positive and nondecreasing,
$K(t)=\int _0^t k(s)\,ds$
satisfies
$$ \begin{align*} \lim\limits_{t\rightarrow 0^+} \frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)=C_k, \end{align*} $$
and
$\psi $
satisfies
$$ \begin{align*} \int_{\psi(t)}^{\infty}\frac{ds}{(4F(s))^{{1}/{4}}}=t. \end{align*} $$
In [Reference Mi28], the boundary behavior of the boundary blow-up viscosity solutions to problem (1-3) was studied under different conditions on the weight function
$b(x)$
and the nonlinear term
$f.$
To obtain the existence of boundary blow-up viscosity solutions of (1-1), we first establish the comparison principle and then combine Perron’s method with compactness arguments. Due to the strong degeneracy of the operator
$\Delta ^h_{\infty } $
and the boundary blow-up condition, it is difficult to study the comparison principle for viscosity solutions of (1-1). To overcome this difficulty, we employ the perturbation method and the logarithmic transformation of functions so that the double variables method can be carried out in the usual way. Then we consider the approximate problems with boundary condition
$u=M$
, where
$M\geq 1$
is a constant. To establish the existence of large solutions, a difficulty with respect to the degenerate operators is the lack of the existence of barriers. Thanks to the particular structure of
$\Delta ^h_{\infty }, $
we can construct ‘good’ barriers and use the standard Perron method to get the existence of approximate solutions. Finally, based on compactness analysis, we establish that the limit function of the approximate solutions is the desired boundary blow-up solution. To conclude that the limit is finite, we use again the comparison principle with a radial large solution obtained by analyzing the corresponding ordinary differential equation.
One should notice that if the regularity assumption of Theorem 1.2 holds, then the distance function is a solution of
$\Delta _{\infty }^h v=0$
near the boundary. Therefore, we can perturb the distance function to analyze the asymptotic behavior near the boundary. Based on the asymptotic estimates and the comparison principle, the uniqueness result of the viscosity solution follows immediately. Let us point out that, unlike the case
$h=1$
, the operator
$ \Delta _{\infty }^h$
is quasi-linear even in dimension 1. Therefore, we must make a subtle analysis.
Due to the high degeneracy of
$\Delta _{\infty }^h ,$
we employ the logarithmic transformation of functions and comparison principle to obtain the nonexistence of the large solution of (1-1) for the case
$q\leq h.$
The outline of this paper is as follows. In Section 2, we give the definition of the viscosity solution to (1-1) and prove the comparison principle for the equation
${\Delta _{\infty }^h u=u^q}$
based on the perturbation method of viscosity solutions. In Section 3, based on the comparison principle, we establish the existence of viscosity solutions to (1-1) by Perron’s method and compactness analysis for
$q>h$
. In Section 4, under some regular assumption of the domain, we give the characteristic of the boundary blow-up solution near the boundary, and then by the comparison principle, we obtain the uniqueness of solutions to (1-1). Finally, in Section 5, we give the nonexistence of viscosity solutions to (1-1) for
$q \leq h$
.
2 Comparison principles
In this section, we first give the definition of the viscosity solutions to the equation
$$ \begin{align} \Delta_{\infty}^h u=u^q \quad \mathrm{in}\; \Omega, \end{align} $$
where
$q>0,$
and then establish the comparison results by the double variables method based on the viscosity solutions theory. Notice that the operator
$\Delta _{\infty }^h $
is singular for
$1<h<3$
at points where the gradient of the function vanishes. To give a reasonable explanation when the gradient vanishes, we use the definition of viscosity solutions based on semi-continuous extension and we refer the reader to [Reference Crandall, Ishii and Lions12, Reference Liu and Yang22, Reference Lu and Wang27], and so on. In fact, the singularity is removable when
$h>1$
since we are not considering here
$h=1.$
Therefore, we can write (2-1) as
$$ \begin{align*} {F}_h (D^2 u, Du)=u^q\quad \mathrm{in}\; \Omega, \end{align*} $$
where
$F_h:\mathbb {S}\times (\mathbb {R}^n\backslash \{0\})\rightarrow \mathbb {R}$
and
$F_h(M, p) := |p|^{h-3}(Mp)\cdot p.$
Here
$ \mathbb {S}$
denotes the set of
$n\times n$
real symmetric matrices. Since
$h>1,$
we have
$\lim \nolimits _{p\rightarrow 0} F_h(M, p)=0 $
for arbitrary
$M\in \mathbb {S}. $
Hence, we can define the continuous extension of
$F_h$
as follows:
$$ \begin{align*} \overline{F}_h (M, p) := \begin{cases} F_h(M, p) \; &\mathrm{if }\, p\neq 0, \\ 0\quad \quad &\mathrm{if }\, p= 0. \end{cases} \end{align*} $$
We remark that due to the strong degeneracy of (2-1), it is not clear that the functions that one would like to call solutions are actually differentiable even once.
Definition 2.1. Suppose that
$u: \Omega \rightarrow \mathbb {R}$
is an upper semi-continuous and nonnegative function. If, for every
$ x_0 \in \Omega $
and test function
$\varphi \in C^{2 }(\Omega )$
such that
$u(x_0 )=\varphi (x_0)$
and
$u(x )\leq \varphi (x )$
for all
$x \in \Omega $
near
$ x_0,$
there holds
$$ \begin{align*} \overline{F}_h (D^2 \varphi(x_0 ),D \varphi(x_0 )) \geq \varphi(x_0 )^q, \end{align*} $$
then we say u is a viscosity subsolution of (2-1).
Similarly, suppose that
$u: \Omega \rightarrow \mathbb {R}$
is a lower semi-continuous and nonnegative function. If, for every
$ x_0 \in \Omega $
and test function
$\varphi \in C^{2 }(\Omega )$
such that
$u(x_0)=\varphi (x_0 )$
and
$u(x )\geq \varphi (x )$
for all
$x \in \Omega $
near
$ x_0 $
, there holds
$$ \begin{align*} \overline{F}_h (D^2 \varphi(x_0 ),D \varphi(x_0 ))\leq \varphi(x_0)^q, \end{align*} $$
then we say u is a viscosity supersolution of (2-1).
If a continuous function u is both a viscosity subsolution and a viscosity supersolution of (2-1), then we say u is a viscosity solution of (2-1).
Remark 2.2. Let
$u,v\in C(\Omega )$
. The two statements below follow easily from the definition of a viscosity sub (super)-solution of (2-1).
Because of the degeneracy at the points where the gradient vanishes and the explosive boundary condition, the general comparison result stated in [Reference Crandall, Ishii and Lions12] does not apply to (2-1). Now we present a comparison principle which is proved via a perturbation argument.
Theorem 2.3. Let
$q\geq h$
. Suppose that u and
$ v $
are nonnegative continuous functions defined in a bounded domain
$\Omega $
and satisfy
$$ \begin{align*}\Delta_{\infty}^h u(x)\ge u(x)^q\quad \text{for all } x\in \Omega\end{align*} $$
and
$$ \begin{align*}\Delta_{\infty}^h v(x) \le v(x) ^q \quad \text{for all } x\in \Omega \end{align*} $$
in the viscosity sense. If
$$ \begin{align} \lim\sup\frac{u( x) }{v( x) }\leq1 \quad \;{\mathrm{as}}\; {\mathrm{dist}} (x,\partial\,\Omega) \rightarrow 0, \end{align} $$
then we have
$ u\leq v $
in
$\Omega $
.
One should notice that assumption (2-2) implies that
$v> 0$
near
$\partial \,\Omega $
so as to make the ratio
${u}/{v}$
well defined.
Proof. We argue it by contradiction. Suppose that
$$ \begin{align*} \sup\limits_{\Omega}{(u-v)}>0. \end{align*} $$
Then, due to (2-2), there exist
$\varepsilon>0$
small enough and
$x_{0} \in \Omega $
such that
$$ \begin{align} M:=u(x_0)-v_{\varepsilon}(x_0)=\sup\limits_{x\in\Omega}(u(x)-v_{\varepsilon}(x))>0, \end{align} $$
where
$ v_{\varepsilon }:=(1+\varepsilon )v.$
Notice that assumptions (2-2) and (2-3) also imply that there exists an open set
$\Omega _0\subset \subset \Omega $
such that
$x_0\in \Omega _0$
and
$$ \begin{align*}M=\sup\limits_{x\in\Omega_0}(u(x)-v_{\varepsilon}(x))> \sup\limits_{x\in\partial\,\Omega_0}(u(x)-v_{\varepsilon}(x)).\end{align*} $$
Since
$q\geq h $
and v is a viscosity supersolution of (2-1), one can verify that
$$ \begin{align*} \Delta_{\infty}^h v_{\varepsilon} =(1+\varepsilon)^h\Delta_{\infty}^h v \leq (1+\varepsilon)^hv^q=(1+\varepsilon)^{h-q}v_{\varepsilon}^q\leq v_{\varepsilon}^q \end{align*} $$
in the viscosity sense, that is,
$v_{\varepsilon } $
is also a viscosity supersolution of (2-1).
Based on the ideas in [Reference Crandall, Ishii and Lions12], we double the variables
$$ \begin{align*} w_j(x ,y ) := u(x )-v_{\varepsilon}(y )-\frac{j}{4}|x-y|^4, \quad (x,y)\in \Omega_0\times\Omega_0,\;j=1,2,\ldots. \end{align*} $$
We denote the maximum point of
$w_j$
over
$ \overline {\Omega }_0 \times \overline {\Omega }_0$
by
$(x_j,y_j )$
and
$ M_j:=w_j(x_j,y_j )$
. According to Proposition 3.7 in [Reference Crandall, Ishii and Lions12], we have
$$ \begin{align*}\lim \limits _{j\rightarrow \infty}{M_j}=\lim \limits _{j\rightarrow \infty}(u(x_j )-v_{\varepsilon}(y_j )-{j}|x_j-y_j|^4/{4}) =M \end{align*} $$
and
$$ \begin{align*} \lim \limits _{j\rightarrow \infty}j|x_j-y_j|^4/{4}=0. \end{align*} $$
Furthermore,
$x_j\rightarrow x_0$
,
$y_j\rightarrow x_0$
as
$j\rightarrow \infty .$
It is obvious that
$M=u(x_0)-v_{\varepsilon }(x_0).$
Due to
$M>\sup \nolimits _{\partial \,\Omega _0}(u-v_{\varepsilon }),$
we have
$x_j$
,
$y_j$
are interior points of
$\Omega _0$
for j large enough.
Set
$$ \begin{align*} \psi(x )=j|x-y_j|^4/4 , \quad \phi(y )=-j|x_j-y|^4/4. \end{align*} $$
It is clear that the functions
$u-\psi $
and
$v_{\varepsilon }-\phi $
have a local maximum at
$ x_j $
and a local minimum at
$y_j$
. We consider the two cases: either
$x_j\ne y_j$
or
$x_j= y_j$
for j large enough.
Case 1: If
$x_j= y_j$
, we have
$D\psi (x_j )=0 $
and
$D^2\psi (x_j )=0$
. Since u is a viscosity subsolution, we have
$$ \begin{align} 0\ge u^q(x_j)=\psi^q(x_j). \end{align} $$
By (2-3), we have
$$ \begin{align} u(x_j)\geq u(x_j)-v_{\varepsilon}(y_j)\geq u(x_j)-v_{\varepsilon}(y_j)-\frac{j}{4}|x_j-y_j|^4=w_j(x_j ,y_j ) \geq u(x_0)-v_{\varepsilon}(x_0)>0. \end{align} $$
Obviously, (2-5) contradicts (2-4).
Case 2: If
$x_j\neq y_j$
, we use jets and the maximum principle for semi-continuous functions, see [Reference Crandall, Ishii and Lions12]. Now we recall the definitions of super-jets and sub-jets. The second-order super-jet of an upper semi-continuous function
$\gamma $
at
$x_0\in \Omega $
is the set
$$ \begin{align*} \mathcal{J}^{2,+}\gamma(x_0)=\{(D\varphi(x_0),D^2\varphi(x_0)) : \varphi\in C^2(\Omega)\; {\mathrm{and}} \; \gamma-\varphi \; {\mathrm{has\; a\; local\; maximum \; at\; }} x_0 \}, \end{align*} $$
and its closure is
$$ \begin{align*} \overline{\mathcal{J}}^{2,+}\gamma(x_0):=& \{(p,M)\in \mathbb{R}^n\times \mathbb{S} :\text{there exists } (x_i,p_i,M_i)\in \Omega\times\mathbb{R}^n\times \mathbb{S}\\ & {\mathrm{such \; that\;}} (p_i, M_i)\in \mathcal{J}^{2,+}\gamma(x_i) \;{\mathrm{and}} \; (x_i,\gamma(x_i),p_i,M_i)\rightarrow (x_0,\gamma(x_0), p, M) \}. \end{align*} $$
Here
$ \mathbb {S}$
denotes the set of
$n\times n$
real symmetric matrices. Similarly, the second-order sub-jet of a lower semi-continuous function
$\gamma $
at
$x_0\in \Omega $
is the set
$$ \begin{align*} \mathcal{J}^{2,-}\gamma(x_0)=\{(D\varphi(x_0),D^2\varphi(x_0)) : \varphi\in C^2(\Omega)\;{\mathrm{ and}} \; \gamma-\varphi \;{\mathrm{ has\; a\; local\; minimum \; at\; }} x_0 \}, \end{align*} $$
and its closure is
$$ \begin{align*} \overline{\mathcal{J}}^{2,-}\gamma(x_0):=& \{(p,M)\in \mathbb{R}^n\times \mathbb{S} :\text{ there exists } \;(x_i,p_i,M_i)\in \Omega\times\mathbb{R}^n\times \mathbb{S}\\ &{\mathrm{ such \; that\; }}(p_i, M_i)\in \mathcal{J}^{2,-}\gamma(x_i) \;{\mathrm{and}} \; (x_i,\gamma(x_i),p_i,M_i)\rightarrow (x_0,\gamma(x_0), p, M) \}. \end{align*} $$
By the maximum principle for semi-continuous functions in [Reference Crandall, Ishii and Lions12], there exist symmetric matrices
$X_j$
,
$ Y_j\in \mathbb {S}$
such that
$Y_j-X_j \ge 0$
and
$$ \begin{align*} (\,j|x_j-y_j|^2(x_j-y_j), X_j) \in \overline{\mathcal{J}}^{2,+} u(x_j ), \end{align*} $$
$$ \begin{align*} (\,j|x_j-y_j|^2(x_j-y_j), Y_j) \in \overline{\mathcal{J}}^{2,-} v_{\varepsilon}(y_j). \end{align*} $$
For simplicity, we denote
$p_j=j|x_j-y_j|^2(x_j-y_j).$
Again since u and
$v_{\varepsilon }$
are a viscosity subsolution and supersolution, respectively, we have
$$ \begin{align*} \begin{array}{ll} 0\!\!\!\!&\leq|p_j|^{h-3} \langle X_jp_j,p_j \rangle -u^q(x_j)\\ &\leq|p_j|^{h-3} \langle Y_jp_j,p_j \rangle -v_{\varepsilon}^q(y_j)+v_{\varepsilon}^q(y_j)-u^q(x_j)\\ &\leq v_{\varepsilon}^q(y_j)-u^q(x_j), \end{array} \end{align*} $$
where we use
$Y_j-X_j \ge 0$
. Passing to the limit and noting
$q\geq h>1$
, we obtain
$v_{\varepsilon }(x_0)\geq u(x_0),$
which is a contradiction to (2-3).
Remark 2.4. In the above proof, the assumption
$q\geq h$
was used to guarantee that
$(1+\varepsilon )v$
is also a supersolution if v is, and the rest of the argument does still work for all
$ q> 0.$
Therefore, we can also establish the following comparison result which will be needed to establish the existence of the large solution in Section 3. We leave its proof to the reader.
Theorem 2.5. Let
$q>0.$
Suppose that u and
$ v $
are nonnegative continuous functions defined in the closure of a bounded domain
$\Omega $
, and satisfy
$$ \begin{align*}\Delta_{\infty}^h u(x)\ge u(x)^q\quad \text{for all } x\in \Omega\end{align*} $$
and
$$ \begin{align*}\Delta_{\infty}^h v(x) \le v(x) ^q \quad \text{for all } x\in \Omega \end{align*} $$
in the viscosity sense. Then
$u\leq v$
on
$\partial \,\Omega $
implies
$ u\leq v $
in
$\Omega $
.
3 Existence of boundary blow-up solutions for
$q>h$
In this section, we focus on the existence of boundary blow-up solutions to (1-1) for the case
$q>h.$
First, we show certain radial solutions to (1-1) on a ball. Then we establish the existence of large solutions of (1-1) based on Perron’s method and the compactness argument.
First, we consider the following boundary blow-up problem in the ball
$ B_R(x_0)$
:
$$ \begin{align} \begin{cases} \Delta_{\infty}^h u=u^q &{\mathrm{in}}\; B_R(x_0), \\ u=\infty &{\mathrm{on}} \;\partial B_R(x_0), \end{cases} \end{align} $$
where
$q>h.$
We look for solutions of the form
$u(x) = g(r),$
where
$r=|x-x_0|.$
By direct calculation, we get
$$ \begin{align*}Du(x )= g'(r)\frac{x-x_0}{|x-x_0|}\end{align*} $$
and
$$ \begin{align*}D^2u(x )= \bigg[g"(r)\frac{(x-x_0)\otimes(x-x_0)}{|x-x_0|^2}+g'(r)\frac{1}{|x-x_0|}I-g'(r)\frac{(x-x_0)\otimes(x-x_0)}{|x-x_0|^3}\bigg]\end{align*} $$
for
$x\neq x_0,$
where
$\otimes $
denotes the tensor product. Then u satisfies
$$ \begin{align} \Delta_{\infty}^h u=u^q,\quad x\in B_R(x_0) \backslash \{x_0\}, \end{align} $$
if and only if
$$ \begin{align} |g'(r)|^{h-1}g"(r)=g^q(r),\quad r\in(0,R), \end{align} $$
which is a 1-dimensional case for (3-2).
Now we consider the nonnegative solution of the following initial value problem:
$$ \begin{align} \begin{cases} |g'(r)|^{h-1}g"(r)=g^q(r),\quad r\in (0,R), \\ g(0)=\tau,\\ g'(0)=0, \end{cases} \end{align} $$
where
$ \tau>0 $
and
$0<R< \infty $
. Our aim is to find a solution satisfying (3-4) and
$$ \begin{align} \lim\limits _{r\rightarrow R} g(r)= +\infty. \end{align} $$
Lemma 3.1. Let
$0 < R < \infty $
and
$q> h > 1$
be given. Then, there exists
$ \tau>0 $
(depending on R,
$ q$
, and h) such that problem (3-4) admits a nonnegative solution
$g\in C^2( ( 0,R) ) \cap C^1( [ 0,R) ) $
satisfying (3-5).
Proof. Notice that the above discussion implies
$g\in C^2( ( 0,R) ) \cap C^1( [ 0,R) ). $
Since
$g^q $
is nonnegative and
${g'(0) = 0}$
, we integrate (3-3) on
$(0,r)$
, and then we get
$g' $
is nonnegative in
$(0,R)$
. Next we multiply both sides of the differential equation (3-3) by
$g' $
and integrate on
$(0,r)$
. We get
$$ \begin{align*} g'(r)= \bigg[ \frac{h+1}{q+1}( g(r)^{q+1}-g(0)^{q+1}) \bigg] ^{{1}/({h+1})}. \end{align*} $$
Therefore, an implicit solution of the initial value problem (3-4) is
$$ \begin{align} \int^{g(r)}_{\tau}\frac{ds}{(s^{q+1}-\tau^{q+1})^{{1}/({h+1})}}=\bigg( \frac{h+1}{q+1}\bigg) ^{{1}/({h+1})} r \quad {\textrm{for every }} r\in (0,R). \end{align} $$
Using the change of variables
$ k^{q+1}=s^{q+1}-\tau ^{q+1} $
, we get
$$ \begin{align*} ds=k^q\cdot (k^{q+1}+ \tau ^{q+1})^{-{q}/({q+1})}\,dk \end{align*} $$
and
$$ \begin{align*} \int^{g}_{\tau}\frac{ds}{(s^{q+1}-\tau^{q+1})^{{1}/({h+1})}}= \int_{0}^{v}\frac{k^{({qh-1})/({h+1})}}{(k^{q+1}+\tau^{q+1})^{{q}/({q+1})}}\,dk, \end{align*} $$
where
$ v=(g^{q+1}-\tau ^{q+1} )^{{1}/({q+1})} $
. Since
$ q>h $
, this integral is finite for any
$v> 0$
. Thus for any
$g> \tau $
, we have
$$ \begin{align*} \begin{array}{ll} \displaystyle\lim \limits _{v\rightarrow \infty} \int_{0}^{v} \frac{k^{({qh-1})/({h+1})}}{(k^{q+1}+\tau^{q+1})^{{q}/({q+1})}}\,dk &\leq \displaystyle\int_{0}^{\tau}\tau^{-q}k^{({qh-1})/({h+1})}\,dk +\lim \limits _{v\rightarrow \infty}\displaystyle \int_{\tau}^{v}k^{-({q+1})/({h+1})}\,dk \\ &=\bigg( \dfrac{h+1}{h(q+1)} +\dfrac{h+1}{q-h}\bigg) \tau^{-({q-h})/({h+1})} \end{array} \end{align*} $$
and
$$ \begin{align*} \begin{array}{ll} \displaystyle\lim\limits _{v\rightarrow \infty} \int_{0}^{v} \frac{k^{({qh-1})/({h+1})}}{(k^{q+1}+\tau^{q+1})^{{q}/({q+1})}}\,dk &\geq \displaystyle\int_{0}^{\tau}\displaystyle\frac{k^{({qh-1})/({h+1})}}{2^{{q}/({q+1})}\tau^q}\,dk \\ &=\displaystyle 2^{-{q}/({q+1})}\frac{h+1}{h(q+1)} \tau^{-({q-h})/({h+1})}. \end{array} \end{align*} $$
By continuity, this means that for any fixed
$ \tau>0 $
, the function
$$ \begin{align*}g\longmapsto\int^{g(r)}_{\tau}\frac{ds}{(s^{q+1}-\tau^{q+1})^{{1}/({h+1})}} \end{align*} $$
is a bijection from
$ [\tau ,\infty ) $
to
$ [0, \ell _{\tau }) $
for some constant
$\ell _{\tau }>0$
. Moreover,
$\ell _{\tau }\rightarrow 0$
as
$\tau \rightarrow \infty $
and
$\ell _{\tau }\rightarrow \infty $
as
$ \tau \rightarrow 0.$
Choosing
$\tau $
such that
$$ \begin{align*} \ell_{\tau}=\bigg( \frac{h+1}{q+1}\bigg) ^{{1}/({h+1})}R, \end{align*} $$
we get the implicit function defined in (3-6) satisfying (3-4) and (3-5).
Remark 3.2. During the above procedure, we have obtained
$$ \begin{align*}C_1\tau^{-({q-h})/({h+1})}\leq\int^{\infty}_{\tau}\frac{ds}{(s^{q+1}-\tau^{q+1})^{{1}/({h+1})}}\leq C_2\tau^{-({q-h})/({h+1})},\end{align*} $$
where
$C_1=2^{-{q}/({q+1})} ({h+1})/{(h(q+1))}$
and
$ C_2=({h+1})/{(h(q+1))} +{({h+1})/({q-h})}.$
This implies
$$ \begin{align*}C_1\tau^{-({q-h})/({h+1})}\leq\ell_{\tau} \leq C_2\tau^{-({q-h})/({h+1})}.\end{align*} $$
Since
$g(0)=\tau $
and
$$ \begin{align*} \ell_{\tau}=\bigg( \frac{h+1}{q+1}\bigg) ^{{1}/({h+1})}R, \end{align*} $$
we get
$$ \begin{align} \widetilde{C_1}R^{-({h+1})/({q-h})}\leq g(0) \leq \widetilde{C_2}R^{-({h+1})/({q-h})}, \end{align} $$
where
$\widetilde {C_1}$
and
$\widetilde {C_2}$
are positive constants, which depend only on
$q.$
Moreover, the solution g obtained as above satisfies
$$ \begin{align*} \widetilde{C_1}( R-r)^{-({h+1})/({q-h})}\leq g(r)\leq \widetilde{C_2} ( R-r )^{-({h+1})/({q-h})}, \quad r\in (0,R). \end{align*} $$
Indeed, for
$0<r<R$
,
$\varepsilon>0$
, let
$\widetilde {g}(s)$
be the solution of (3-4) in the interval
$ (0, r+R_{\varepsilon }),$
where
$R_{\varepsilon }=R-r-\varepsilon .$
Since
$\widetilde {g}(s)$
blows up on the boundary of the ball
$B_{r+R_{\varepsilon }}(x_0),$
that is,
$$ \begin{align*}\lim\limits _{s\rightarrow {r+R_{\varepsilon}}} \widetilde{g} (s)= +\infty,\end{align*} $$
the comparison principle implies
$g \leq \widetilde { g} $
in the interval
$ (0,r+R_{\varepsilon }).$
Especially, we have
$$ \begin{align*}g(r) \leq \widetilde{g}(r)\leq \widetilde{C_2} R_{\varepsilon}^{-({h+1})/({q-h})}, \end{align*} $$
where we use (3-7). Letting
$\varepsilon \rightarrow 0,$
we have
$g(r) \leq \widetilde {C_2} (R-r)^{-({h+1})/({q-h})}$
. For the lower bound estimate, see Remark 3.4.
Lemma 3.3. There exists a positive viscosity solution of problem (3-1).
Proof. Set
$$ \begin{align} U(x):=g(|x-x_0|), \end{align} $$
where g and
$\tau $
are defined in Lemma 3.1. We want to show that the function U in (3-8) is the desired viscosity solution of the problem (3-1).
Noticing that
$U\in C^2( B_R(x_0)\setminus \lbrace x_0\rbrace ) \cap C^1(B_R(x_0) ), $
we have that U is a classical solution in
$B_R(x_0)\setminus \lbrace x_0\rbrace $
, which implies U is also a viscosity solution in the ball
$B_R(x_0)$
except
$x_0$
.
Now we show that U is indeed a viscosity solution at
$x_0$
. Let
$\varphi \in C^2(B_R(x_0))$
be such that
$\varphi $
touches U from below at
$x_0\in B_R(x_0)$
. Noting that
$D\varphi (x_0)=DU(x_0)=0,$
we have
$\Delta ^h_{\infty }\varphi (x_0)=0\leq U(x_0)^q$
. Therefore, U is a viscosity supersolution at
$x_0$
.
Next we show that U is a viscosity subsolution at
$x_0$
. Consider the test function
$\varphi \in C^2(B_R(x_0))$
such that
$U-\varphi $
has a local maximum at
$x_0\in B_R(x_0)$
. Since
$D\varphi (x_0)=DU(x_0)=0$
, we have
$$ \begin{align*}&U(x)-U(x_0)\leq \varphi(x)-\varphi(x_0)\\&\quad=\tfrac{1}{2}\langle D^2\varphi(x_0)(x-x_0),x-x_0\rangle +o(|x-x_0|^2)\quad {\mathrm{as}} \; |x-x_0|\rightarrow 0. \end{align*} $$
Taking
$x-x_0=t\vec {e}$
, where
$\vec {e}$
is a unit vector and
$t>0$
, we get
$$ \begin{align*} U(x_0+t\vec{e})-U(x_0)\leq \tfrac{1}{2}\langle D^2\varphi(x_0)\vec{e},\vec{e}\rangle t^2 +o(t^2)\quad {\mathrm{as}} \; t\rightarrow 0. \end{align*} $$
This is equivalent to
$$ \begin{align} g(t)-g(0)\leq \tfrac{1}{2}\langle D^2\varphi(x_0)\vec{e},\vec{e}\rangle t^2 +o(t^2)\quad {\mathrm{as}} \;t\rightarrow 0. \end{align} $$
Multiplying by
$g'(r)$
in (3-3) and integrating on
$(0,r)$
, we get
$$ \begin{align*}\frac{1}{h+1}|g'(r)|^{h+1}=\int_{\tau}^{g(r)}u^qdu\geq \tau^q(g(r)-\tau).\end{align*} $$
Since
$g'(r)$
is nonnegative in
$(0,R)$
, one has
$$ \begin{align*} g'(r)[(h+1)(g(r)-\tau)]^{-{1}/({h+1})}\geq \tau^{{q}/({h+1})}.\end{align*} $$
Integrating this again on
$(0,r)$
, we have
$$ \begin{align*} g(r)-\tau\geq Cr^{({h+1})/{h}}, \quad r\in (0,R), \end{align*} $$
where C is a positive constant. Then we have
$$ \begin{align*}Ct^{({h+1})/{h}}\leq g(t)-\tau=g(t)-g(0), \end{align*} $$
which contradicts (3-9). Hence, U is a viscosity subsolution at
$x_0$
.
Remark 3.4. One can verify that
$$ \begin{align*} u_R(x):=\bigg( \bigg(\frac{h+1}{q-h}\bigg) ^h \frac{q+1}{q-h}\bigg)^{{1}/({q-h})}( R-|x-x_0|) ^{-({h+1})/({q-h})} \end{align*} $$
is a classical solution of (3-2) in
$ B_R(x_0)\setminus {\lbrace x_0 \rbrace } $
. Noticing
$q>h, $
we have that
$ \lim \nolimits _{x\rightarrow \partial B_R(x_0) } u_R(x)=\infty $
. Since
$u_R$
is not differentiable at
$ x=x_0 ,$
we have
$u_R $
is a classical boundary blow-up solution of (3-1) in
$ B_R(x_0)\setminus {\lbrace x_0 \rbrace }. $
Additionally,
$ u_R $
is a viscosity subsolution of (3-1) in
$ B_R(x_0) $
since there are no test functions touching
$ u_R $
from above at
$x_0$
. However, it is not a viscosity supersolution at
$x_0$
because one can test it from below by test functions with small but nonzero gradient and any Hessian.
Remark 3.5. If u is a viscosity solution of (3-1) in
$ B_R(x_0), $
then
$u_R\leq u.$
In fact, for any
$\varepsilon>0, $
we have
$u_{R+\varepsilon }\leq u$
in
$ B_R(x_0) $
based on the comparison principle. Therefore,
$u_{R+\varepsilon }$
converges locally uniformly to u as
$\varepsilon \rightarrow 0.$
Next, we study the following problem:
$$ \begin{align} \begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u(x)=M &\mathrm{on} \;\partial\,\Omega, \end{cases} \end{align} $$
where
$M> 1$
is a fixed constant. Since we want to study the boundary blow-up solution of the problem (1-1), in fact, we are interested in M sufficiently large.
Theorem 3.6. For any
$M>1,$
there exists a unique, nonnegative viscosity solution
$ u_M \in C(\overline {\Omega }) $
to (3-10).
Proof. The uniqueness follows immediately by the comparison principle, Theorem 2.5. The proof of the existence relies on Perron’s method applied to viscosity solutions. It is obvious that
$ \overline {u}= M $
is a viscosity supersolution of (3-10).
Next we want to construct a viscosity subsolution with appropriate boundary value. For
$z \in \partial \,\Omega , $
let
$$ \begin{align*} v_z(x)=M-C|x-z|^{\alpha},\quad x\in \Omega, \end{align*} $$
where
$\alpha \in (0,1)$
and
$ C\geq 1 $
is to be determined. One can verify that
$$ \begin{align*} \Delta_{\infty}^h v_z(x)=(1-\alpha)(\alpha C)^h|x-z|^{h(\alpha -1)-1}\geq(1-\alpha)\alpha^h|x-z|^{h(\alpha -1)-1}. \end{align*} $$
Choosing
$ 0<\delta = {M^q}/({(1-\alpha )\alpha ^h})^{{1}/({h(\alpha -1)-1})}<1 $
and
$C=({\delta }/{2}) ^{-\alpha }M, $
we get
$ v_z\leq 0 $
outside
$B_{\delta /2}(z)$
and
$$ \begin{align*} \Delta_{\infty}^h v_z(x)\geq M^q\geq v^q_z(x)\quad \text{for all } x\in B_{\delta}(z). \end{align*} $$
Then the function
$$ \begin{align*} \underline{u}(x)=\max \lbrace 0,\sup\limits_{z\in\partial\,\Omega} ( M-C|x-z|^{\alpha}) \rbrace \end{align*} $$
is the desired viscosity subsolution. Hence, using the standard Perron method, we can obtain the existence of a nonnegative viscosity solution
$u_M$
to the problem (3-10).
Remark 3.7. From the above proof, we can see that Theorem 3.6 still holds for any
$q> 0,$
not just for
$q>h.$
We use this fact in the proof of the nonexistence of boundary blow-up solutions to (1-1) for
$0 < q \leq h $
in Section 5.
With the aid of Theorem 3.6 and the compactness method, we can establish the existence of boundary blow-up viscosity solutions to (1-1). To establish the existence of large solutions, a difficulty with respect to the degenerate operators is the lack of existence of barriers. However, thanks to the special structure of the operator
$\Delta _{\infty }^h, $
we can construct appropriate barriers without assuming any regularity of the boundary of the domain.
Proof of Theorem 1.1.
By the comparison principle, we have that the sequence
$\{u_M \}$
obtained in Theorem 3.6 is increasing. Hence, the limit function
$u_{\infty }:\overline {\Omega }\rightarrow [ 0,\infty ]$
$$ \begin{align*} u_{\infty}(x):= \lim\limits_{M\rightarrow\infty}u_M(x)\end{align*} $$
exists. Next we show that
$ u_{\infty } $
is finite in
$\Omega $
and gives the desired solution to (1-1) for
$q>h$
.
Fix
$x_0\in \Omega $
and
$r>0$
so that
$B_r(x_0)\subset \Omega $
. By Lemma 3.3, there exists a radial function U as constructed in (3-8) satisfying
$$ \begin{align*} \begin{cases} \Delta_{\infty}^h U(x)=U^q(x), &x\in B_r(x_0),\\ \lim\limits_{x\rightarrow z}U(x)=\infty , &z\in \partial B_r(x_0). \end{cases} \end{align*} $$
The comparison principle implies
$u_M \leq U$
in
$B_r(x_0)$
for every
$M>1$
. Hence,
$u_{\infty } \leq U$
in
$B_r(x_0).$
This means that
$ u_{\infty } $
is locally bounded in
$\Omega $
.
Since
$\Delta _{\infty }^h u_M=u_M^q\geq 0 $
in the viscosity sense and
$\lbrace u_M\rbrace $
is locally uniformly bounded, the Lipschitz continuity of
$u_M$
(see for example Lemma 2.9 in [Reference Aronsson, Crandall and Juutinen3]) implies that
$\lbrace u_M\rbrace $
is also equicontinuous. Therefore,
$ u_M$
converges to
$ u_{\infty }$
locally uniformly as
$M\rightarrow \infty .$
Then we have
$u_{\infty }$
is a viscosity solution of
$\Delta _{\infty }^h u=u^q$
in
$\Omega $
by the stability theory of viscosity solutions [Reference Crandall, Ishii and Lions12].
Next we show that
$\lim \nolimits _{x\rightarrow z}u_{\infty }(x)=\infty $
for all
$ z\in \partial \,\Omega $
. Consider the barrier function
$$ \begin{align*} w(x)=H_q( |x-z|+\varepsilon) ^{\alpha}, \quad x\in \Omega,\end{align*} $$
where
$z\in \partial \,\Omega , \varepsilon>0$
,
$\alpha =-({h+1})/({q-h})$
, and
$$ \begin{align*} H_q = \bigg( \bigg(\frac{h+1}{q-h}\bigg)^h \frac{q+1}{q-h}\bigg)^{{1}/{(q-h)}}. \end{align*} $$
Then, a direct calculation yields
$$ \begin{align*} Dw (x)=\alpha H_q( |x-z|+\varepsilon) ^{\alpha-1} \frac{x-z}{|x-z|}\end{align*} $$
and
$$ \begin{align*} \begin{cases} D^2 w(x) &\!\!\!\!=\alpha( \alpha-1) H_q( |x-z|+\varepsilon) ^{\alpha-2}\displaystyle\frac{x-z}{|x-z|}\otimes \frac{x-z}{|x-z|}\\[8pt] &+\,\alpha H_q( |x-z|+\varepsilon) ^{\alpha-1}\bigg(\displaystyle\frac{I}{|x-z|}-\frac{(x-z)\otimes(x-z)}{|x-z|^3}\bigg). \end{cases} \end{align*} $$
Hence, for any
$ x\in \Omega ,$
we have
$$ \begin{align}\begin{cases} \Delta_{\infty}^h w (x)\!\!\!\! & =(| \alpha| H_q( |x-z|+\varepsilon) ^{\alpha-1} )^{h-1} \alpha( \alpha-1) H_q( |x-z|+\varepsilon) ^{\alpha-2} \\ & =|\alpha|^{h-1}\alpha ( \alpha-1) H_q^h( |x-z|+\varepsilon) ^{(\alpha-1)h-1} \\ & = \displaystyle \bigg( \frac{h+1}{q-h}\bigg) ^h \frac{q+1}{q-h} H_q^h( |x-z|+\varepsilon) ^{\alpha q} \\ & = H_q^q ( |x-z|+\varepsilon) ^{\alpha q} \\ & = w^q (x). \end{cases} \end{align} $$
By the comparison principle, we get
$u_M\geq w$
in
$\Omega $
for all
$M\geq H_q\varepsilon ^{-({h+1})/({q -h})}$
. Letting first
$ M\rightarrow \infty $
and then
$\varepsilon \rightarrow 0, $
we have
$$ \begin{align} u_{\infty} (x)\geq H_q |x-z| ^{-({h+1})/({q-h})}, \quad x\in \Omega. \end{align} $$
Noticing
$q>h,$
we obtain
$\lim \nolimits _{x\rightarrow z}u_{\infty }(x)=\infty $
for all
$ z\in \partial \,\Omega $
. By (3-12), it is obvious that the boundary blow-up solution
$u_{\infty }$
is positive in
$\Omega .$
This finishes the proof.
4 Boundary asymptotic estimate and uniqueness
In this section, we investigate the boundary asymptotic estimate of the large solutions to problem (1-1). The lower growth estimate can be easily obtained by the proof of the existence, Theorem 1.1. Hence, the key is to establish the upper growth estimate. To reach to this goal, we use the perturbation of the distance function and comparison principle. The idea comes from [Reference Juutinen and Rossi19]. However, for
$h>1,$
the operator
$ \Delta _{\infty }^h$
is quasi-linear even in dimension 1. Therefore, we must make a subtle analysis.
Lemma 4.1. Let
$q>h$
,
$H_q=( ( ({h+1})/({q-h})) ^h ({q+1})/({q-h}))^{{1}/({q-h})}$
and u be a viscosity solution to (1-1) in a bounded domain
$ \Omega $
. Assume that there exists a neighborhood
$\mathcal {N}$
of
$ \partial \,\Omega $
such that
${\mathrm {dist}}(x,\partial \,\Omega )\in C^1(\mathcal {N}\cap \Omega ).$
Then there are constants
$\gamma>0$
and
$\mu>0$
depending only on the domain
$ \Omega $
such that
$$ \begin{align} u(x)\leq H_q{\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})}+\gamma, \quad x\in \Omega_{\mu}, \end{align} $$
where
$\Omega _{\mu }:= \{x |\; x\in \Omega \; \mathrm { and } \; {\mathrm {dist}}( x,\partial \,\Omega )<\mu \}.$
Proof. Let
$ \mu>0$
such that
$ \Omega _{\mu }\subset (\mathcal {N}\cap \Omega )$
. Denote
$$ \begin{align*} D_{\mu } :=\{x \; | \; x\in \Omega \; \mathrm { and }\; {\mathrm {dist}}( x, \partial \,\Omega )=\mu \}. \end{align*} $$
Then there exists some
$x_0\in D_{\mu }$
such that
$$ \begin{align*} u(x_0)=\sup\limits_{x\in D_{\mu} }u(x). \end{align*} $$
For any
$ 0<r<\mu , $
let
$v_r$
be the radial function satisfying
$$ \begin{align*} \begin{cases} \Delta_{\infty}^h v_r(x)=v_r^q(x), & x\in B_r(x_0),\\ \lim\limits_{x\rightarrow z}v_r(x)=\infty, & z\in\partial B_r(x_0), \end{cases} \end{align*} $$
in the viscosity sense. The comparison principle implies
$u(x_0)\leq v_r(x_0)$
for all
$r>0 $
. Therefore, we have
$$ \begin{align} \tau:=u(x_0)=\sup\limits_{x\in D_{\mu} }u(x)\leq v_{\mu}(x_0):=\gamma. \end{align} $$
For simplicity, we denote
$d(x):={\mathrm {dist}}( x,\partial \,\Omega ).$
Then we have
$|Dd(x)|=1$
for all
$x\in \Omega _{\mu }$
. Furthermore,
$d(x)$
is a viscosity solution of
$ \Delta _{\infty }^h u=0$
in
$\Omega _{\mu }$
(see for example [Reference Aronsson, Crandall and Juutinen3]).
Now for
$0<\varepsilon <\mu ,$
set
$$ \begin{align*} w_{\varepsilon}(x):=H_q(d(x)-\varepsilon)^{\alpha}, \quad x\in\Omega_{\mu}\setminus\overline{\Omega}_{\varepsilon},\end{align*} $$
where
$\alpha =-({h+1})/({q-h})<0$
. By direct calculations, we have
$$ \begin{align*} \begin{cases} \Delta_{\infty}^h w_{\varepsilon}(x) & =(| \alpha| H_q( d(x)-\varepsilon) ^{\alpha-1} )^{h-1} \alpha( \alpha-1) H_q( d(x)- \varepsilon) ^{\alpha-2} \\ &\, + (| \alpha| H_q( d(x)-\varepsilon) ^{\alpha-1} )^{h-1} \alpha H_q( d(x)- \varepsilon) ^{\alpha-1} \Delta_{\infty}^h d(x) \\ & =|\alpha|^{h-1}\alpha ( \alpha-1) H_q^h(d(x)- \varepsilon) ^{(\alpha-1)h-1} \\ & = \displaystyle \bigg( \frac{h+1}{q-h}\bigg) ^h \frac{q+1}{q-h} H_q^h( d(x)- \varepsilon) ^{\alpha q} \\ & = H_q^q ( d(x)- \varepsilon) ^{\alpha q} \\ & =w_{\varepsilon}^q(x) \end{cases} \end{align*} $$
in the viscosity sense in
$ \Omega _{\mu }\setminus \overline {\Omega }_{\varepsilon }.$
Indeed, one can see that if
$w_{\varepsilon }(x)-\varphi (x) $
has a local maximum (respectively minimum) at some point
$\hat {x}\in \Omega _{\mu }\setminus \overline {\Omega }_{\varepsilon }, $
then
$d(x)-\psi (x) $
with
$\psi (x)=({1}/{H_q}\varphi (x)) ^{1/\alpha }+\varepsilon $
has a local maximum (respectively minimum) at
$\hat {x}. $
Since
$d(x)$
is a viscosity solution of
$ \Delta _{\infty }^h u=0$
in
$\Omega _{\mu }$
, we obtain that
$ w_{\varepsilon } $
verifies
$ \Delta _{\infty }^h w_{\varepsilon }= w_{\varepsilon }^q $
in the viscosity sense in
$ \Omega _{\mu }\setminus \overline {\Omega }_{\varepsilon }$
.
Noticing
$q>h>1$
, we get
$$ \begin{align*} \Delta_{\infty}^h (w_{\varepsilon}+\gamma) \leq (w_{\varepsilon}+\gamma)^q \end{align*} $$
in the viscosity sense in
$ \Omega _{\mu }\setminus \overline {\Omega }_{\varepsilon }$
. By (4-2), we have
$$ \begin{align*} u(x)\leq \tau\leq \gamma\leq w_{\varepsilon} (x)+\gamma, \quad x\in D_{\mu}. \end{align*} $$
Since
$$ \begin{align*} \lim\limits_{x\rightarrow z\in D_{\varepsilon}} w_{\varepsilon} (x)=\infty, \end{align*} $$
the comparison principle implies
$$ \begin{align*} u(x)\leq w_{\varepsilon}(x)+\gamma, \quad x\in\Omega_{\mu}\setminus\overline{\Omega}_{\varepsilon}. \end{align*} $$
Letting
$ \varepsilon \rightarrow 0$
, we have
$$ \begin{align*} u(x)\leq H_q{\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})}+\gamma, \quad x\in \Omega_{\mu}. \\[-36pt] \end{align*} $$
Remark 4.2. By Remark 3.2 and the proof above, we actually have that
$\gamma \approx \mu ^{-({h+1})/({q-h})}.$
In particular, there is a constant
$C> 0$
depending only on q such that
$\gamma \leq C \mu ^{-({h+1})/({q-h})}.$
With Lemma 4.1 in hand, we can immediately establish the asymptotic estimate.
Proof of Theorem 1.2.
By Lemma 4.1, we have
$$ \begin{align} u(x)\leq H_q{\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})}+\gamma, \quad x\in \Omega_{\mu}. \end{align} $$
Recall that in the proof of Theorem 1.1, we have (3-12). That is, if u is a viscosity solution to (1-1) in a bounded domain
$\Omega $
, then there holds
$$ \begin{align*} u(x)\geq H_q|x-z|^{-({h+1})/({q-h})} \quad \text{for all } x\in \Omega, \end{align*} $$
where
$z\in \partial \,\Omega $
and
$H_q=( ( ({h+1})/({q-h})) ^h ({q+1})/({q-h}))^{{1}/({q-h})}$
. Hence, we can immediately get the lower growth estimate
$$ \begin{align} u(x)\geq H_q{\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})} \quad \text{for all } x\in \Omega. \end{align} $$
By the inequalities (4-3), (4-4), and Remark (4.2), we get the asymptotic behavior estimate (1-2).
Finally, we give the uniqueness of the boundary blow-up viscosity solution. We argue it by contradiction. Suppose that u and v are both viscosity solutions to (1-1). By the inequality (4-4) and Lemma 4.1, we have
$$ \begin{align*} \frac{u(x)}{v(x)} \leq\frac{H_q {\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})}+\gamma}{H_q {\mathrm{dist}}( x,\partial\,\Omega) ^{-({h+1})/({q-h})}},\quad x\in \Omega_{\mu}. \end{align*} $$
Therefore,
$$ \begin{align*} \lim \sup \frac{u(x)}{v(x)}\leq 1\quad \mathrm{ as}\ {\mathrm{dist}}(x,\partial\,\Omega)\rightarrow 0. \end{align*} $$
By the comparison principle, Theorem 3.6, we obtain
$u(x)\leq v(x)$
in the whole
$\Omega $
. Similarly, we can obtain
$v(x)\leq u(x)$
in the whole
$\Omega $
. Hence, the uniqueness is completed.
Remark 4.3. One should notice that the assumption condition on the domain in Lemma 4.1 holds for all
$C^2$
-domains [Reference Gilbarg and Trudinger17] and Lipschitz domains satisfying a uniform interior ball condition [Reference Juutinen18]. In fact, the condition
${\mathrm {dist}}( x,\partial \,\Omega )\in C^1(\mathcal {N}\cap \Omega )$
is equivalent to the geometric condition that for any
$x \in \mathcal {N}\cap \Omega $
, there exists a unique
$z\in \partial \,\Omega $
such that
${\mathrm {dist}}( x,\partial \,\Omega )=|x-z|.$
5 Nonexistence for
$q \leq h$
In this section, we show that problem (1-1) has no positive viscosity solution when
$q \leq h$
. We consider two cases,
$q\leq 0$
and
$0<q\leq h.$
Theorem 5.1. Let
$ \Omega \subset \mathbb {R}^n$
be a bounded domain and
$q\leq 0$
. Then problem (1-1) has no positive viscosity solution.
Proof. Suppose that there is a positive solution u to (1-1). Since
$q\leq 0,$
there exists
$C>0$
such that
$$ \begin{align*} 0\leq u^q(x)\leq C, \quad x\in \Omega. \end{align*} $$
In particular, we have that u is a viscosity supersolution of
$\Delta _{\infty }^h u = C $
in
$ \Omega .$
We introduce
$$ \begin{align*} w(x):=L+\frac{h}{h+1}( {Ch}) ^{{1}/{h}}|x-x_0|^{({h+1})/{h}}, \quad x\in \Omega, \end{align*} $$
where
$L\in \mathbb {R}$
is an arbitrary constant and
$x_0\in \mathbb {R}^n\backslash \Omega $
. Then we can verify that
$\Delta _{\infty }^h w= C $
in
$ \Omega $
. Since
$\lim \nolimits _{x\rightarrow z}u(x)=\infty $
for all
$z\in \partial \,\Omega ,$
we have
$w\leq u$
in
$\Omega $
by the comparison principle (see [Reference Lu and Wang25]). Letting
$L\rightarrow \infty ,$
we get
$u\equiv \infty $
which is a contradiction.
To obtain the nonexistence of a large solution for the case
$0<q\leq h$
, we need the following maximum/minimum principle which can be derived from Harnack’s inequality [Reference Aronsson, Crandall and Juutinen3, Reference Crandall10].
Lemma 5.2. Let
$u\in C(\overline {\Omega })$
satisfy
$\Delta _{\infty } u \geq 0 \;(\leq 0)$
in the viscosity sense. Then
$$ \begin{align*} \sup\limits_{\Omega} u= \sup\limits_{\partial\,\Omega} u \quad (\inf\limits_{\Omega} u= \inf\limits_{\partial\,\Omega} u). \end{align*} $$
Moreover, the supremum (infimum) occurs only on the set
$\partial \,\Omega $
unless u is a constant.
Theorem 5.3. Let
$ \Omega \subset \mathbb {R}^n$
be a bounded domain and
$0<q\leq h$
. Then problem (1-1) has no positive viscosity solution.
Proof. We argue it by contradiction. Assume that there is a viscosity solution u to (1-1) in
$\Omega $
. For
$x_0\in \partial \,\Omega ,$
there exist
$r>0$
and
$c>1 $
such that
$u>c $
in
$\Omega _0:= B(x_0,r)\cap \Omega .$
Denote
$$ \begin{align*} U(x)=\log u(x), \quad x\in \Omega_0. \end{align*} $$
By direct calculation, we have
$$ \begin{align*} DU(x) =\frac{1}{u(x)}Du(x), \end{align*} $$
$$ \begin{align*} D^2U(x)=- \frac{1}{u^2(x)} Du(x)\otimes Du(x)+\frac{1}{u(x)}D^2u(x), \end{align*} $$
where
$\otimes $
denotes the tensor product. Then one can verify that
$$ \begin{align*} \Delta_{\infty}^h U(x)=-\bigg( \frac{1}{u(x)}\bigg) ^{h+1}|Du(x)|^{h+1}+\bigg( \frac{1}{u(x)}\bigg) ^h\Delta_{\infty}^h u(x) \leq \bigg( \frac{1}{u(x)}\bigg)^h (u(x))^q \leq 1 \quad \mathrm{ in}\; \Omega_0 \end{align*} $$
in the viscosity sense, where we use
$0<q \leq h.$
Since
$u=\infty $
on
$\partial \,\Omega $
, we have that
$U=\infty $
on
$B(x_0,r)\cap \partial \,\Omega $
.
Let g be a continuous function supported on
$\partial \,\Omega _0$
and
$$ \begin{align*} g(x)= \begin{cases} 1, &x\in B\bigg(x_0,\dfrac{r}{3}z\bigg)\cap\partial\,\Omega, \\[3pt] 0, &x\in\partial\,\Omega_0\setminus\bigg(B\bigg(x_0,\dfrac{r}{2}\bigg)\cap\partial\,\Omega\bigg). \end{cases} \end{align*} $$
The existence and uniqueness of a viscosity solution v of the following problem:
$$ \begin{align*} \begin{cases} \Delta_{\infty}^h v=1 &{\mathrm{in}}\; \Omega_0, \\ v=g &{\mathrm{on}} \;\partial\,\Omega_0, \end{cases} \end{align*} $$
were obtained in [Reference Liu and Yang22, Reference Lu and Wang25]. Due to the h-homogeneity of the operator
$\Delta _{\infty }^h ,$
for any constant
$k\geq 1,$
the function
$v_k(x):=kv(x)$
satisfies
$$ \begin{align*} \begin{cases} \Delta_{\infty}^h v_k=k^h\geq 1 \quad &{\mathrm{in}}\; \Omega_0, \\ v_k=kg \quad &{\mathrm{on}} \;\partial\,\Omega_0, \end{cases} \end{align*} $$
in the viscosity sense. By the comparison principle in [Reference Lu and Wang25], there holds
$v_ k\leq U$
in
$\Omega _0$
for any
$k\geq 1$
. Since
$\Delta _{\infty }^h v= |Dv|^{h-3} \Delta _{\infty } v=1>0$
, by Lemma 5.2, there exists
$x_1\in \Omega _0$
such that
$v(x_1)>0$
. Then we have
$v_ k(x_1)=kv(x_1)\leq U(x_1)$
for all
$k\geq 1$
which leads to a contradiction.
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading of the manuscript and useful suggestions and comments.
