Singular elliptic equations with directional diffusion

To Italo with great esteem and friendship.

1.   Introduction and main results

In this paper we are interested to the Dirichlet problem

in a bounded domain of $\mathbb R^n$, for an integer $k \in\{1, \dots, n\}$, where

and $f(u)$ is a positive nonlinearity on $(0, \infty)$, which can go to infinity as $u \to 0$.

The above problem has a positive solution for uniformly elliptic equations ($k = n$) with a positive polynomial nonlinearity with negative exponent:

See for instance Lazer-McKenna ^[31] and Crandall-Rabinowitz-Tartar ^[24]. For $n = 1$ equations of this kind arise in the theory of pseudoplastic fluids, appearing as generalized Emden-Fowler equations with negative exponent. See Nachman-Callegari ^[32].

Existence and uniqueness results results have also been obtained for equations with advection terms. See Giarrusso-Porru ^[27] and Porru-Vitolo ^[34].

Partial diffusion operators have been considered with respect to the maximum principle in cylindrical domains in ^[19,20,21], which had been previously investigated for linear and fully nonlinear uniformly elliptic equations in ^{[4,9,10,16,19,37,38,39]}.

In the present paper, we will show that similar results hold in the case of degenerate, not uniform ellipticity ($k < n$). For nonlinearities as in equation (1.3) in degenerate elliptic cases see also ^[5].

We need in our case a geometric condition $(\bf G)$ introduced by Blanc and Rossi ^[8], which says for the present use that Ω is a strictly convex domain. See Section 4.

Theorem 1.1. Let $\Omega$ be a bounded domain of $\mathbb R^n$ satisfying condition $(\bf G)$. Let $f:(0, \infty) \to (0, \infty)$ be a continuous non-increasing function. Then there exists an unique solution, positive in $\Omega$, of the Dirichlet problem $(1.1)$.

It is plain that Theorem 1.1 holds more generally in the case of the Dirichlet problem

for anisotropic partial diffusion operators

with $\delta_i > 0$, $i = 1, \dots, k$.

The paper is organized as follows. In Section 2 we introduce the different notions of ellipticity and the viscosity solutions, then we establish comparison principles and recall existence, uniqueness and regularity results. In Section 3 we study an associate ODE and construct radial solutions of partial Laplace equations with positive non-increasing reaction terms. In Section 4 we prove the existence of Dirichlet problems for the smallest and the largest Hessian eigenvalue equation with positive non-increasing reaction terms. In Section 5 we finally prove Theorem 1.1.

2.   Definitions and auxiliary results

Let $\mathcal F$ be a mapping from $\Omega\times\mathbb R\times\mathbb R^n\times \mathcal S^n$ to $\mathbb R$, where $\Omega$ is an open connected set of $\mathbb R^n$ and $\mathcal S^n$ the vector space of real $n\times n$ symmetric matrices. The fully nonlinear operator $\mathcal F$ acts on $u\in C^2(\Omega)$ as follows:

where $Du$ is the gradient and $D^2u$ the Hessian matrix of the function $u$.

As usual, $\mathcal S^n$ is endowed with the partial order relationship: $X \le Y$ if and only if $X-Y$ is semidefinite positive. Moreover, if $\mathcal F$ is constant with respect to a variable, we omit such a variable.

For instance, the operator considered in Eq (1.1) can be represented as

where

Definition 1. We say that $\mathcal F$ is degenerate elliptic in $\Omega$ if and only if

for all $X, Y \in \mathcal S^n$ such that $X \le Y$ and $(x, t, \xi) \in \Omega \times\mathbb R\times\mathbb R^n$.

Therefore the operator (2.3) under consideration is degenerate elliptic for all $k = 1, \dots, n$. Furthermore, let

be the projection matrix on the vector subspace generated by the first $k$ vectors $e_1, \dots, e_k$ of the canonical basis of $\mathbb R^n$, so that in particular $\Pi_n = I$, the $n\times n$ identity matrix. Setting

then

Definition 2. We say that $\mathcal F$ is proper in $\Omega$ if and only if it is degenerate elliptic and

for all $s, t \in \mathbb R$ such that $s \le t$ and $(x, \xi) \in \Omega\times\mathbb R^n\times \mathcal S^n$.

Assuming that $f$ is a non-increasing function, the operator (2.7) is proper. Under the same condition on $f$, the operator (2.7) is proper when considering any matrix $A\ge 0$ instead of $\Pi_k$.

Definition 3. Let $\lambda$ and $\Lambda$ be positive constants such that $\lambda \le \Lambda$. We say that $\mathcal F$ is uniformly elliptic in $\Omega$ with ellipticity constants $\lambda$ and $\Lambda$ if and only if

for all $X, Y \in \mathcal S^n$ such that $X \le Y$ and $(x, t, \xi) \in \Omega \times\mathbb R\times\mathbb R^n$.

It is plain that the uniform ellipticity is stronger than the degenerate ellipticity. Moreover, let $\mathcal M^\pm_{\lambda, \Lambda}(X)$, $X \in \mathcal S^n$, be the extremal Pucci operators:

Then $\mathcal M^\pm_{\lambda, \Lambda}$ are uniformly elliptic and (2.10) can be restated as

It is easy to check that the operators (2.7) are uniform elliptic only for $k = n$, when the second-order part is the Laplace operator $\Delta u$.

For $k < n$ (directional diffusion) the operators (2.6) are degenerate elliptic (proper if $f$ is non-increasing), but not uniformly elliptic. See ^[41].

Even though a degenerate elliptic fully nonlinear operator $\mathcal F[u]$ is defined in the classical sense only when $u$ is twice differentiable, nonetheless the equation $\mathcal F[u] = 0$ makes (a weaker) sense also when $u$ is less regular.

Here we will consider the viscosity sense. Let $D$ be a locally compact subset of $\mathbb R^n$, the spaces of upper and lower semicontinuous functions in $D$ will be indicated with usc$(D)$ and lsc$(D)$, respectively.

Let $x_0\in D$ and $u: D \to \mathbb R$. We denote by $J_{D}^{2, \pm}u(x_0)$ the second order superjet and subjet of $u$ at $x_0$:

Definition 4. Let $u$ be in usc$(D)$, resp. in lsc$(D)$. We say that $u$ is a viscosity subsolution, resp. supersolution, in $\Omega$ of the equation $\mathcal F[u] = 0$ if and only if for all $x_0 \in D$

resp.

A function $u \in C(D)$, which is both a subsolution and a supersolution of the equation $\mathcal F[u] = 0$ is called a viscosity solution of such equation.

We will also use the notations $\mathcal F[u] \ge 0$, resp. $\mathcal F[u] \ge 0$, to say that $u$ is a subsolution $u$, resp. a supersolution, of the equation $\mathcal F[u] = 0$.

For more properties of viscosity solutions we refer to ^{[12,22,23,30]}.

An important role is played by the maximum principle for subsolutions and, more generally, the comparison principle between upper semicontinuous subsolutions and lower semicontinuous supersolutions. See ^[23,25,41].

The comparison principle for the operators considered in this paper depends on the non-totally degenerate ellipticity, which is intermediate between the degenerate and the uniform ellipticity. See ^[3,41].

Definition 5. Let $\mathcal F$ be a degenerate elliptic fully nonlinear operator. We say that $\mathcal F$ is non-totally degenerate elliptic in $\Omega$ if and only if there exists a continuous function $\lambda(x) > 0$ such that

for all $\varepsilon > 0$ and $(x, t, \xi, X) \in \Omega \times \mathbb R \times \mathbb R^n \times\mathcal S^n$.

It is easy to check, by linearity, that the operators (2.5) are non-totally degenerate elliptic with $\lambda(x) = k$: in fact,

We will also consider below other non-totally degenerate elliptic operators, the partial traces

where $\lambda_1(X) \le \dots \le \lambda_n(X)$ are the eigenvalues of the matrix $X \in \mathcal S^n$, and the dual partial trace operators

Such operators arise for geometric problems of partial mean curvature ^[35,36,42] and in stochastic differential games ^[7,8]. They have been firstly investigated by Harvey-Lawson ^[28] and Caffarelli-Li-Nireneberg ^[13,14,15], and then in subsequent papers among which for instance ^{[1,6,7,8,25,26,40]}.

Is is also easy to check as well that the operators (2.17) and (2.18) are non-totally degenerate elliptic with $\lambda(x) = k$. For instance

Note also that all operators $\mathcal D_k$ and $\mathcal P_k^\pm$ coincide with the Laplace operator when $k = n$. Only in this case such operators are uniformly elliptic.

Lemma 2.1. Let $\mathcal F(t, X) = \mathcal M(X)+f(t)$ be a degenerate elliptic operator, where $f$ is a continuous and non-increasing positive function in $(0, \infty)$.

Let $u\in$ usc $(\overline\Omega)$ and $v\in$ lsc $(\overline\Omega)$ be non-negative functions such that $\mathcal F[u]\ge 0$ and $\mathcal F[v]\le 0$ in a bounded domain $\Omega$. If $u \le v$ on $\partial \Omega$, then $u \le v$ in $\Omega$.

Proof. Since the function $u-v$ is upper semicontinuous, then $u-v$ has a maximum at a point $\hat x \in \overline\Omega$. We have to prove that, $u \le v$ on $\partial \Omega$, then $u(\hat x) \le v(\hat x)$.

Suppose, by contradiction, that $\max_{\overline \Omega}(u-v) = \delta > 0$., namely

Let $B_d$ a ball centered at the origin such that $\overline \Omega \subset B_d$, and where $0 < \varepsilon < \frac{\delta}{d^2}$. Then set

By the choice of $\varepsilon$, we have for $x \in \partial\Omega$:

so that the usc function $u_\varepsilon-v$ has a positive maximum at a point of $\Omega$.

Following the proof of [23,Theorem 3.3], let $x_\alpha, y_\alpha \in \overline \Omega$ be such that

Let $\hat x_\varepsilon \in \overline \Omega$ be a limit point of $x_\alpha$. From [23,Lemma 3.1] we have

From the above limits:

By the upper semicontinuity of $u_\varepsilon$ and the lower semicontinuity of $v$, on a subsequence:

By (2.22) we have $\max_{\partial\Omega}(u_\varepsilon-v) < u_\varepsilon(\hat x_\varepsilon)-v(\hat x_\varepsilon)$. Therefore $\hat x_\varepsilon \in \Omega$, and $x_\alpha, y_\alpha \in \Omega$ for large $\alpha$, by the upper semicontinuity of $u_\varepsilon-v$.

From [23,Theorem 3.2] we deduce that matrices $X_\alpha, Y_\alpha \in \mathcal S^n$ such that

and

Since $v$ is a viscosity subsolution, then

On the other hand, by the non-totally degenerate ellipticity, $u_\varepsilon$ is a viscosity solution of the differential inequality

and therefore

Combining (2.29) and (2.31), by the degenerate ellipticity:

so that

Let $\alpha \to \infty$ using (2.26). Then

From this, by the non-increasing monotonicity of $f$ we have therefore

On the other hand, recalling the assumption $u(\hat x)-v(\hat x) = \delta > 0$, we have:

Together with (2.36), this implies $0 < \delta \le \tfrac\varepsilon2\, |\hat x_\varepsilon-\hat x|^2$, which cannot hold for small $\varepsilon > 0$. So the assumption $u(\hat x) > v(\hat x)$ leads to a contradiction, and it turns aout that $u(\hat x)\le v(\hat x)$.

The above theorem works for all non-totally degenerate elliptic operators, in particular when $\mathcal M$ is the partial diffusion operator $\mathcal D_k$ as well as for the partial trace operators $\mathcal P_k^\pm$.

We will use in the sequel the interior Lipschitz estimate of [25,Lemma 5.5] for $\mathcal P_{1}^-(X) = \lambda_1(X)$ and the dual operator $\mathcal P_{1}^+(X) = \lambda_n(X)$. See also ^[6].

Lemma 2.2. Let $u\in usc(B_1)$ be a viscosity subsolution of the equation $\lambda_1(D^2u) = g(x)$ in $B_1$, a ball of unit radius. If $g$ is a continuous function, bounded below in $B_1$, then $u$ in $B_1$ is Lipschitz-continuous and the following interior Lipschitz estimate holds:

where $B_{1/2}$ is the ball of radius $1/2$ concentric with $B_1$ and $C$ a positive costant depending on $n$.

By duality, let $u\in lsc(B_1)$ be a supersolution of the equation $\lambda_n(D^2u) = g(x)$ in $B_1$. If $g$ is a continuous function, bounded above in $B_1$, then $u \in C^\alpha(B_1)$ and the following interior $C^\alpha$ estimate holds:

It is also worth to recall a very interesting $C^{1, \alpha}$ regularity result proved in ^[33] for the highly degenerate elliptic operator $\lambda_1$, if we consider that, also in the uniform elliptic case, we have basically the $C^\alpha$ regularity, and this is the best we can have in the general case. For Hölder estimates in the uniform and degenerate elliptic case we refer for instance to ^[2,11,25]. For Lipschitz estimates see ^[6,25,29].

3.   On the equation $\lambda_j(D^2u)+f(u)=0$: radial solutions

In this section we investigate the qualitative properties of the radial solutions of the equation $\lambda_j(D^2u)+f(u) = 0$, where $f$ is a continuous and non-increasing positive function in $(0, \infty)$.

Asin the case of entire solutions, we preliminarly discuss an associated ODE with suitable initial conditions. See ^[17,18].

In the present case, for $j = 1$ we consider the Cauchy problem

and for $j \ge2$

Here below we state some useful properties of the classical solutions.

Lemma 3.1. Let $f:(0, \infty) \to (0, \infty)$ be a continuous and non-increasing function. Then the local classical solution of problem $(3.1)$ or $(3.2)$ is strictly decreasing and concave.

In case $(3.1)$ the solution $\varphi$ is $C^2$ and

In case $(3.2)$ the solution $\varphi$ is $C^1$. Assuming in addition that $f$ is $C^1$, then $\varphi$ is $C^2$, and (3.3) holds as well.

Proof. For problem (3.1) we refer for instance Lemma 2.1 of ^[27] and Lemma 3 of ^[34]. Here we limit the discussion to problem (3.2).

Since $f$ is positive and non-increasing, we have:

Supposing that $\varphi$ is $C^2$,

Lemma 3.2. Let $f:(0, \infty) \to (0, \infty)$ be a non-increasing function. Let $[0, R)$ be the maximal positivity interval of the classical solution $\varphi$ of problem $(3.1)$ or $(3.2)$. Then $R_f(t_0) = R$ is finite. Moreover

(i) $t_1 < t_2$ $\Rightarrow$ $R_f(t_1) < R_f(t_2)$;

(ii) $\lim_{ t_0 \to 0^+}R_f(t_0) = 0$;

(iii) if $g:(0, \infty) \to (0, \infty)$, then $f \le g$ $\Rightarrow$ $R_g(t_0)\le R_f(t_0)$.

Proof. We discuss the case of problem (3.2), referring to ^[27] and ^[34] for (3.1).

Since $\varphi'(0) = 0$ and $\varphi(r)$ is concave, then $R_f(t_0) < \infty$.

The remaining properties (ⅰ), (ⅱ) and (ⅲ) can be deduced observing that:

4.   On the equation $\lambda_j(D^2u)+f(u)=0$: the Dirichlet problem

In this section we investigate the Dirichlet problem

with $f$ strictly decreasing in $(0, \infty)$.

We start solving by the Perron's method the approximate problems

with $\varepsilon > 0$.

In order to do this, we need some geometric property of the boundary $\partial\Omega$. Consider the Dirichlet problem

Definition 6. (condition $\bf(G)$) . Given $y \in \partial \Omega$, for every $r > 0$ there exists $\delta > 0$ such that

for every $x\in B_\delta(y)$ and direction $v \in \mathbb R^n$ $(|v| = 1)$.

This is a necessary and sufficient to solve the Dirichlet problem (4.3) for all $j = 1, \dots, n$ and all continuous boundary data $g$.

Such condition can be weakened to require that $(4.4)$ holds for only one direction $v$ in each subspace of dimension $j$ and $n-j$ to solve (4.3) for a singòe $j$ such that $1 < j < n$. See Blanc and Rossi [8,Theorem 1].

Lemma 4.1. Let $\Omega$ be a bounded open set satisfying the geometric condition $\bf(G)$, and $f:(0, \infty) \to (0, \infty)$ be a continuous and strictly decreasing function. For every $\varepsilon > 0$ there exists an unique continuous viscosity solution of problem $(4.2)$ such that $u(x)\ge \varepsilon$.

Proof. As announced before, we use the Perron's method [23,Theorem 4.1].

The comparison principle is provided by Lemma 2.1. We need to find a subsolution $\underline u$ and a supersolution $\overline u$ such that $\underline u = \varepsilon = \overline u$ on $\partial \Omega$.

The function $\underline u\equiv \varepsilon$ is plainly a subsolution.

To find a supersolution we solve the problem

See for instance ^[8,28,40]. By duality $\overline u(x) = -v(x)-\frac12\, f(\varepsilon)|x|^2$ is a viscosity solution of the problem

Since $\lambda_n(D^2\overline u) \le 0$ in $\Omega$ and $\overline u \ge \varepsilon$ on $\partial \Omega$, by the maximum principle ^[1,26,28] we have $\overline u \ge \varepsilon$ in $\Omega$. By the non-increasing monotonicity of $f$, then $f(\overline u) \le f(\varepsilon)$. On the other hand $\lambda_j \le \lambda_n$, so that $\overline u$ can be chosen as the supersolution we are searching for:

By [23,Theorem 4.1] we conclude that there exists an unique continuous viscosity solution of problem (4.2).

Furthermore, by the comparison principle, being $u\equiv\varepsilon$ a subsolution, we have $u \ge \varepsilon$ in $\Omega$.

From the solutions of problems (4.2), we obtain as limit the solution of problem (4.1) for $j = 1$ and $j = n$.

Firstly we show that such limit is a positive function.

Lemma 4.2. Let $\Omega$ be a bounded open set satisfying the geometric condition $\bf(G)$, and $f:(0, \infty) \to (0, \infty)$ be a continuous non-increasing function.

For $h \in \mathbb N$, let $u_h$ be the solutions of problems $(4.2)$ with $\varepsilon = \frac1h$ for $j = 1$. Then for every compact subset $K$ of $\Omega$ there exists a number $t_K > 0$ such that

Suppose in addition that $f$ is $C^1$. Then the same holds for the solutions of problems $(4.2)$ with $\varepsilon = \frac1h$ for $j = 2, \dots, n$.

Proof. Let $K$ be a compact subset of $\Omega$.

In the case $j = 1$, we consider a maximal positive solution $\varphi$ of the Cauchy problem (3.1). We choose an initial condition $\varphi(0) = t_0 > 0$ such that the maximal positivity radius $R$ is small enough (see Lemma 3.2), say $0 < R \le \frac12$ dist$(K, \partial\Omega)$.

For any $x_0 \in K$, let $\phi(x) = \varphi(|x-x_0|)$. Then $\phi$ is $C^2$, and by (3.3) :

Thus $\phi$ is a classical solution of the Dirichlet problem:

For $j\ge2$, we consider a maximal positive solution $\varphi$ of the Cauchy problem (3.2), and choose the initial condition $\varphi(0) = t_0 > 0$ as before, depending on the maximal positivity radius $R$.

In this case, we need to assume that $f$ is $C^1$ in order that the radial function $\phi(x) = \varphi(|x-x_0|)$ is $C^2$, according to Lemma 3.1, and by (3.3) :

Thus we have found as well a classical solution $\phi$ of the Dirichlet problem:

From now on, we can proceed with the same argument for all $j = 1, \dots, n$.

We note that $u_h \ge 1/h > 0 = \phi$ on $\partial B_R(x_0)$, for every $h \in\mathbb N$. Comparing the solutions $\phi$ and $u_h$ we have $u_h(x) \ge \phi_(x) = \varphi(|x-x_0|)$ in $B_R(x_0)$.

In particular $u_h(x_0)\ge \varphi(0) = t_0$, which proves the assert with $t_K = t_0$.

Theorem 4.3. Let $\Omega$ be a bounded open set satisfying the geometric condition $\bf(G)$, and $f:(0, \infty) \to (0, \infty)$ be a continuous non-increasing function. Then for $j = 1$ there exists an unique continuous positive viscosity solution of problem $(4.1)$.

Proof. For $h \in \mathbb N$, let $u_h$ be the solutions of the Dirichlet problems (4.2) with $\varepsilon = \frac1h$ for $j = 1$. By the comparison principle, this is a non-increasing sequence. Therefore the $u_h$ converge pointwise, as $h \to \infty$, to a function $u$ in $\Omega$ such that $u = 0$ on $\partial \Omega$.

From Lemma 4.2 the function $u$ has a positive lower bound every compact subset $K$ of $\Omega$, namely $u(x) \ge t_K > 0$ for all $x \in K$.

We will show that the $u_h$ converge locally uniformly in $\Omega$. Thus, by the stability theorems for viscosity solutions ^[12,23], $u$ is a solution of problem (4.1), which is unique by the comparison principle.

We are left therefore with proving the local uniform convergence of the $u_h$.

To this end, we firstly observe that, being $u_1 \ge u_h > 0$ on $\Omega$ for all $h \in \mathbb N$, the $u_h$ are equi-bounded in $\Omega$: setting $M = \max_{\Omega}u_1$,

Next, let us fix a compact subset $K$ of $\Omega$, and consider a finite covering of $K$ with balls of type $B_{R/4}(x_0)$, where $x_0\in K$ and $0 < R \le \frac12$ dist$(K, \partial\Omega)$.

As in the proof of Lemma 4.2, we choose $t_0 > 0$ be such that $R$ is the maximal positivity radius of Cauchy problem (3.1). Letting $\varphi$ be the maximal positive solution, we obtain a radial function $\phi(x) = \varphi(|x-x_0|)$, which is a classical solution of the Dirichlet problem (4.12).

By the comparison principle $u_h(x) \ge \phi(x)$ in $B_R(x_i)$. In particular, setting $t_K^* = \varphi(R/2)$, and using the decreasing monotonicity of $\varphi$, we have:

As a consequence:

Then, for all $x_0 \in K$:

Therefore, by (4.16), (2.37) and (4.13):

This inequality, together with (4.13), shows that the $u_h$ are equi-continuous and equi-bounded on $K$. By Ascoli-Arzelà therefore $u_h \to u$ as $h \to\infty$ uniformly on $K$, as it was to be proved.

We prove the same result for the Dirichlet problem (4.1) with $j = n$. We follow the same lines of the proof of Theorem 4.3 but an additional approximation argument is needed since $\varphi$ is $C^2$ provided $f$ is $C^1$, and we want to show the result under the weaker assumption that $f$ is $C^0$.

Theorem 4.4. Let $\Omega$ be a bounded open set satisfying the geometric condition $\bf(G)$, and $f:(0, \infty) \to (0, \infty)$ be a continuous non-increasing function. Then for $j = n$ there exists an unique continuous positive viscosity solution of problem $(4.1)$.

Proof. Case 1: $f$ is $C^1$ non-increasing

We can repeat step by step the proof of Theorem 4.3 if we assume that $f$ is $C^1$, referring to (4.2) for $j = n$ instead of $j = 1$. We construct as there a non-increasing sequence $u_h$, converging to a function $u$ in $\Omega$.

The sequence $u_h$ is equi-bounded as (4.13) in $\Omega$.

We obtain easier way for all $x_0\in \Omega$ the inequality

Let $K$ be a compact subset of $\Omega$, and choose $R$ as in the proof of Theorem 4.3.

From (4.18), by (2.38) and (4.13), we deduce for all $x_0 \in K$:

We conclude as before that the $u_h$ are equi-bounded and equi-continuous on $K$, so that $u$ is a continuous positive solution of problem (4.4) in this case.

Case 2: $f$ is continuous non-increasing

We approximate $f$ with a sequence $f_i$ of $C^1$ non-increasing functions such that $f_i\to f$ as $i \to \infty$ locally uniformly in $(0, \infty)$ and $f_{i+1}\le f_i$ for all $i \in \mathbb N$.

Then we solve, by Case 1, the following Dirichlet problems:

Recall that the viscosity solutions $u_i$ are positive in $\Omega$, for $i \in \mathbb N$.

Since $f_{i+1}\le f_i$, then $\lambda_n(D^2u_{i+1})+f_i(u_{i+1}) \ge \lambda_n(D^2u_{i+1})+f_{i+1}(u_{i+1}) = 0$, so that by comparison $u_i \ge u_{i+1}$.

Then the $u_i$ converge to a function $u$ in $\Omega$ such that $u = 0$ on $\partial \Omega$.

The $u_i$ are equi-bounded in $\Omega$. In fact

where $M = \max_{\Omega}u_1$.

Let $K$ be a compact subset of $\Omega$.

Since $\lambda_n(D^2u_i)\le 0$ in $B_R(x_0)$, then (2.38) and (4.21) imply, as in the proof of Case 1,

Therefore the $u_i$ are equi-bounded and equi-continuous on $K$, and by Ascoli-Arzelà the $u_h$ converge uniformly to $u$.

Next, choose $t_0 > 0$ such that the maximal positive radius $R = R_f(t_0)$ of the Cauchy problem (3.2) is small enough, in order that $0 < R < \frac12$ dist$(K, \partial \Omega)$.

Let $\varphi$ be the maximal positive solution, and let $\varphi_i$ be the maximal positive solutions of the following Cauchy problems

Since $f \le f_i \le f_1$, then $R_1 \le R_i\equiv R_{f_i}(t_0) \le R$ and $\varphi_1 \le \varphi_i \le \varphi$ in $[0, R_1]$.

For $x_0 \in K$, then $\phi_i(x) = \varphi_i(|x-x_0|)$ is a classical radial solution of the Dirichlet problem

Since $u_i > 0$ in $\Omega$, comparing $u_i$ and $\phi_i$ on $B_{R_i}(x_0)$, we get $u_i \ge \phi_i$ in $B_{R_i}(x_0)$.

In particular, recalling that the $\varphi_i$ are decreasing, we get

Therefore the $f_i$ uniformly converge to $f$ in $\{u_i(x) : x \in K, i \in \mathbb N\} \subset [t_K', M]$. We have already shown that the $u_i$ converge uniformly on $K$.

By the aforementioned stability results for viscosity solutions, then $u$ is a viscosity solution of the Dirichlet problem (4.2) with $j = n$, positive in $\Omega$ by (4.25).

5.   Proof of Theorem 1.1

We have to solve problem (1.1) with $f$ positive and non-increasing in $(0, \infty)$.

Thanks to the results of the previous section, we will use once again the Perron's method of [23,Theorem 4.1].

The comparison principle is provided by Lemma 2.1, recalling that $\mathcal D_k$ is non-totally degenerate elliptic.

Next we solve with Theorems 4.3 and 4.4 the following Dirichlet problems:

and

Observe that

Then $\underline u$ and $\overline u$ are continuous viscosity subsolution and supersolution, respectively, positive on $\Omega$, of the equation

such that $\underline u = 0 = \overline u$ on $\partial\Omega$.

Theorem 4.1 of ^[23] implies therefore that there exists an unique positive viscosity solution of the problem

as wanted.

Conflict of interest

The authors declare no conflict of interest.