1 Introduction

This note deals with the Hölder continuity issue of solutions of degenerate elliptic equations of the form

$$\begin{aligned} \mathcal {M}_{\mathbf {a}}(D^2u)=f(x)\quad \;\text {in }\Omega , \end{aligned}$$
(1.1)

where \(\Omega \subset \mathbb {R}^N\) is a domain, \(f\in C(\Omega )\), and \(\mathcal {M}_{\mathbf {a}}\) is the weighted partial trace operator defined, for any symmetric matrix X, by the formula

$$\begin{aligned} \mathcal {M}_{\mathbf {a}}(X)=\sum _{i=1}^Na_i\lambda _i(X). \end{aligned}$$
(1.2)

In (1.2), \(\lambda _1(X)\le \cdots \le \lambda _N(X)\) are the ordered eigenvalues of \(X\in \mathbb S^N\), where \(\mathbb S^N\) denotes the space of \(N\times N\) real symmetric matrices, and \(\mathbf {a}=\left( a_1,\ldots ,a_N\right) \) is such that \(a_i\ge 0\) for any \(i=1,\ldots ,N\). It is plain that \(\mathcal {M}_{\mathbf {a}}\) reduces to the classical Laplace operator when \(\mathbf {a}=\left( 1,\ldots ,1\right) \) and that it falls out the class of uniformly elliptic operators as soon as \(a_i=0\) for some \(i=1,\ldots ,N\). Such operators include, as particular cases, significant examples of degenerate operators, for instance,

$$\begin{aligned} \mathcal {P}^-_k(X)=\sum _{i=1}^k\lambda _i(X)\quad \;\text {and}\quad \; \mathcal {P}^+_k(X)=\sum _{i=1}^k\lambda _{N-k+i}(X), \end{aligned}$$

which arise in the study of various geometric and elliptic problems, see, e.g., [1, 2, 9,10,11, 16,17,18, 22, 23], as well as the operators \(\lambda _k(X)\), for some \(k\in \left\{ 1,\ldots ,N\right\} \), whose interest has been developed in the framework of differential games theory, see [3,4,5,6].

In [15], the authors studied qualitative properties of solutions of (1.1) under the further assumption that \(a_1>0\) and \(a_N>0\), having in mind as prototype the Isaacs operator

$$\begin{aligned} \lambda _1(X)+\lambda _N(X)=\min _{|\xi |=1}\max _{|\eta |=1}\left( \left\langle X\xi ,\xi \right\rangle +\left\langle X\eta ,\eta \right\rangle \right) , \end{aligned}$$
(1.3)

which is neither uniformly elliptic nor convex/concave (in dimension \(N\ge 3\)). Among other results, they in particular obtained an Alexandov-Bakelman-Pucci (ABP) type inequality following the scheme of the proof showed in [7], starting from the fact that (see [7, Section 2.2] for the notation) if u is a viscosity solution of (1.1), then

$$\begin{aligned} u\in \underline{\mathcal {S}}(\frac{a_*}{N-1},|\mathbf {a}|_\infty ,f)\cap \overline{\mathcal {S}}(\frac{a_*}{N-1},|\mathbf {a}|_\infty ,f), \end{aligned}$$
(1.4)

where from now on we denote \(a_*=\min \left\{ a_1,a_N\right\} \), \(|\mathbf {a}|_1=a_1+\cdots +a_N,\) and \(|\mathbf {a}|_\infty =\text {max} \left\{ a_1,\ldots ,a_N\right\} \). As a byproduct, they obtain, in the same way as in the uniformly elliptic case, that viscosity solutions of (1.1) are \(C_{\text {loc}}^{0,\alpha }(\Omega )\), where \(\alpha \in (0,1)\), which is not explicitly known, depends on the constant that appears in the ABP estimate. They did not obtain any further result about a possible lower bound on \(\alpha \) or, possibly, a sharper result about the regularity of solutions due to the lack of structure in the nonlinear equation.

The goal of this note is to provide an explicit lower bound for \(\alpha \), only depending on \(a_1\) and \(a_N\) which therefore are assumed to be both positive. Let us point out that our result does not apply to \(\mathcal {P}^\pm _k\) when \(k<N\) since in these cases \(a_1\) or \(a_N\) are zero. Nevertheless, some regularity results concerning such operators, in particular for \(k=1\), can be found in [19] and [2, Propositions 3.1-3.2]-[15, Theorem 1.4]. Applying the Ishii-Lions approach to the problem (see [20]), we manage to prove that viscosity solutions of (1.1) are \(C^{0,\beta }_{\text {loc}}(\Omega )\), where

$$\begin{aligned} \beta =1-\frac{a_1+a_N}{\left( \sqrt{a_1}+\sqrt{a_N}\right) ^2}\,. \end{aligned}$$
(1.5)

From this, we infer that \(\alpha \ge \beta \) and, concerning the main example (1.3), we in particular obtain that \(\alpha \ge \frac{1}{2}\). It is worth to point out that the fundamental assumption in the strategy of Ishii-Lions, in order to prove the Lipschitz continuity of solutions, is the uniformly ellipticity of the equation which clearly is outside our setting. Nevertheless, using the assumption \(a_1>0\) and \(a_N>0\), we are still able to detect some useful information encoded in the structure of the operator, so leading to the \(\beta \)-regularity of solutions, where \(\beta \), defined in (1.5), is strictly less than one. In addition, this approach can be applied to a larger class of operators with first order terms. Thus, for stating our main result, we introduce the class of the equations we are going to consider. Let

$$\begin{aligned} \mathcal {M}_{\mathbf {a}}(D^2u)+H(\nabla u)=f(x)\quad \;\text {in}~ \Omega , \end{aligned}$$
(1.6)

where \(\Omega \subseteq \mathbb R^N\) is a domain, f is continuous in \(\Omega \), and \( \mathcal {M}_{\mathbf {a}}\) is the fully nonlinear operator that we have introduced in (1.2).

Our assumptions are:

  1. (H1)

    \(\mathcal {M}_{\mathbf {a}}\in \mathcal A=\{\mathcal {M}_{\mathbf {a}}(X):=\sum _{i=1}^Na_i\lambda _i(X):\,\, a_i\ge 0,\, i=1,\dots ,N,\,\, a_1>0,\,a_N>0,\,X\in \mathbb S^N \}.\)

  2. (H2)

    \(H\in C(\mathbb R^N)\) and there exists a nonnegative constant \(C_H\) such that

    $$\begin{aligned} |H(p+q)-H(p)|\le C_H(1+|p|+|q|)|q| \end{aligned}$$
    (1.7)

    for every \(p,q\in \mathbb R^N\).

A typical example of H satisfying (1.7) is \(H(p)=A|p|^2+B|p|^\tau \), where \(\tau \in [0,2]\) and \(A,B\in \mathbb R\). Although we shall allow H to have a quadratic growth in the gradient variable, the prototype equation to be kept in mind is still the one obtained when \(H\equiv 0\), e.g.,

$$\begin{aligned} a_1\lambda _1(D^2u)+a_N\lambda _N(D^2u)=f(x)\quad \;\text {in }\Omega , \end{aligned}$$

with \(a_1,a_N>0\).

Now, we are in position to state our main result.

Theorem 1.1

Let \(u\in C(\Omega )\) be a viscosity solution of (1.6). If (H1)–(H2) hold, then

$$\begin{aligned} u\in C_{\text {loc}}^{0,\beta }(\Omega ),\quad \qquad \beta =1-\frac{a_1+a_N}{\left( \sqrt{a_1}+\sqrt{a_N}\right) ^2}, \end{aligned}$$

and the following estimate holds: for any \(\omega \subset \subset \omega '\subset \subset \Omega \), one has

$$\begin{aligned} \left\| u\right\| _{C^{0,\beta }(\omega )}\le C=C\left( a_1,a_N,\text {dist}(\omega ,\omega '), C_H, \left\| u\right\| _{L^\infty (\omega ')},\left\| f\right\| _{L^\infty (\omega ')}\right) . \end{aligned}$$

The main consequence of Theorem 1.1 is a lower bound of the expected regularity of viscosity solutions to a large class of operators that are not uniformly elliptic. We point out that very few results are known about the sharp regularity of solutions to fully nonlinear equations that are not convex/concave and that are not uniformly elliptic. In particular, we recall the fundamental result [21]. Concerning the regularity issues of viscosity solutions of degenerate equations closely related to ours, we refer to [8, 13,14,15].

We conclude this introduction by pointing out that if we drop the assumption (H1), in the sense that \(a_1=a_N=0\), then there exist viscosity solutions of

$$\begin{aligned} a_2\lambda _2(D^2u)+\cdots +a_{N-1}\lambda _{N-1}(D^2u)=0\quad \text {in}~ B_1, \end{aligned}$$

which do not belong to any \(C_{\text {loc}}^{0,\alpha }(B_1)\) for \(\alpha \in (0,1)\), even if \(a_i>0\) for any \(i=2,\ldots ,N-1\). We present a simple example at the end of this note.

2 Hölder regularity

We start with the following elementary lemma.

Lemma 2.1

Let \(\delta >0\) and let A, B, C, D be nonnegative constants such that \(A\in (0,1)\) and \(C>0\). Then there exists \(\varphi \in C^2((0,\delta ])\cap C\left( [0,\delta ]\right) \), depending on \(A,B,C,D,\delta \), which is a positive solution of

$$\begin{aligned} \varphi ''(r)+\left( \frac{A}{r}+B\right) \varphi '(r)=-C,\qquad r\in (0,\delta ], \end{aligned}$$
(2.1)

and satisfies the following conditions:

$$\begin{aligned}&\varphi '(r)>0 \quad \text {and}\quad \varphi ''(r)<0\quad \text {for any}\quad r\in (0,\delta ], \end{aligned}$$
(2.2)
$$\begin{aligned}&\varphi ''(r)-\frac{\varphi '(r)}{r}\le -C\quad \text {for any}\quad r\in (0,\delta ], \end{aligned}$$
(2.3)
$$\begin{aligned}&\varphi (\delta )\ge D, \end{aligned}$$
(2.4)
$$\begin{aligned}&\sup _{0<r\le \delta }\frac{\varphi (r)}{r^{1-A}}<+\infty . \end{aligned}$$
(2.5)

Proof

By a straightforward computation, for any \(K\in \mathbb R\), the function

$$\begin{aligned} \varphi (r)=\int \limits _0^r\psi (s)\,ds\qquad \text {where}\qquad \psi (s)=\frac{e^{-Bs}}{s^A}\left( K-C\int \limits _0^st^Ae^{Bt}\,dt\right) \end{aligned}$$

is a solution of (2.1). Pick \(K=K(A,B,C,D,\delta )\) such that \(\psi (\delta )=\frac{D}{\delta }\). Hence \(\varphi '(r)>0\) for \(r\in (0,\delta ]\) and, since \(\varphi (0)=0\), then \(\varphi (r)>0\) for any \(r\in (0,\delta ]\). Moreover, just using the equation (2.1), we infer that (2.2) holds. Condition (2.3) easily follows by (2.2) and again using (2.1). Since \(\psi \) is a decreasing function in \((0,\delta ]\), we have

$$\begin{aligned} \varphi (\delta )=\int \limits _0^\delta \psi (s)\,ds\ge \delta \psi (\delta )=D \end{aligned}$$

by the choice of K. This shows (2.4). To conclude, it is sufficient to observe that, for any \(r\in (0,\delta ]\),

$$\begin{aligned} \varphi (r)\le K\int \limits _0^r\frac{1}{s^A}\,ds=\frac{K}{1-A}r^{1-A}. \end{aligned}$$

\(\square \)

Proof of Theorem 1.1

Take \(\delta >0\) small enough such that \(\omega _{2\delta }=\big \{x\in \mathbb R^N:\,\text {dist}(x,\omega )<2\delta \big \}\subset \omega '\). Fix \(z\in \omega \) and let

$$\begin{aligned} \Delta _z=\left\{ (x,y)\in \Omega \times \Omega :\;|x-y|<\delta ,\,|x-z|<\delta \right\} . \end{aligned}$$

Note that if \((x,y)\in \Delta _z\), then both x and y belong in particular to \(\omega '\). For \((x,y)\in \overline{\Delta }_z\), let

$$\begin{aligned} \phi (x,y)=u(x)-u(y)-\varphi (|x-y|)-L|x-z|^2, \end{aligned}$$

where \(\varphi (r)\) is, for \(r\in [0,\delta ]\), the function provided by Lemma 2.1 and depending on the parameters A, B, C, D. Let \(A=1-\beta \in (0,1)\). We claim that for an appropriate choice of B, C, D and L, then

$$\begin{aligned} \max _{(x,y)\in \overline{\Delta }_z}\phi (x,y)\le 0. \end{aligned}$$
(2.6)

This will imply the desired result, taking first \(x=z\), then making z vary and using (2.5).

Set

$$\begin{aligned} \begin{aligned} L&=\frac{2\left\| u\right\| _{L^\infty (\omega ')}}{\delta ^2},\\ D&=2\left\| u\right\| _{L^\infty (\omega ')},\\ C&=\frac{2\left( L\left( |\mathbf {a}|_1+C_H\delta (1+2L\delta )\right) +\left\| f\right\| _{L^\infty (\omega ')}+1\right) }{\left( \sqrt{a_1}+\sqrt{a_N}\right) ^2},\\ B&=\frac{2L\delta C_H}{\left( \sqrt{a_1}+\sqrt{a_N}\right) ^2}\,. \end{aligned} \end{aligned}$$
(2.7)

By contradiction, we suppose that (2.6) does not hold. Let \(({\hat{x}},{\hat{y}})\in \overline{\Delta }_z\) be such that

$$\begin{aligned} \max _{(x,y)\in \overline{\Delta }_z}\phi (x,y)=\phi ({\hat{x}},{\hat{y}})>0. \end{aligned}$$
(2.8)

By (2.8), it is clear that \({\hat{x}}\ne {\hat{y}}\). Moreover, using (2.7), we exclude that \(|{\hat{x}}-{\hat{y}}|=\delta \) or \(|{\hat{x}}-z|=\delta \). Hence \(({\hat{x}},{\hat{y}})\in \Delta _z\). By a standard result in theory of viscosity solutions, see [12, Theorem 3.2 and Remark 3.8], for any \(\varepsilon >0\), there exist matrices \(X_\varepsilon ,Y_\varepsilon \in \mathbb S^N\) such that

$$\begin{aligned}&\left( \nabla \varphi (|{\hat{x}}-{\hat{y}}|)+2L({\hat{x}}-z),X_\varepsilon +2LI\right) \in \overline{J}^{2,+} u({\hat{x}}), \end{aligned}$$
(2.9)
$$\begin{aligned}&\left( \nabla \varphi (|{\hat{x}}-{\hat{y}}|),Y_\varepsilon \right) \in \overline{J}^{2,-} u({\hat{y}}), \end{aligned}$$
(2.10)

and

$$\begin{aligned} \left( \begin{array}{cc} X_\varepsilon &{} 0\\ 0 &{} -Y_\varepsilon \end{array}\right) \le \left( \begin{array}{rr} \Theta _\varepsilon &{} -\Theta _\varepsilon \\ -\Theta _\varepsilon &{} \Theta _\varepsilon \end{array}\right) , \end{aligned}$$
(2.11)

where \(I\in \mathbb S^N\) is the identity matrix and \(\Theta _\varepsilon \in \mathbb S^N\) is given by

$$\begin{aligned}&\Theta _\varepsilon =\varphi ''(|{\hat{x}}-{\hat{y}}|)\left( 1+2\varepsilon \varphi ''(|{\hat{x}}-{\hat{y}}|)\right) P\\&\,\,\qquad +\frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\left( 1+2\varepsilon \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) (I-P) \end{aligned}$$

where \(P=\frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\otimes \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\). Note that the eigenvalues of \(\Theta _\varepsilon \) are \(\varphi ''(|{\hat{x}}-{\hat{y}}|)\big (1+2\varepsilon \varphi ''(|{\hat{x}}-{\hat{y}}|)\big )\), which is simple, and \(\frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\left( 1+2\varepsilon \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) \) with multiplicity \(N-1\). In view of (2.3), we can assume that, for \(\varepsilon \) sufficiently small, one has

$$\begin{aligned} \lambda _1(\Theta _\varepsilon )=\varphi ''(|{\hat{x}}-{\hat{y}}|)\left( 1+2\varepsilon \varphi ''(|{\hat{x}}-{\hat{y}}|)\right) \end{aligned}$$

and that

$$\begin{aligned} \lambda _2(\Theta _\varepsilon )=\cdots =\lambda _{N}(\Theta _\varepsilon )=\frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\left( 1+2\varepsilon \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) . \end{aligned}$$

Using (2.9)-(2.10), and the equation (1.6), we then obtain

$$\begin{aligned} \begin{aligned} -2\left\| f\right\| _{L^\infty (\omega ')}&\le a_1\lambda _1(X_\varepsilon )+a_N\lambda _N(X_\varepsilon )-a_1\lambda _1(Y_\varepsilon )-a_N\lambda _N(Y_\varepsilon )\\&\qquad +\sum _{i=2}^{N-1}a_i\left( \lambda _i(X_\varepsilon )-\lambda _i(Y_\varepsilon )\right) +2L|\mathbf {a}|_1\\&\qquad +H\left( \nabla \varphi (|{\hat{x}}-{\hat{y}}|)+2L({\hat{x}}-z)\right) -H\left( \nabla \varphi (|{\hat{x}}-{\hat{y}}|)\right) \,. \end{aligned} \end{aligned}$$

Since \(X_\varepsilon \le Y_\varepsilon \) and \(a_i\ge 0\), then using (1.7) and (2.2), we have

$$\begin{aligned} \begin{aligned} -2\left\| f\right\| _{L^\infty (\omega ')}&\le a_1\lambda _1(X_\varepsilon )+a_N\lambda _N(X_\varepsilon )-a_1\lambda _1(Y_\varepsilon )-a_N\lambda _N(Y_\varepsilon )\\&\qquad +2L|\mathbf {a}|_1+2L\delta C_H(1+\varphi '(|{\hat{x}}-{\hat{y}}|)+2L\delta )\,. \end{aligned} \end{aligned}$$
(2.12)

In order to reach a contradiction, we now estimate the right hand side of (2.12) using the inequality

$$\begin{aligned} \left( \begin{array}{cc} X_\varepsilon &{} 0\\ 0 &{} -Y_\varepsilon \end{array}\right) \left( \begin{array}{c}v\\ w\end{array}\right) \cdot \left( \begin{array}{c}v\\ w\end{array}\right) \le \left( \begin{array}{rr} \Theta _\varepsilon &{} -\Theta _\varepsilon \\ -\Theta _\varepsilon &{} \Theta _\varepsilon \end{array}\right) \left( \begin{array}{c}v\\ w\end{array}\right) \cdot \left( \begin{array}{c}v\\ w\end{array}\right) \quad \forall v,w\in \mathbb R^N\nonumber \\ \end{aligned}$$
(2.13)

and choosing v, w in a suitable way. With the choice

$$\begin{aligned} v=\sqrt{a_1}\,\frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\;,\quad w=-\sqrt{a_N}\,\frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}, \end{aligned}$$

then (2.13) yields

$$\begin{aligned} \begin{aligned} a_1\lambda _1(X_\varepsilon )-a_N\lambda _N(Y_\varepsilon )&\le a_1X_\varepsilon \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\cdot \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}-a_NY_\varepsilon \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\cdot \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\\&\le \left( \sqrt{a_1}+\sqrt{a_N}\right) ^2\Theta _\varepsilon \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\cdot \frac{{\hat{x}}-{\hat{y}}}{|{\hat{x}}-{\hat{y}}|}\\&=\left( \sqrt{a_1}+\sqrt{a_N}\right) ^2\varphi ''(|{\hat{x}}-{\hat{y}}|)\left( 1+2\varepsilon \varphi ''(|{\hat{x}}-{\hat{y}}|)\right) . \end{aligned} \end{aligned}$$
(2.14)

On the other hand, taking

$$\begin{aligned} v=\sqrt{a_N}\,\xi \;,\quad w=0, \end{aligned}$$

where \(|\xi |=1\) and \(X_\varepsilon \xi =\lambda _N(X_\varepsilon )\xi \), we have

$$\begin{aligned} a_N\lambda _N(X_\varepsilon )\le a_N\,\Theta _\varepsilon \xi \cdot \xi \le a_N\frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\left( 1+2\varepsilon \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) . \end{aligned}$$
(2.15)

In a similar way, we also obtain that

$$\begin{aligned} -a_1\lambda _1(Y_\varepsilon )\le a_1\Theta _\varepsilon \xi \cdot \xi \le a_1\frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\left( 1+2\varepsilon \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) . \end{aligned}$$
(2.16)

Putting together (2.12), (2.14)-(2.16), we infer that

$$\begin{aligned} \begin{aligned} -2\left\| f\right\| _{L^\infty (\omega ')}&\le \left( \sqrt{a_1}+\sqrt{a_N}\right) ^2\varphi ''(|{\hat{x}}-{\hat{y}}|)+\left( \frac{a_1+a_N}{|{\hat{x}}-{\hat{y}}|}+2 L\delta C_H\right) \varphi '(|{\hat{x}}-{\hat{y}}|)\\ {}&\qquad +2L\left( |\mathbf {a}|_1+C_H\delta (1+2L\delta )\right) \\ {}&\qquad +2\varepsilon \left[ \left( \sqrt{a_1}+\sqrt{a_N}\right) ^2(\varphi ''(|{\hat{x}}-{\hat{y}}|))^2+(a_1+a_N)\left( \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) ^2\right] \,. \end{aligned}\nonumber \\ \end{aligned}$$
(2.17)

By (2.7) and Lemma 2.1, the function \(\varphi (r)\) is a solution, for \(r\in (0,\delta ]\), of the ordinary differential equation

$$\begin{aligned} \begin{aligned} \left( \sqrt{a_1}+\sqrt{a_N}\right) ^2\varphi ''(r)&+\left( \frac{a_1+a_N}{r}+2 L\delta C_H\right) \varphi '(r)=\\ {}&=-2\left( L\left( |\mathbf {a}|_1+C_H\delta (1+2L\delta )\right) +\left\| f\right\| _{L^\infty (\omega ')}+1\right) . \end{aligned}\nonumber \\ \end{aligned}$$
(2.18)

Coupling (2.17)-(2.18), then

$$\begin{aligned} 1\le \varepsilon \left[ \left( \sqrt{a_1}+\sqrt{a_N}\right) ^2(\varphi ''(|{\hat{x}}-{\hat{y}}|))^2+(a_1+a_N)\left( \frac{\varphi '(|{\hat{x}}-{\hat{y}}|)}{|{\hat{x}}-{\hat{y}}|}\right) ^2\right] , \end{aligned}$$

leading to a contradiction for \(\varepsilon \) small enough.\(\square \)

Remark 2.2

We note that in the case \(H\equiv 0\), the function \(\varphi (r)\) used in the proof of Theorem 1.1 is more explicit, in fact it is given by \(\varphi (r)=Ar^\beta -Br^2\) for a suitable choice of \(A,B>0\).

2.1 Lack of regularity

Let \(N\ge 3\) and consider the equation

$$\begin{aligned} a_2\lambda _2(D^2u)+\cdots +a_{N-1}\lambda _{N-1}(D^2u)=0\quad \text {in}~ B_1. \end{aligned}$$
(2.19)

We are going to exhibit a continuous function u which is a solution of (2.19) for any \(a_i\ge 0\) and \(i=2,\ldots ,N-1\), but which does not belong to \(C^{0,\alpha }_{\text {loc}}(B_1)\) for any possible choice of \(\alpha \in (0,1]\).

Let \(f:(-1,1)\mapsto \mathbb R\) be the function

$$\begin{aligned} f(t)=\left\{ \begin{array}{cl} \frac{1}{2-\log |t|} &{} \text {if } t\ne 0,\\ 0 &{} \text {if }t=0, \end{array} \right. \end{aligned}$$

and consider it as a function of N variables just by setting \(u(x)=f(x_1)\) for \(x\in B_1\). It is clear that \(u\in C(B_1)\) but \(u\notin C^{0.\alpha }_{\text {loc}}(B_1)\) for any \(\alpha \in (0,1]\). We claim that u is a viscosity solution of (2.19).

The function u is smooth for \(x\in B_1\backslash \left\{ x\in B_1:\,x_1=0\right\} \) and

$$\begin{aligned} D^2u(x)=\text {diag}(f''(x_1),0,\ldots ,0). \end{aligned}$$

Since \(f''(t)\le 0\) for any \(t\in (-1,1)\backslash \left\{ 0\right\} \), we infer that u is in fact a classical solution of (2.19) in the set \(B_1\backslash \left\{ x\in B_1:\,x_1=0\right\} \). Now we prove that u satisfies (in viscosity sense) the equation (2.19) also in \(\left\{ x\in B_1:\,x_1=0\right\} \). For \(x\in \mathbb R^N\) such that \(x_1=0\), we adopt the notation \(x=(0,x')\) with \(x'\in \mathbb R^{N-1}\). Let \(x_0=(0,x'_0)\in B_1\). Since there are no test functions \(\phi \in C^2(B_1)\) touching u from above at \(x_0\), we infer that u is a viscosity subsolution of (2.19). As far as the supersolution property is concerned, it is sufficient to prove that if \(\phi \in C^2(B_1)\) is such that

$$\begin{aligned} 0=u(0,x'_0)=\phi (0,x'_0)\quad \text {and}\quad u(x)\ge \phi (x)\;\;\;\forall x\in B_1, \end{aligned}$$
(2.20)

then \(\lambda _{N-1}(D^2\phi (0,x'_0))\le 0\). Set \(\psi (x')=\phi (0,x')\) for \(|x'|<1\). From (2.20), we deduce that \(\psi (x')\) attains its maximum at \(x'_0\). Hence

$$\begin{aligned} \left\langle D^2\psi (x'_0)v,v\right\rangle \le 0\quad \;\forall v\in \mathbb R^{N-1}. \end{aligned}$$
(2.21)

Using the Courant-Fischer formula

$$\begin{aligned} \lambda _{N-1}(D^2\phi (0,x'_0))=\min _{\dim W=N-1}\max _{{\mathop {|w|=1}\limits ^{w\in W}}}\left\langle D^2\phi (0,x'_0)w,w\right\rangle , \end{aligned}$$

with the particular choice of \(W=\left\{ \left( 0,v\right) :\,v\in \mathbb R^{N-1}\right\} \), and (2.21), we then obtain

$$\begin{aligned} \lambda _{N-1}(D^2\phi (0,x'_0))\le \max _{{\mathop {|v|=1}\limits ^{v\in \mathbb R^{N-1}}}}\left\langle D^2\psi (x'_0)v,v\right\rangle \le 0. \end{aligned}$$

This shows that u(x) is a viscosity solution of (2.19), for any \(a_i\ge 0\) and \(i=2,\ldots ,N-1\).

We end this note by pointing out that the regularity issue for solutions of (1.1) in the case \(a_1=0\) and \(a_N>0\) (or \(a_1>0\) and \(a_N=0\)) is an open problem and only partial results are known: Lipschitz regularity for \({\mathcal P}^\pm _1\), see [2], Hölder estimates for \(\mathcal {M}_{\mathbf {a}}\) in the case of asymmetric distributions of weights concentrated on the smallest or on the largest eigenvalue, namely \(a_1>a_2+\cdots +a_N\) or \(a_N>a_1+\cdots +a_{N-1}\), see [15] and the references therein.