A limiting problem for local/non-local p-Laplacians with concave–convex nonlinearities

Abstract

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by

$$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega ,\\ u(x)&{} =&{} 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset \mathbb {R}^n$ is a bounded and smooth domain, $s\in (0,1)$, $2 \le p < \infty$, $0<q(p)<p-1<r(p)<\infty$ and $0<\lambda _p< \infty$, being $\Delta _p$ and $(-\Delta )_p^s$ the p-Laplace and fractional p-Laplace operators, respectively. We study existence and global uniform and explicit boundedness results to weak solutions. Then, we perform an asymptotic analysis for the limit of a family of weak solutions $\{u_p\}_{p\ge 2}$ as $p \rightarrow \infty$, which converges, up to a subsequence (under suitable assumptions on the problem data), to a non-trivial profile with uniform and explicit bounds, enjoying of a universal Lipschitz modulus of continuity, and verifying a nonlinear limiting PDE in the viscosity sense, which exhibits both local/non-local character.

Appendices

Appendix A: The (s, p)-eigenvalue problem and its limit setting

In this section, we state some results on the first eigenvalue and eigenfunction of $\mathcal {P}_{s, p}$ with Dirichlet boundary condition. Given a bounded and open regular domain $\Omega \subset \mathbb {R}^n$, for any $s\in (0,1)$ and $p>1$, the first eigenvalue, namely $\lambda _1(s,p)$, is variationally characterized as the minimizer of following Rayleigh’s quotient (see [8, Section 1.2] for details):

Moreover, if $\varphi _1 = \varphi _1(s,p)\in W^{1, p}_0(\Omega )$ is an associated eigenfunction, then it fulfills the equation

$$\begin{aligned} \left\{ \begin{array}{rclcl} \mathcal {P}_{s, p} \,\varphi _1 &{} = &{} \lambda _1(s,p) |\varphi _1 |^{p-2}\varphi _1 &{} \text{ in } &{}\Omega \\ \varphi _1 &{} = &{} 0 &{} \text{ on } &{} \mathbb {R}^n{\setminus } \Omega , \end{array} \right. \end{aligned}$$

in the weak sense.

By arguing as in [8, Theorems 1.1 and 1.2], it can be proved that $\lambda _1(s,p)$ is simple and its corresponding eigenfunctions are bounded and have constant sign. Furthermore, by supposing (without loss of generality) that $\Vert \varphi _1\Vert _{L^p(\Omega )} = 1$, we have the following limiting characterization (see [8, Theorem 1.3] for more details):

$$\begin{aligned} \displaystyle \lim _{p\rightarrow \infty } \root p \of {\lambda _1(s,p)} \mathrel {\mathop :}=\Lambda _1(\Omega ) = \displaystyle \inf _{u \in W^{1,\infty }_0(\Omega ) \atop {u \not \equiv 0} }\max \left\{ \Vert \nabla u\Vert _{L^\infty (\Omega )}, [u_{\infty }]_{C^{0, s}(\mathbb {R}^n)} \right\} . \end{aligned}$$

(A.1)

Another interesting piece of information is that such an “$\infty$-eigenvalue” enjoys the following (useful and simple) geometric characterization (cf. [38, 43] for similar results in the local and non-local scenarios):

Theorem 1

([8, Theorem 1.3]) Under the above definition, we have that

where $\displaystyle \mathfrak {R}_{\Omega } \mathrel {\mathop :}=\max _{x \in \Omega } \text {dist}(x, \partial \Omega )$. Moreover, there exists $u_{\infty } \in W_0^{1,\infty }(\Omega )$, such that $\Vert u_{\infty }\Vert _{L^{\infty }(\Omega )}=1$, $u_p \rightarrow u_{\infty }$ uniformly in $\Omega$ (up to subsequences) and

$$\begin{aligned} \Lambda _1(\Omega ) = \mathcal {J}_{\infty }(u_{\infty }) \mathrel {\mathop :}=\displaystyle \inf _{w \in W_0^{1,\infty }(\Omega ) \atop {w \not \equiv 0}} \max \left\{ \Vert \nabla u\Vert _{L^\infty (\Omega )}, [u_{\infty }]_{C^{0, s}(\mathbb {R}^n)} \right\} . \end{aligned}$$

Finally, by using the same ideas as in Theorem 1.5 we are able to deduce the corresponding limit operator governing the limiting eigenvalue problem (cf. [29, Theorem 1.3]).

Theorem 2

([8, Theorem 1.4]) Assume that condition (A.1) holds. Then, any uniform limit of (normalized, i.e., $\Vert \varphi \Vert _{L^{\infty }(\Omega )}=1$) eigenfunctions corresponding to the first (s, p)-eigenvalue $\lambda _1(s,p)$, is a viscosity solution to

$$\begin{aligned} \left\{ \begin{array}{rcrcl} &{}&{}\mathcal {G}_{\infty }(\varphi _{\infty }, \nabla \varphi _{\infty }, D^2 \varphi _{\infty }) = 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\varphi _{\infty } > 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\varphi _{\infty } = 0 &{} \text {on} &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \mathcal {G}_{\infty }(v, \nabla v, D^2 v) = \max \{\mathcal {G}_1[v], \, \mathcal {G}_2[v]\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {G}_1[v]&\mathrel {\mathop :}=\min \left\{ \mathcal {L}_{\infty } v,\, \mathcal {L}_{\infty }^{+} v - \max \left\{ \frac{1}{\mathfrak {R}_{\Omega }} , \frac{1}{\mathfrak {R}^s_{\Omega }}\right\} v,\, \mathcal {L}_{\infty }^{+} v -|\nabla v|\right\} \\ \mathcal {G}_2[v]&\mathrel {\mathop :}=\min \left\{ -\Delta _{\infty } v, \,|\nabla v| - \max \left\{ \frac{1}{\mathfrak {R}_{\Omega }} , \frac{1}{\mathfrak {R}^s_{\Omega }}\right\} v, \,|\nabla v| - \mathcal {L}_{\infty }^{+} v, \,|\nabla v| +\mathcal {L}_{\infty }^{-} v\right\} . \end{aligned}$$

Appendix B: The (s, p)-torsional creep problem and its limit counterpart

In this section, we study several properties of the (s, p)-torsional creep-type problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rclcl} &{}&{}\mathcal {P}_{s,p} \, \tau _p = 1 &{} \text{ in } &{} \Omega , \\ &{}&{}\tau _p > 0&{}\text { in } &{} \Omega ,\\ &{}&{}\tau _p = 0&{}\text { on } &{} \mathbb {R}^n {\setminus } \Omega . \end{array} \end{array}\right. } \end{aligned}$$

(B.1)

Notice that existence of a weak solution of (B.1) follows from standard variational arguments. The Euler–Lagrange (or energy) functional $\mathrm {H}_{s, p}:W^{1,p}_0(\Omega ) \rightarrow \mathbb {R}$ associated with such a problem is defined as follows:

$$\begin{aligned} \mathrm {H}_{s, p}(v)=\frac{1}{2p}[v]_{W^{s,p}(\mathbb {R}^n)}^p + \frac{1}{p}\Vert \nabla u\Vert _{L^p(\Omega )}^p -\Vert v\Vert _{L^1(\Omega )}. \end{aligned}$$

We are now in a position to present the existence/uniqueness’ result to (B.1).

Lemma B.1

Let $s\in (0,1)$ and $p>\frac{n}{s}$ be fixed. Then, there exists a unique weak solution of (B.1), which is a minimizer of $\mathrm {H}_{s, p}$ on $W^{1,p}_0(\Omega )$.

Proof

First of all, observe that $\mathrm {H}_{s, p}$ is coercive in $W^{1,p}_0(\Omega )$. Indeed, by using Hölder’s inequality and the characterization of ((s, p)-Eigenvalue) we get that

$$\begin{aligned} \mathrm {H}_{s, p}(v)&\ge \frac{1}{2p}[v]_{W^{s,p}(\mathbb {R}^n)}^p + \frac{1}{p}\Vert \nabla v\Vert _{L^p(\Omega )}^p -\frac{1}{\root p \of {\lambda _1(s,p)}}|\Omega |^{1-\frac{1}{p}}\Vert v\Vert _{W^{1,p}_0(\Omega )}\\&\ge \frac{1}{p}\Vert v\Vert _{W^{1,p}_0(\Omega )}^p -\frac{1}{\root p \of {\lambda _1(s,p)}}|\Omega |^{1-\frac{1}{p}}\Vert v\Vert _{W^{1,p}_0(\Omega )}. \end{aligned}$$

At this point, we stress that the function

$$\begin{aligned} \mathrm {g}_p(t)\mathrel {\mathop :}=\frac{1}{p}t^p-\frac{1}{\root p \of {\lambda _1(s,p)}}|\Omega |^{1-\frac{1}{p}}t \end{aligned}$$

is bounded from below, and in addition, $\mathrm {g}_p(t)\rightarrow \infty$ when $t\rightarrow \infty$, thereby providing the desired property. Moreover, since it is clear that $\mathrm {H}_{s, p}$ is also lower semi-continuous, existence of a minimizer can be ensured by applying the direct methods in the calculus of variations.

Finally, uniqueness follows from the strict convexity of $\mathrm {H}_{s, p}$, which, in turn, follows from the strict monotonicity of the operators $-\Delta _p$ and $(-\Delta )_p^s$, respectively. Lastly, it is clear that any minimizer to $\mathrm {H}_{s, p}$ is a weak solution to (B.1). This concludes the proof. $\square$

In the sequel, we establish a key result which ensures that any family of solutions to (B.1) is uniformly bounded in $W^{1,p}_0(\Omega )$.

Lemma B.2

Let $\tau _p$ be a weak solution of (B.1). Then, there exists $C_p>0$ such that

$$\begin{aligned} \displaystyle \Vert \tau _p\Vert _{W^{1,p}_0(\Omega )}\le C_p \quad \text {with}\quad \lim _{p\rightarrow \infty }C_p=1. \end{aligned}$$

Proof

Remember that a weak solution $\tau _p$ of (B.1) satisfies the following integral relation:

$$\begin{aligned} \Vert \nabla \tau _p\Vert _{L^p(\Omega )}^p+[\tau _p]_{W^{s,p}(\mathbb {R}^n)}^p=\int _\Omega \tau _p(x)\mathrm{d}x. \end{aligned}$$

One more time, by using Hölder’s inequality and ((s, p)-Eigenvalue) we obtain the following:

$$\begin{aligned} \begin{array}{rcl} \displaystyle \Vert \tau _p\Vert _{W^{1,p}_0(\Omega )}^p &{} \le &{} \Vert \nabla \tau _p\Vert _{L^p(\Omega )}^p + [\tau _p]_{W^{s,p}(\mathbb {R}^n)}^p \\ &{} = &{} \displaystyle \int _\Omega \tau _p(x)\mathrm{d}x \\ &{}\le &{} \displaystyle |\Omega |^{1-\frac{1}{p}} \Vert \tau _p\Vert _{L^p(\Omega )}\\ &{} \le &{} \displaystyle \frac{1}{\root p \of {\lambda _1(s,p)}}|\Omega |^{1-\frac{1}{p}}\Vert \tau _p\Vert _{W^{1,p}_0(\Omega )}. \end{array} \end{aligned}$$

Therefore, by defining $\mathrm {h}_p(t)=t^p - \frac{1}{\root p \of {\lambda _1(s,p)}}\mathcal {L}^n(\Omega )^{1-\frac{1}{p}}t$ we get

$$\begin{aligned} \Vert \tau _p\Vert _{W^{1,p}_0(\Omega )}\le C_p:=\sup \{ t>0:\mathrm {h}_p(t)\le 0\}. \end{aligned}$$

Now, notice that

$$\begin{aligned} \mathrm {h}_p(t)\le 0 \quad \text {for} \quad 0<t\le \sqrt{[p-1]{\frac{1}{\root p \of {\lambda _1(s,p)}}\mathcal {L}^n(\Omega )^{1-\frac{1}{p}}}} \qquad \text {and} \qquad \mathrm {h}_p(t)\rightarrow \infty \quad \text {as} \quad t\rightarrow \infty . \end{aligned}$$

Particularly, this implies that

$$\begin{aligned} \lim _{p\rightarrow \infty } \mathrm {h}_p(t)\le 0 \quad \text {for} \quad 0<t\le 1 \qquad \text {and} \qquad \lim _{p\rightarrow \infty } \mathrm {h}_p(t)= \infty \quad \text {for} \quad t>1, \end{aligned}$$

which concludes the proof. $\square$

Next, we prove a convergence result for any family of solutions to (B.1).

Lemma B.3

Let $\{\tau _p\}_{p \ge 2}$ be a family of weak solution to (B.1). Then, up to subsequence,

$$\begin{aligned} \tau _p\rightarrow \tau _\infty \quad \quad \text {uniformly in} \quad \mathbb {R}^n \quad \text {as} \quad p\rightarrow \infty . \end{aligned}$$

In addition, $\tau _\infty \in W^{1,\infty }_0(\Omega )$ and the following (uniform) estimates hold true

$$\begin{aligned} \Vert \tau _{\infty }\Vert _{L^{\infty }(\Omega )} \le C(n) \quad \text {and} \quad \Vert \tau _\infty \Vert _{W^{1,\infty }_0(\Omega )}\le 1. \end{aligned}$$

(B.2)

Proof

Let $\{\tau _p\}_{p \ge 2}$ be any sequence of weak solutions of (B.1). By Morrey’s inequality (Proposition 2.7) and Hölder embedding (Theorem 2.8), we have that

$$\begin{aligned} \begin{array}{rcl} \Vert \tau _p\Vert _{L^{\infty }(\Omega )} &{} \le &{} \max \left\{ \Vert \tau _p\Vert _{C^{0,1-\frac{n}{p}}(\Omega )}, \Vert \tau _p\Vert _{C^{0, s-\frac{n}{p}}(\mathbb {R}^n)}\right\} \\ &{} \le &{} \max \left\{ \mathbf{C}_H\Vert \nabla \tau _p\Vert _{L^p(\Omega )}, \mathbf{C}^{*}_H\Vert \tau _p\Vert _{W^{s, p}(\mathbb {R}^n)}\right\} \\ &{} \le &{} \max \left\{ \mathbf{C}_H, \mathbf{C}^{*}_H\right\} \Vert \tau _p\Vert _{W^{s, p}(\mathbb {R}^n)}\\ &{}\le &{} \max \left\{ \mathbf{C}_H\mathbf{C}_p, \mathbf{C}^{*}_HC_p \right\} , \end{array} \end{aligned}$$

which implies the uniform boundedness and pre-compactness for $p\gg 1$ sufficiently large. Hence, by Arzelá–Ascoli compactness criterion, there exists a subsequence of $\{\tau _p\}_{p \ge 2}$ such that

$$\begin{aligned} \lim _{p\rightarrow \infty } \tau _p=\tau _\infty \quad \text { uniformly in } \quad \mathbb {R}^n \end{aligned}$$

and such that $\tau _\infty =0$ in $\mathbb {R}^n{\setminus } \Omega$.

Finally, notice that by sending $p\rightarrow \infty$, since $\displaystyle \lim _{p \rightarrow \infty } C_p = 1$ (see Lemma B.2), we conclude that $\tau _\infty \in W^{1, \infty }_0(\Omega )$ (extended to zero outside $\Omega$) and the desired estimates in (B.2) hold true. $\square$

From Lemmas B.2 and B.3, there exists a function $\tau _\infty$ such that $\tau _p\rightarrow \tau _\infty$ uniformly in ${\bar{\Omega }}$. Therefore, applying mutatis mutandis the arguments in the proof of Theorem 1.5 we obtain a characterization of the limit equation fulfilled by the limit profile (in the viscosity sense). We leave the interested reader to verify the details.

Theorem B.4

Let $\Omega \subset \mathbb {R}^n$ be a bounded domain and $p>n$. Then, the sequence $\{\tau _p\}_{p\ge 2}$ of weak solutions of (B.1) converge uniformly as $p\rightarrow \infty$ to a viscosity solution to

$$\begin{aligned} \left\{ \begin{array}{rcrcl} &{}&{}\mathcal {T}_{\infty }(\tau _\infty , \nabla \tau _\infty , D^2 \tau _\infty ) = 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\tau _\infty > 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\tau _\infty = 0 &{} \text {on} &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

(B.3)

where

$$\begin{aligned} \mathcal {T}_{\infty }(v, \nabla v, D^2 v) = \max \{\mathcal {T}_1[v], \, \mathcal {T}_2[v]\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {T}_1[v]&\mathrel {\mathop :}=\min \left\{ \mathcal {L}_{\infty } v, \mathcal {L}_{\infty }^{+} v -1, \mathcal {L}_{\infty }^{+} v -|\nabla v|\right\} \\ \mathcal {T}_2[v]&\mathrel {\mathop :}=\min \left\{ -\Delta _{\infty } v, |\nabla v| - 1, |\nabla v| - \mathcal {L}_{\infty }^{+} v, |\nabla v| +\mathcal {L}_{\infty }^{-} v\right\} . \end{aligned}$$

Appendix C: The (s, p)-concave problem and its limit scenario

In this final section, we deal with the (s, p)-concave problem given by

$$\begin{aligned} \left\{ \begin{array}{rclcl} &{}&{}\mathcal {P}_{s, p} v = \lambda _p v^q &{}\quad \text { in } &{} \Omega ,\\ &{}&{}v> 0 &{}\quad \text { in }&{} \Omega ,\\ &{}&{}v= 0 &{}\quad \text { on } &{} \mathbb {R}^n{\setminus } \Omega ,\\ \end{array} \right. \end{aligned}$$

(C.1)

where $q=q(p)$ is assumed to satisfy (LC).

We start by proving existence/uniqueness of weak solutions of the concave problem (C.1).

Lemma C.1

Let $\Omega \subset \mathbb {R}^n$ be a bounded and regular domain, $p>n$ and $0<q<p-1$. Then, for any $\lambda _p>0$ there exists a unique weak solution $\kappa _p \in W^{1,p}_0(\Omega )$ of (C.1).

Proof

In light of Proposition 2.6, existence of a weak solution will follow if we can find a sub-solution $\underline{v}$ and a super-solution $\overline{v}$ of (C.1) such that $\underline{v}\le \overline{v}$.

In order to construct a sub-solution, according to Appendix 4, let $\lambda _1(s,p)$ be the first eigenvalue of $\mathcal {P}_{s, p}$ in $\Omega$ with (zero) Dirichlet boundary condition, and let $\varphi _1(s,p)$ be the corresponding eigenfunction normalized such that $\Vert \varphi _1\Vert _{L^{\infty }(\Omega )}=1$. Now, we define $\underline{v}=\theta \varphi _1 \in W^{s, p}_0(\mathbb {R}^n)$ with $\theta >0$ to be chosen a posteriori. Then, for any nonnegative $\psi \in W^{1,p}_0(\Omega )$ it holds that

$$\begin{aligned} \begin{array}{rcl} \displaystyle \mathcal {E}_p(\underline{v},\psi ) + \mathcal {E}_{s,p}(\underline{v},\psi ) &{} = &{} \displaystyle \lambda _1(s, p) \int _\Omega \underline{v}^{p-1}\psi \,\mathrm{d}x \\ &{} \le &{} \displaystyle \frac{\lambda _1(s, p)\theta ^{p-1}}{\lambda _p \theta ^{q}} \int _\Omega \lambda _p \underline{v}^q \psi \,\mathrm{d}x. \end{array} \end{aligned}$$

Therefore, $\underline{v}$ is the required weak sub-solution provided we choose $\theta$ such that

$$\begin{aligned} 0<\theta \le \min \left\{ 2, \left( \frac{\lambda _p}{\lambda _1(s,p)}\right) ^\frac{1}{p-1-q}\right\} . \end{aligned}$$

In the sequel, let us construct a weak super-solution. For this purpose, we choose a larger (regular) domain ${\tilde{\Omega }} \supset \Omega$ with eigenvalue ${\tilde{\lambda }}_1 = {\tilde{\lambda }}_1(s,p)$ such that the corresponding positive eigenfunction ${\tilde{\varphi }}_1 = {\tilde{\varphi }}_1(s,p)$ is normalized with $\displaystyle \min _{x\in \Omega } {\tilde{\varphi }}_1=1$.

Now, we define $\overline{v}=\iota (1+{\tilde{\varphi }}_1)$. Observe that $\overline{v}\ge 2\iota$ in $\Omega$. For this reason, by taking $\iota$ large enough (to be explicitly chosen a posteriori), we ensure that $\overline{v}\ge \underline{v}$ in $\Omega$. Moreover, for any nonnegative ${\tilde{\psi }}\in W^{1, p}_0({\tilde{\Omega }})$ it holds that

$$\begin{aligned} \mathcal {E}_p({\overline{v}}, {\tilde{\psi }}) + \mathcal {E}_{s,p}({\overline{v}}, {\tilde{\psi }})&= \iota ^{p-1}(\mathcal {E}_p({\tilde{\varphi }}_1, {\tilde{\psi }}) + \mathcal {E}_{s,p}({\tilde{\varphi }}_1, {\tilde{\psi }})) =\iota ^{p-1} {\tilde{\lambda }}_1 \int _{{\tilde{\Omega }}} {\tilde{\varphi }}_1^{p-1} {\tilde{\psi }}\,\mathrm{d}x\\&\ge \iota ^{p-1} {\tilde{\lambda }}_1 \int _{{\tilde{\Omega }}} {\tilde{\varphi }}_1^q {\tilde{\psi }}\,\mathrm{d}x \ge \frac{\iota ^{p-1}}{2^q} {\tilde{\lambda }}_1 \int _{{\tilde{\Omega }} } ({\tilde{\varphi }}_1+1)^q {\tilde{\psi }}\,\mathrm{d}x\\&\ge \frac{ {\tilde{\lambda }}_1 \iota ^{p-1}}{ (2\iota )^q\lambda _p} \int _\Omega \lambda _p \overline{v}^q \psi \,\mathrm{d}x. \end{aligned}$$

As a result, we find that $\overline{v}$ is the desired weak super-solution (in $\Omega$) provided $\iota >0$ fulfills:

$$\begin{aligned} \iota \ge \max \left\{ 1, \left( \frac{2^q \lambda _p}{{\tilde{\lambda }}_1(s,p)} \right) ^\frac{1}{p-1-q}\right\} , \end{aligned}$$

where we have used that $W^{1,p}_0(\Omega ) \subset W^{1,p}_0({\tilde{\Omega }})$.

In conclusion, we remark that for $p \gg 1$ large enough, in view of (LC) and (A.1) we can choose both $\theta$ and $\iota$ independent of p as follows:

$$\begin{aligned} 0< \theta \le \min \left\{ 2, \frac{1}{2} \left( \frac{\lambda _\infty }{\Lambda _1(\Omega )} \right) ^\frac{1}{1-\mathcal {Q}}\right\} \qquad \text {and} \qquad \iota \ge \max \left\{ 1, 2.2^{\frac{1}{\mathcal {Q}-1}}\left( \frac{\lambda _\infty }{\Lambda _1({\tilde{\Omega }})} \right) ^\frac{1}{1-\mathcal {Q}}\right\} . \end{aligned}$$

The proof is now concluded. $\square$

Finally, we derive the corresponding limiting operator governing the limit of any family to the associated concave problem C.1.

Theorem C.2

Let $\Omega \subset \mathbb {R}^n$ be a bounded and regular domain. Suppose that assumptions (LC) are in force. Then, the sequence of weak solutions $(\kappa _p)_{p\ge 2}$ of (C.1) converges uniformly to $\kappa _{\infty }$, a viscosity solution of

$$\begin{aligned} \left\{ \begin{array}{rcrcl} &{}&{}\mathcal {K}_{\infty }(\kappa _{\infty }, \nabla \kappa _{\infty }, D^2 \kappa _{\infty }) = 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\kappa _{\infty } > 0 &{} \text {in} &{} \Omega ,\\ &{}&{}\kappa _{\infty } = 0 &{} \text {on} &{} \mathbb {R}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$

(C.2)

where

$$\begin{aligned} \mathcal {K}_{\infty }(v, \nabla v, D^2 v) = \max \{\mathcal {K}_1[v], \, \mathcal {K}_2[v]\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {K}_1[v]&\mathrel {\mathop :}=\min \left\{ \mathcal {L}_{\infty }\, v,\, \mathcal {L}_{\infty }^{+}\, v - \lambda _{\infty }v^{\mathcal {Q}},\, \mathcal {L}_{\infty }^{+} \,v -|\nabla v|\right\} \\ \mathcal {K}_2[v]&\mathrel {\mathop :}=\min \left\{ -\Delta _{\infty }\, v, |\nabla v| - \lambda _{\infty } v^{\mathcal {Q}},\, |\nabla v| - \mathcal {L}_{\infty }^{+}\, v, |\nabla v| +\mathcal {L}_{\infty }^{-}\, v\right\} . \end{aligned}$$

Proof

The uniform convergence of the sequence follows with the same arguments that in the proofs of Theorem 1.3 and Proposition 4.1. The equation satisfied by the limit profile can be deduced with the same procedure that in Theorem 1.5. $\square$