tangential PDE of the limit formed by the “curve” of fully nonlinear operators
$\mathcal{F}_{\tau}$, provided the flatness degree $\tau$ of $u$, as well as
the data, are universally under small control.

######  Lemma 7.4 (Approximation Lemma).

Let $F$ satisfy (A5), and $f\in L^{p}(\Omega)$ for $p>n$. Then there exists
$\tau_{0}>0$ depending on $n,m,\lambda,\Lambda,p,q,\varrho,\omega$, such that
if $v$ is an $L^{p}$-viscosity solution to

$\tau^{-1}F(x,\tau
D^{2}v)+b(x)|Dv|+\mu(x)|Dv|^{m}=f(x)\quad\mbox{in}\;B_{2}\quad\mbox{with}\;\;\|v\|_{L^{\infty}(B_{2})}\leq
1,$

for $\tau\in(0,\tau_{0})$, and

$\max\left\\{\tau,\,\,\,\|\beta_{F}(0,\cdot)\|_{L^{p}(B_{2})},\,\,\|\mu\|_{L^{q}(B_{2})}\,,\,\|b\|_{L^{\varrho}(B_{2})}\,,\,\|f\|_{L^{p}(B_{2})}\right\\}\leq\tau_{0}$

then we can find $\gamma\in(0,1)$, depending only on
$n,m,p,q,\varrho,\lambda,\Lambda,\omega$, and a harmonic function
$\mathfrak{h}$ so that

$\Delta\mathfrak{h}(x)=0\;\;\;\text{in}\;B_{1},\quad\|\mathfrak{h}\|_{L^{\infty}(B_{1})}\leq\mathrm{C},\quad\sup_{B_{\gamma}}|v-\mathfrak{h}(0)-D\mathfrak{h}(0)\cdot
x|\leq\gamma^{1+\alpha}.$ (7.5)

In the sequel, we proceed with Caffarelli’s iterative scheme used in Section
5.2 in order to obtain a sequence of linear functions
$\mathfrak{a}_{k}+\mathfrak{b}_{k}\cdot x$ satisfying the geometric decay
estimate

$\textstyle\sup_{B_{r_{k}}}|v-\mathfrak{a}_{k}-\mathfrak{b}_{k}\cdot x|\leq
r_{k}^{1+\alpha},\qquad r_{k}\mathrel{\mathop{:}}=\gamma^{k}.$

Finally, by coming back to $u$ via such a geometric tangential path $u=v\tau$
we show that the graph of a solution to (7.2) can be approximated in
$B_{\sigma}$, for certain $\sigma\ll 1$, by an appropriated harmonic
polynomial profile, whose error is of the order
$\sim\mbox{O}(\tau\sigma^{1+\alpha})$, cf. [40, Lemma 3.2].

Hence, the previous estimate and a covering argument imply, via Campanato’s
embedding, that flat solutions are $C_{\text{loc}}^{1,\alpha}(B_{2})$.
Therefore, we obtain the following sharp estimate:

######  Theorem 7.5.

Assume (A5) and let $u$ be an $L^{p}$ viscosity solution to (7.2). Then there
exists $\tau>0$ depending only upon $n,m,p,q,\varrho,\lambda,\Lambda,\omega$,
such that if $\|u\|_{L^{\infty}(B_{2})}\leq\tau$, then $u\in
C_{\text{loc}}^{1,\hat{\alpha}}\left(B_{2}\right)$ and

$\|u\|_{C^{1,\hat{\alpha}}(B_{1})}\leq\mathrm{C}(n,p,q,\varrho,\lambda,\Lambda,m)\,\tau,\;\;\textrm{
where $\hat{\alpha}$ is given by }\eqref{sharp hat alpha}.$

###  7.3 Fully nonlinear singular PDEs

In this section, one considers an application to fully nonlinear PDEs, which
are degenerate elliptic in the sense that they become singular when the
gradient vanishes.

Some of our inspirations in this direction are the works [3, 4, 23]. The model
one might have in mind is the $\alpha$-Poisson equation. Thus, as in [23] we
consider the following class of equations,

$\displaystyle
G(x,Du,D^{2}u)\mathrel{\mathop{:}}=|Du|^{\alpha-2}F(x,D^{2}u)+b(x)|Du|^{\alpha-1}+f(x)|Du|^{\alpha-2}+\mu(x)=0\;\;\mathrm{in}\;\Omega,$
(7.6)

where $1<\alpha<2$, $b\in L^{\varrho}_{+}(\Omega)$, $\mu\in
L^{q}_{+}(\Omega)$, $f\in L^{p}_{+}(\Omega)$, and for $s>0$, $x\in\Omega$,
$X,Y\in\mathbb{S}^{n}$,

$\displaystyle\;\mathcal{M}^{-}(X-Y)\leq\;F(x,X)-F(x,Y)\leq\mathcal{M}^{+}(X-Y),\quad
F(x,sX)=sF(x,X).$ (7.7)

######  Definition 7.6 (bf Julin-Juutinen).

A function $u\in C(\Omega)$ is called an $L^{p}$-viscosity JJ-subsolution
(resp. JJ-supersolution) of (7.6) if for $\phi\in
W^{2,p}_{\mathrm{loc}}(\Omega)$, it follows that

$\displaystyle\mathrm{ess.}\varlimsup_{\epsilon\to
0}\,\\{G(y,D\phi(y),D^{2}\phi(y))\,|\,y\in B_{\epsilon}(x),D\phi(y)\neq
0\\}\geq 0$ (7.8) $\displaystyle(\textrm{resp., ess.}\varliminf_{\epsilon\to
0}\,\\{G(y,D\phi(y),D^{2}\phi(y))\,|\,y\in B_{\epsilon}(x),D\phi(y)\neq
0\\}\leq 0)$ (7.9)

whenever $u-\phi$ assumes its local maximum (resp. minimum) at $x\in\Omega$
and $D\phi\neq 0$ a.e. in $B_{r}(x)\setminus\\{x\\}$ for some $r>0$. On the
other hand, $u\in C(\Omega)$ is called a $C$-viscosity JJ-subsolution (resp.
JJ-supersolution) of (7.6) if (7.8) (resp. (7.9)) holds provided that $\phi\in
C^{2}(\Omega)$, $u-\phi$ takes its local maximum (resp., minimum) at
$x\in\Omega$ and $D\phi\neq 0$ in $B_{r}(x)\setminus\\{x\\}$ for some $r>0$.

######  Remark 7.7.

We observe that any notion of viscosity solution for the singular equation
(7.6) requires $f$ to be zero whenever $u$ is locally constant, as pointed out
in [2, 3].

Through Definition 7.6, the authors in [23] proved that solutions of (7.6)
satisfy

$\displaystyle\mathcal{L}^{+}_{m}[u]\geq
f,\;\;\quad\;\mathcal{L}^{-}_{m}[u]\leq f\;\;\;\mathrm{in}\;\Omega,\quad
m=2-\alpha,$

in the $C$-viscosity JJ-sense, and further in the usual $C$-viscosity sense,
under continuity on the data $b,\mu,f$, as we state in the following lemma.

######  Lemma 7.8 ([23]).

Assume (A2) with $m=2-\alpha\in(0,1)$, and (7.7) for $F$ continuous,
$b,\mu,f\in C(\Omega)$. Let $u\in C(\Omega)$ be an $L^{p}$-viscosity JJ-
subsolution (resp., JJ-supersolution) of (7.6). Then, $u$ is a $C$-viscosity
JJ-subsolution and also a standard $C$-viscosity subsolution (resp.
supersolution) of

$\displaystyle
F(x,D^{2}u)+b(x)|Du|+\mu(x)|Du|^{m}+f(x)=0\;\;\mathrm{in}\;\Omega.$ (7.10)

The proof of Lemma 7.8 follows exactly the same arguments used in [23, Lemmas
5.2 and 5.4], by replacing the Pucci operator there by $F(x,D^{2}u)$ and using
Remark 7.7; so we do not repeat it here.

######  Theorem 7.9.

Let $\Omega\subset{\mathbb{R}^{n}}$ be a bounded $C^{1,1}$ domain. Let $u$ be
an $L^{p}$-viscosity JJ-solution of (7.6), with $b\in L^{\varrho}(\Omega)\cap
C(\Omega)$, $\mu\in L^{q}(\Omega)\cap C(\Omega)$, and $f\in L^{p}(\Omega)\cap
C(\Omega)$. Then, the results of Theorems 1.3, 1.5, and 1.6 in the cases
$p_{0}<p<n$, $p=n$, and $n<p<\infty$ respectively are satisfied.

In particular, this generalizes the regularity results in [3, Theorem 1.1] and
[4, Theorem 1.1], since the coefficients are not supposed to be bounded – they
might be unbounded near the boundary.

###  7.4 Perron method and multiplicity phenomena

In this section, we give an application of the viscosity equivalence in
Section 3 in what refers to a Perron method for our equations. In [22,
Proposition 2.1] it is proven a version of Perron method for $L^{p}$-viscosity
solutions of coercive equations related to quadratic growth and bounded
coefficients. Here we extend this result to any nonlinear gradient regime and
unbounded data.

A fundamental tool in order to apply the Perron method consists of showing
that a supremum of subsolutions is still a subsolution. It usually presupposes
that this supremum is properly defined, by considering the set where $u^{*}$
is locally bounded. This auxiliary result plays an important role in
multiplicity problems, see [38]. More precisely, with this result at hands and
the very recent Vázquez strong maximum principle developed in [44], one may
extend all multiplicity results in [38] to unbounded drift $b(x)$; as well as
to an unbounded noncoercive term $c(x)$. The latter is possible due to
$C^{1,\alpha}$ regularity theory for explicit unbounded zero order terms from
[36, Theorem 1, Remark 1.2] – indeed, in this case one may view the term
$c(x)u$ as the source term (by putting it to the RHS of the equation).

Let us first introduce the kind of solution we have in mind. We observe that,
in both $L^{p}$ or $C$–viscosity senses, we only need to define a subsolution
as an upper semicontinuous function, and a supersolution being lower
semicontinuous, which is sufficient to ensure the attainment of a maximum or
minimum, respectively, over compact sets. Moreover, if the function $u$ is not
semicontinuous, we say that $u$ is a viscosity subsolution (resp.
supersolution) provided $u^{*}$ (resp. $u_{*}$) is. Here, $u^{*}$ and $u_{*}$
are the upper and lower semicontinuous envelopes of $u$ respectively, namely

$\displaystyle\textstyle
u^{\ast}(x)\mathrel{\mathop{:}}=\varlimsup_{y\rightarrow x}u(y),\qquad
u_{\ast}(x)\mathrel{\mathop{:}}=\varliminf_{y\rightarrow x}u(y),$

see [12, p. 22], [21, p. 40]. Note that $u^{\ast}\in USC(\Omega)$ and
$u_{\ast}\in LSC(\Omega)$.

######  Proposition 7.10.

Let $\Omega\subset{\mathbb{R}^{n}}$ be a bounded open set, and $F$ satisfying
(A1), (A2). Let $\mathcal{A}\subset C(\Omega)$ be a nonempty set of
$L^{p}$-viscosity subsolutions $($supersolutions$)$ of
$F(x,Du,D^{2}u)+c(x)u\geq f(x)$ in $\Omega$, for $c,f\in
L^{p}_{\mathrm{loc}}(\Omega)$, $c\geq 0$ a.e. in $\Omega$. Assume that
$u(x)\mathrel{\mathop{:}}=\sup_{v\in\mathcal{A}}v(x)$ $($resp.
$u(x)\mathrel{\mathop{:}}=\inf_{v\in\mathcal{A}}v(x))$ is locally bounded in
$\Omega$. Then, $u$ is a viscosity subsolution $($supersolution$)$ of the same
equation in $\Omega$.

###### Proof.

The reasoning is similar to the classical proof in [21, Theorem 4.2], see also
[22]. The main difference comes from the fact that one needs to use Lemma 3.3
as far as unbounded coefficients are concerned; we sketch the proof for the
sake of completeness. Since $u$ can be a discontinuous function, it is
accomplished through the semicontinuous envelopes $u_{\ast}$, $u^{\ast}$.

Next, we note that, since $c\geq 0$, we have $F[v]\geq f(x)-c(x)v\geq
f(x)-c(x)u=:g(x)$ for all $v\in\mathcal{A}$. Then it is enough to prove that
$u$ is a subsolution of $F[u]\geq g(x)$ in $\Omega$. We assume on the contrary
that there exist $\varepsilon,r>0$, $x_{0}\in\Omega$ and $\phi\in
W^{2,p}_{\mathrm{loc}}(\Omega)$ such that $u^{\ast}-\phi$ attains a local
maximum at $x_{0}$ but

$\displaystyle F(x,D\phi(x),D^{2}\phi(x))-g(x)\leq-\varepsilon\;\textrm{ a.e.\
in }B_{r}(x_{0}).$ (7.11)

By Lemma 3.3 this maximum can supposed to be strict, i.e. we assume

$(u^{\ast}-\phi)(x)<(u^{\ast}-\phi)(x_{0})=0$ in
$\overline{B}_{r}(x_{0})\setminus\\{x_{0}\\}$.

Further, $\phi(x_{0})=\lim_{x_{k}\rightarrow
x_{0}}u(x_{k})=\lim_{k\rightarrow\infty}v_{k}(x_{k})$, for some
$x_{k}\in\overline{B}_{r}(x_{0}$), $x_{k}\rightarrow x_{0}$. Here,
$u(x_{k})=\sup_{v\in\mathcal{A}}v(x_{k})=\lim_{m\rightarrow\infty}v_{m}^{k}(x_{k})$,
where $v_{m}^{k}$ is a maximizing sequence for each $k\in\mathbb{N}$. Set
$v_{k}=v_{k}^{k}\in\mathcal{A}$.

Let $z_{k}\in\overline{B}_{r}(x_{0})$ so that
$\max_{\overline{B}_{r}(x_{0})}(v_{k}-\phi)=(v_{k}-\phi)(z_{k})$, with
$z_{k}\rightarrow z\in\overline{B}_{r}(x_{0})$ up to a subsequence. By the
definition of $v_{k}$ as an $L^{p}$-viscosity subsolution we have

$\displaystyle\textrm{ess}\varlimsup_{x\rightarrow
z_{k}}\;\\{F(x,D\phi(x),D^{2}\phi(x))-g(x)\\}\geq 0.$ (7.12)

Then, as in [21] one shows that $z=x_{0}$ and $v_{k}(z_{k})\rightarrow
u^{\ast}(x_{0})$ from the fact that $x_{0}$ is the unique strict global
maximum point in the ball $\overline{B}_{r}(x_{0})$. This fact and (7.12)
contradicts (7.11). ∎

The following Perron’s type existence result is now a consequence of [22,
proof of Theorem 3.3], by making use of Lemma 3.3 and Proposition 3.1.

######  Proposition 7.11.

Assume $F$ satisfies (A1), (A2), (3.2). Let $\underline{u},\,\overline{u}\in
C(\Omega)$ be a pair of viscosity sub and supersolutions of $F[u]=f(x)$ in
$\Omega$ respectively, with $\underline{u}\leq\overline{u}$ in $\Omega$. Then
the function $u$ defined as $u(x)=sup_{v\in\mathcal{A}}\,v(x)$ for
$x\in\Omega$, where

$\mathcal{A}\mathrel{\mathop{:}}=\left\\{v\in C(\Omega);\;v\textrm{ is a
viscosity subsolution of $F[u]=f(x)$ in $\Omega$,\; $\underline{u}\leq
v\leq\overline{u}$ in $\Omega$}\right\\},$

is a viscosity solution of $F[u]=f(x)$ in $\Omega$.

As an application, we consider a class of problems with quadratic gradient
growth given by

$\displaystyle-F(x,Du,D^{2}u)=\sigma c(x)u+\langle
M(x)Du,Du\rangle+h(x)\textrm{ in }\Omega,\quad u=0\textrm{ on
}\partial\Omega,$ ($P_{\sigma}$)

under the following assumption:

  * (A6)

(Multiplicity requirements) $\Omega$ is a bounded $C^{1,1}$ domain in
${\mathbb{R}^{n}}$, $\sigma\in\mathbb{R}$, $n\geq 1$, $F$ satisfies (A1)–(A3)
for $m=1$ and $p>n$, also $c,h\in L^{p}(\Omega)$, $c\geq 0$ a.e. in $\Omega$
with $c\not\equiv 0$, in addition to:

    1. (a)

$M$ satisfies $\mu_{1}I\leq M(x)\leq\mu_{2}I$ a.e. in $\Omega$, for some
$\mu_{1},\mu_{2}>0$;

    2. (b)

the problem $(P_{0})$ has a strong solution $u_{0}$;

    3. (c)

for $f\in L^{p}(\Omega)$, there exists a unique viscosity solution of
$F[u]=f(x)$ in $\Omega$, with $u=0$ on $\partial\Omega$;

    4. (d)

($P_{\sigma}$) enjoys $W^{2,p}$ estimates, in the sense of Definition 6.4.

We observe that item (b) in (A6) holds true via [42, Theorem 1 (ii)] if, for
instance $Mh$ has small $L^{p}$ norm. Likewise, by [42, Theorem 1 (iii)], such
function $u_{0}$ is the unique viscosity solution of $(P_{0})$. Recall that
(c) and (d) are both true if $F$ satisfies for example the hypotheses in
Proposition 6.7.

The supplementary hypothesis (d) in (A6) for the uniqueness or comparison
results is unavoidable, since in the universe of $L^{p}$-viscosity solutions
they are only achieved in the presence of a strong solution; see [42] and
references therein. One observes that if $h$ has a sign, then $u_{0}$ has the
same sign via the maximum principle, see [38, Remark 6.25].

In this case, a priori bounds and multiplicity results in [38, 37] become
available for unbounded coefficients, in view of Proposition 7.10 and the
recent Vázquez’s strong maximum principle in [44], see the discussion in [38].

######  Theorem 7.12.

Assume items (a)–(c) in (A6), and let $\Lambda_{1},\Lambda_{2}>0$ so that
$0<\Lambda_{1}<\Lambda_{2}$. Then any viscosity solution $u$ of ($P_{\sigma}$)
satisfies the uniform a priori bounds:

$\|u^{-}\|_{L^{\infty}(\Omega)}\leq C\,,\;\textrm{ for all
}\sigma\in[0,\Lambda_{2}],\qquad\|u^{+}\|_{L^{\infty}(\Omega)}\leq
C\,,\;\textrm{ for all }\sigma\in[\Lambda_{1},\Lambda_{2}],$

where $C$ depends on
$n,p,\varrho,\mu_{1},\Lambda_{1},\Lambda_{2},\|b\|_{\varrho},\|c\|_{p},\|h\|_{p},\|u_{0}\|_{\infty},\lambda,\Lambda$,
$\partial\Omega$, and the set where $c>0$.

Recall that solutions of ($P_{\sigma}$) are $C^{1,\alpha}$ up to the boundary.
Then it makes sense to consider the set
$E\mathrel{\mathop{:}}=C^{1}(\overline{\Omega})$ with the following ordering:
For $u,v\in E$, we say that $u\ll v$ if for every $x\in\Omega$ it holds
$u(x)<v(x)$, and for $x_{0}\in\partial\Omega$ we have either
$u(x_{0})<v(x_{0})$, or $u(x_{0})=v(x_{0})$ and
$\partial_{\nu}u(x_{0})<\partial_{\nu}v(x_{0})$, where $\vec{\nu}$ is the
interior unit normal to $\partial\Omega$ at $x_{0}$.

######  Theorem 7.13.

Assume items (a)–(c) in (A6). Then the set
$\Sigma=\\{\,(\sigma,u)\in\real\times E\,;u\;\textrm{solves
\eqref{Plambda}}\,\\}$ possesses an unbounded component
$\mathcal{C}^{+}\subset[0,+\infty]\times E$ with
$\mathcal{C}^{+}\cap(\\{0\\}\times E)=\\{u_{0}\\}$, such that:

  1. 1.

either it bifurcates from infinity to the right of the axis $\sigma=0$ with
the corresponding solutions having a positive part blowing up to infinity in
$C(\overline{\Omega})$ as $\sigma\rightarrow 0^{+}$; or its projection on the
$\sigma$ axis is $[0,+\infty)$.

  2. 2.

There exists $\bar{\sigma}\in(0,+\infty]$ such that, for every
$\sigma\in(0,\bar{\sigma})$, the problem ($P_{\sigma}$) has at least two
viscosity solutions, $u_{\sigma,1}$ and $u_{\sigma,2}$, satisfying
$u_{\sigma,1}\rightarrow u_{0}$ in $E$ and
$\max_{\overline{\Omega}}u_{\sigma,2}\to+\infty$ as $\sigma\rightarrow 0^{+}$.
When $\,\bar{\sigma}<+\infty$ the problem $(P_{\bar{\sigma}})$ has at least
one viscosity solution; which is unique if $F(x,\xi,X)$ is convex in
$(\xi,X)$.

  3. 3.

Assume in addition that item (d) holds in (A6). Then $u_{\sigma}\,$ for
$\sigma\leq 0$ are unique among viscosity solutions; whereas the solutions in
item 2 for $\sigma>0$ are ordered, i.e. $u_{\sigma,1}\ll u_{\sigma,2}$.
Moreover:

    1. 3.1

If (b) in (A6) is such that $u_{0}\leq 0$, $cu_{0}\lneqq 0$, then every
nonpositive viscosity solution of ($P_{\sigma}$) with $\sigma>0$ satisfies
$u\ll u_{0}$. Also, for every $\sigma>0$, ($P_{\sigma}$) has at least two
nontrivial strong solutions $u_{\sigma,1}\ll u_{\sigma,2}\,$ so that
$u_{\sigma_{2},1}\ll u_{\sigma_{1},1}\ll u_{0}$ if
$\,0<\sigma_{1}<\sigma_{2}$, with $u_{\sigma,1}\to u_{0}$ in $E$ and
$\max_{\overline{\Omega}}u_{\sigma,2}\to+\infty$ as $\sigma\rightarrow 0^{+}$.
If $F(x,\xi,X)$ is convex in $(\xi,X)$ then
$\max_{\overline{\Omega}}\,u_{\sigma,2}>0$ for all $\sigma>0$.

    2. 3.2

If (b) in (A6) is such that $u_{0}\geq 0$, $cu_{0}\gneqq 0$, then every
nonnegative viscosity solution of ($P_{\sigma}$) with $\sigma>0$ satisfies
$u\gg u_{0}$. Further, there exists $\bar{\sigma}_{1}\in(0,+\infty)$
satisfying:

      1. (i)

for every $\sigma\in(0,\bar{\sigma}_{1})$, ($P_{\sigma}$) has at least two
nontrivial strong solutions $u_{\sigma,1}\ll u_{\sigma,2}\,$, where $u_{0}\ll
u_{\sigma_{1},1}\ll u_{\sigma_{2},1}$ if $\,0<\sigma_{1}<\sigma_{2}$, with
$u_{\sigma,1}\to u_{0}$ in $E$ and
$\max_{\overline{\Omega}}u_{\sigma,2}\to+\infty$ as $\sigma\rightarrow 0^{+}$;

      2. (ii)

$(P_{\bar{\sigma}_{1}})$ has at least one strong solution
$u_{\bar{\sigma}_{1}}$, which is unique if $F$ is convex in $(\xi,X)$;

      3. (iii)

for $\sigma>\bar{\sigma}_{1}$, ($P_{\sigma}$) has no nonnegative viscosity
solution.

In addition there exists $\delta>0$ so that if
$\mu_{2}\|h\|_{L^{p}(\Omega)}\leq\delta$, $h\gneqq 0$, then there is
$\bar{\sigma}_{2}>\bar{\sigma}_{1}$ such that:

      1. (iv)

($P_{\sigma}$) has at least two strong solutions for
$\sigma>\bar{\sigma}_{2}$, with $u_{\sigma,1}\ll 0$ in $\Omega$ and
$\min_{\overline{\Omega}}u_{\sigma,2}<0$;

      2. (iiv)

$(P_{\bar{\sigma}_{2}})$ has at least one nonpositive strong solution, which
is unique if $F$ is convex in $(\xi,X)$;

      3. (iiiv)

for $\sigma<\bar{\sigma}_{2}$, the problem ($P_{\sigma}$) has no nonpositive
viscosity solution.

We highlight that the definition of $L^{p}$-viscosity solutions is equivalent
to the $L^{n}$-sense for instance, by Proposition 3.1 with $m=2$. In other
words, the results in [37, 38] become intrinsically depending on the data.

Acknowledgment. João Vitor da Silva was partially supported by CNPq-Brazil
grant 310303/2019-2. Gabrielle Nornberg was supported by FAPESP grants
2018/04000-9 and 2019/03101-9, São Paulo Research Foundation.

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