Throughout the entire paper we suppose that $\mathrm{\Omega}\subseteq {\mathbb{R}}^{n}$ is a bounded open set with ${C}^{2}$ boundary. In this work it will be convenient to use the following notations. Given $\delta >0$,

${\mathrm{\Omega}}_{\delta}:=\{x\in \mathrm{\Omega}:d(x)<\delta \}$

and

${\mathrm{\Omega}}^{\delta}:=\{x\in \mathrm{\Omega}:d(x)>\delta \},$

where $d(x)$ denotes the distance of $x\in \mathrm{\Omega}$ to the boundary $\partial \mathrm{\Omega}$. Since Ω is a bounded ${C}^{2}$ domain, we note that there is $\mu >0$ such that $d\in {C}^{2}({\overline{\mathrm{\Omega}}}_{\mu})$ and $|\nabla d(x)|=1$ on ${\mathrm{\Omega}}_{\mu}$. See [25, Lemma 14.16] for a proof.
In fact, by modifying the distance function *d* appropriately, we can suppose that *d* is a positive ${C}^{2}$ function on Ω. For instance one can use $(1-\phi )d+\phi $ instead of *d*, where $\phi \in {C}_{c}^{2}(\mathrm{\Omega})$ is a cut-off function with $0\le \phi \le 1$ on Ω, $\phi \equiv 0$ on ${\mathrm{\Omega}}_{{\mu}_{0}}$ for some $0<{\mu}_{0}<\mu $ and $\phi \equiv 1$ on ${\mathrm{\Omega}}^{\mu}$. Therefore hereafter, we will always suppose that *d* is this modified distance function and that *d* is in ${C}^{2}(\overline{\mathrm{\Omega}})$ with $|Dd|\equiv 1$ on ${\mathrm{\Omega}}_{{\mu}_{0}}$.

It is helpful to keep in mind the following alternative description of the Pucci extremal operators:

${\mathcal{\mathcal{P}}}_{\lambda ,\mathrm{\Lambda}}^{+}(X)=\mathrm{\Lambda}\mathrm{tr}({X}^{+})-\lambda \mathrm{tr}({X}^{-})=\mathrm{\Lambda}{\displaystyle \sum _{{e}_{i}(X)>0}}{e}_{i}(X)+\lambda {\displaystyle \sum _{{e}_{i}(X)<0}}{e}_{i}(X),$${\mathcal{\mathcal{P}}}_{\lambda ,\mathrm{\Lambda}}^{-}(X)=\lambda \mathrm{tr}({X}^{+})-\mathrm{\Lambda}\mathrm{tr}({X}^{-})=\lambda {\displaystyle \sum _{{e}_{i}(X)>0}}{e}_{i}(X)+\mathrm{\Lambda}{\displaystyle \sum _{{e}_{i}(X)<0}}{e}_{i}(X),$

where ${X}^{+}$ and ${X}^{-}$ are the positive and negative parts of *X*, respectively, and ${e}_{i}(X)$, $i=1,\mathrm{\dots},n$, are the eigenvalues of *X*, counted according multiplicity, in non-decreasing order.

The positive homogeneity, duality, sub-additive and super-additive properties of the Pucci extremal operators (see [8]) lead to the following useful properties of ${\mathcal{\mathcal{M}}}^{\pm}$:

${\mathcal{\mathcal{M}}}^{\pm}(x,c(t,p,X))=c{\mathcal{\mathcal{M}}}^{\pm}(x,t,p,X),$(3.1)${\mathcal{\mathcal{M}}}^{-}(x,t,p,X)=-{\mathcal{\mathcal{M}}}^{+}(x,-t,-p,-X),$(3.2)${\mathcal{\mathcal{M}}}^{+}(x,t,p,X)+{\mathcal{\mathcal{M}}}^{-}(x,s,q,Y)\le {\mathcal{\mathcal{M}}}^{+}(x,t+s,p+q,X+Y)\le {\mathcal{\mathcal{M}}}^{+}(x,t,p,X)+{\mathcal{\mathcal{M}}}^{+}(x,s,q,Y),$(3.3)${\mathcal{\mathcal{M}}}^{-}(x,t,p,X)+{\mathcal{\mathcal{M}}}^{-}(x,s,q,Y)\le {\mathcal{\mathcal{M}}}^{-}(x,t+s,p+q,X+Y)\le {\mathcal{\mathcal{M}}}^{+}(x,t,p,X)+{\mathcal{\mathcal{M}}}^{-}(x,s,q,Y)$(3.4)

for all $c\ge 0$ and $(x,t,p,X),(x,s,q,Y)\in \mathrm{\Omega}\times \mathbb{R}\times {\mathbb{R}}^{n}\times {\mathcal{\mathcal{S}}}_{n}$.

Given $k\in C(\mathrm{\Omega}\times \mathbb{R})$, a function $u\in {C}^{2}(\mathrm{\Omega})$ is said to be a classical solution of equation $H[u]=k(x,u)$ in Ω if and only if

$H(x,u(x),Du(x),{D}^{2}u(x))=k(x,u(x))\mathit{\hspace{1em}}\text{for all}x\in \mathrm{\Omega}.$(3.5)

However, in this paper we consider functions $u\in C(\mathrm{\Omega})$ which are solutions in the viscosity sense, according to the following definition.

Let $u\in \mathrm{USC}(\mathrm{\Omega})$ (upper semicontinuous in Ω), resp. $u\in \mathrm{LSC}(\mathrm{\Omega})$ (lower semicontinuous in Ω). Then *u* is said to be a viscosity subsolution (resp., supersolution) in Ω of (3.5) if and only if for each $x\in \mathrm{\Omega}$ and $\phi \in {C}^{2}(\mathrm{\Omega})$ such that $u-\phi $ has a local maximum (resp. minimum) at *x* we have

$H(x,u(x),D\phi (x),{D}^{2}\phi (x))\ge k(x,u(x))\hspace{1em}(\text{resp.,}H(x,u(x),D\phi (x),{D}^{2}\phi (x))\le k(x,u(x))).$

A function $u\in C(\mathrm{\Omega})$ that is both a viscosity subsolution and viscosity supersolution in Ω of (3.5) is called a viscosity solution in Ω.

We note the following consequence of condition (H-1):

${\mathcal{\mathcal{M}}}^{-}[u]\le H[u]\le {\mathcal{\mathcal{M}}}^{+}[u]$(3.6)

for any function $u\in {C}^{2}(\mathrm{\Omega})$, where ${\mathcal{\mathcal{M}}}^{\pm}[u]:={\mathcal{\mathcal{M}}}^{\pm}(x,u,Du,{D}^{2}u)$.

In the sequel we will make an extensive use of a fundamental tool for pointwise estimates of viscosity solutions of elliptic equations, known as the Alexandroff–Bakelman–Pucci maximum principle (see, for instance, [6, 9, 3, 43]). For the convenience of the reader we recall below the version needed here.

For this, we first remark that if $k\in C(\mathrm{\Omega})$ and $H[w]\ge k(x)$ for some $w\in C(\mathrm{\Omega})$, then by (3.6) it follows that ${\mathcal{\mathcal{M}}}^{+}[w]\ge k(x)$. Note also that the latter implies that ${w}^{+}(x)=\mathrm{max}(w(x),0)$ satisfies ${\mathcal{\mathcal{M}}}^{+}[{w}^{+}]\ge -{k}^{-}(x)$. Therefore, setting

${\mathcal{\mathcal{M}}}_{\gamma}^{+}[{w}^{+}]:={\mathcal{\mathcal{P}}}_{\lambda ,\mathrm{\Lambda}}^{+}({D}^{2}{w}^{+})+\gamma |D{w}^{+}|,$

we also have

${\mathcal{\mathcal{M}}}_{\gamma}^{+}[{w}^{+}]\ge -{k}^{-}(x).$

Consequently, the standard Alexandroff–Bakelman–Pucci maximum principle (see [6, Proposition 2.12]) leads to the following.

#### Proposition 3.3 (ABP estimate).

*Let $\mathcal{O}\mathrm{\subseteq}{\mathrm{R}}^{n}$ be a bounded domain with diameter **R*. Suppose that *H* satisfies condition (H-1), assuming $\gamma \mathrm{=}{\mathrm{\parallel}{\gamma}^{\mathrm{+}}\mathrm{\parallel}}_{{L}^{\mathrm{\infty}}\mathit{}\mathrm{(}\mathcal{O}\mathrm{)}}\mathrm{<}\mathrm{\infty}$. For $k\mathrm{\in}C\mathit{}\mathrm{(}\mathcal{O}\mathrm{)}\mathrm{\cap}{L}^{n}\mathit{}\mathrm{(}\mathcal{O}\mathrm{)}$, let $w\mathrm{\in}C\mathit{}\mathrm{(}\overline{\mathcal{O}}\mathrm{)}$ be a viscosity subsolution of equation $H\mathit{}\mathrm{[}w\mathrm{]}\mathrm{=}k\mathit{}\mathrm{(}x\mathrm{)}$ in $\mathcal{O}$. There is a non-negative constant *C*, depending only on $n\mathrm{,}\lambda \mathrm{,}\mathrm{\Lambda}\mathrm{,}$ and $\gamma \mathit{}R$, such that

$\underset{\mathcal{\mathcal{O}}}{sup}w\le \underset{\partial \mathcal{\mathcal{O}}}{sup}{w}^{+}+CR{\parallel {k}^{-}\parallel}_{{L}^{n}(\mathcal{\mathcal{O}})}.$

In particular, under the assumptions of Proposition 3.3, the following sign propagation property holds:

$H[w]\ge 0\text{in}\mathcal{\mathcal{O}},w\le 0\text{on}\partial \mathcal{\mathcal{O}}\u27f9w\le 0\text{in}\mathcal{\mathcal{O}}.$

One then obtains a useful comparison principle by combining Proposition 3.3 and the following result which is based on [14, Proposition 2.1]. A justification for the reformulation presented below is sketched in [33, Lemma 2.5].

#### Lemma 3.4.

*Let $\mathcal{O}\mathrm{\subseteq}{\mathrm{R}}^{n}$ be a bounded domain, and let $a\mathit{}\mathrm{(}t\mathrm{)}\mathrm{,}b\mathit{}\mathrm{(}t\mathrm{)}$ be continuous functions on $\mathrm{R}$. Suppose that **H* satisfies (H-1) and (H-2). If $H\mathit{}\mathrm{[}u\mathrm{]}\mathrm{\ge}a\mathit{}\mathrm{(}u\mathrm{)}$ and $H\mathit{}\mathrm{[}v\mathrm{]}\mathrm{\le}b\mathit{}\mathrm{(}v\mathrm{)}$ for some $u\mathrm{,}v\mathrm{\in}C\mathit{}\mathrm{(}\overline{\mathcal{O}}\mathrm{)}$, then

${\mathcal{\mathcal{M}}}^{+}[u-v]\ge a(u)-b(v)\mathit{\hspace{1em}}\mathit{\text{in}}\stackrel{~}{\mathcal{\mathcal{O}}}:=\{x\in \mathcal{\mathcal{O}}:u(x)v(x)\}.$

As mentioned in the Introduction, we will assume throughout the paper that $f:\mathbb{R}\to \mathbb{R}$ satisfies conditions (f-1) and (f-2). We recall some useful consequences of these assumptions.

The non-increasing function $\varphi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ defined in (2.1) satisfies

$\underset{t\to 0}{lim}\varphi (t)=\mathrm{\infty},$

and

${\varphi}^{\prime}(t)=-\sqrt{2F(\varphi (t))},{\varphi}^{\prime \prime}(t)=f(\varphi (t)).$

Here we mention some easy, but useful consequences of the Dindoš’ condition (f-theta).

We also recall the following two lemmas from [36], and [39], respectively.

#### Lemma 3.9.

*Suppose that **f* satisfies (f-1), (f-2) and (f-3). Then:

$\underset{t\to \mathrm{\infty}}{lim\; sup}{\displaystyle \frac{\sqrt{F(t)}}{f(t){\int}_{t}^{\mathrm{\infty}}F{(s)}^{-\frac{1}{2}}\mathit{d}s}}<\mathrm{\infty},$(i)$\underset{t\to \mathrm{\infty}}{lim\; sup}{\displaystyle \frac{t}{f(t){\left({\int}_{t}^{\mathrm{\infty}}F{(s)}^{-\frac{1}{2}}\mathit{d}s\right)}^{2}}}<\mathrm{\infty}.$(ii)

The next result, a consequence of Lemma 3.9, will prove useful in establishing the existence of solutions to problem (1.1).

#### Corollary 3.11.

*Suppose that condition (f-1), (f-2), (f-3) are satisfied. Assuming, in addition, *(C-γ)*, we have
*

$\underset{d\to 0}{lim}\frac{\sqrt{F(\varphi (d))}}{f(\varphi (d))}\gamma (x)=0.$(3.8)

*Assuming, in addition, *(C-χ)*, we have*

$\underset{d\to 0}{lim}\frac{\varphi (d)}{f(\varphi (d))}\chi (x)=0.$

#### Proof.

To show (3.8), observe that

$\frac{\sqrt{F(\varphi (d(x)))}}{f(\varphi (d(x)))}|\gamma (x)|=\frac{\sqrt{F(\varphi (d(x)))}}{d(x)f(\varphi (d(x)))}|\gamma (x)|d(x)=\frac{\sqrt{F(\varphi (d(x)))}}{f(\varphi (d(x))){\int}_{\varphi (d(x))}^{\mathrm{\infty}}\frac{ds}{\sqrt{2F(s)}}}|\gamma (x)|d(x).$

Therefore, in light of Lemma 3.9 (i) and condition (C-γ), recalling that $\varphi (\delta )\to \mathrm{\infty}$ as $\delta \to 0$, the right-hand side tends to zero as $d(x)\to 0$.
In a similar way, using Lemma 3.9 (ii) and condition (C-χ), we get

$\frac{\varphi (d(x))}{f(\varphi (d(x)))}\chi (x)=\frac{\varphi (d(x))}{f(\varphi (d(x))){d}^{2}(x)}\chi (x){d}^{2}(x)=\frac{\varphi (d(x))}{f(\varphi (d(x))){\left({\int}_{\varphi (d(x))}^{\mathrm{\infty}}\frac{ds}{\sqrt{2F(s)}}\right)}^{2}}\chi (x){d}^{2}(x)\to 0\mathit{\hspace{1em}}\text{as}d(x)\to 0\text{.}\mathit{\u220e}$

The next lemma will be useful in the proof of Theorem 6.2, and hence Theorem 2.4.

#### Lemma 3.12.

*Let **f* satisfy conditions (f-1) and (f-3). Then:

(i)

*Given any *
$\kappa >0$
*, there are positive constants *
${t}_{\kappa}$
* and *
${c}_{\kappa}$
* such that *
$f(\kappa t)\ge {c}_{\kappa}f(t)$
* for all *
$t>{t}_{\kappa}$.

(ii)

*If, in addition, *
(f-theta)
* holds, then given *
$\varrho >1$
*, there are constants *
${\delta}_{\varrho}>0$
* and *
${c}_{\varrho}>0$
* such that *
$\varphi (\varrho t)\ge {c}_{\varrho}\varphi (t)$
* for all *
$0<t<{\delta}_{\varrho}$.

We should point out that the constants ${c}_{\kappa}$ and ${\delta}_{\kappa}$ in Lemma 3.12 (i) depend on the parameter α in condition(f-3), while the constants ${c}_{\varrho}$ and ${\delta}_{\varrho}$ Lemma 3.12 (ii) also depend on θ and ${\mathrm{\ell}}_{\theta}$ in condition (f-theta). See [35, Lemmas 2.12, 2.13 and 2.15].

The following condition which holds for any odd function *f* that satisfies (f-1) will be needed in one of our existence results:

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