# Applications of P-functions to Quasi-Linear equations: Gradient Bounds and
Liouville-type properties

Dimitrios Gazoulis  Department of Mathematics and Applied Mathematics,
University of Crete, 70013 Heraklion, Greece, and Institute of Applied and
Computational Mathematics, FORTH, 700 13 Heraklion, Crete, Greece
<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of $P-$functions for fully-nonlinear equations and
obtain some abstract consequences. We study $P-$functions for a class of
quasi-linear equations and establish some general criterion for obtaining such
quantities. Some applications are gradient bounds, De Giorgi-type properties
of entire solutions and Liouville-type theorems. As a special case we obtain a
gradient bound that differs from the Modica inequality. In addition, we
provide examples of such quantities for the Monge-Ampere and for higher order
nonlinear equations. One application for equations of order greater than two
is pointwise estimates for the Laplacian.

##  1\. Introduction

$P-$functions can be thought as quantities of a function $u$ and its higher
order derivatives that is related to an elliptic or more general differential
equation or to a differential inequality and has the property that satisfies
the maximum principle. Perhaps the most well-known example is
$P(u,x)=\frac{1}{2}|\nabla u|^{2}-W(u)$ that is related to the Allen-Cahn
equation

(1.1) $\Delta
u=W^{\prime}(u)\;\;,\;u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$

and Modica in [14] proved the well-known gradient bound

(1.2) $\frac{1}{2}|\nabla u|^{2}\leq W(u)$

for every solution of (1.1).

Later, Caffarelli et al in [6] generalized this gradient bound for a class of
varational quasi-linear equations and proved Liouville-type and De Giorgi-type
properties for a particular choise of $P-$function related to the equation
$div(\Phi^{\prime}(|\nabla u|^{2})\nabla u)=F^{\prime}(u)$. This bound was
generalized for anisotropic partial differential equations in [9].

Furthermore, $P-$functions had been already studied by Sperb in [18], Payne
and Philippin in [15] and [16] who studied other types of quasilinear
equations for the form $\\\ div\>(A(u,|\nabla u|^{2})\nabla u)=B(u,|\nabla
u|^{2})$, which are not necessarily Euler-Lagrange equations of an elliptic
integrand. They derived maximum principles for some appropriate $P-$functions.
Due to the greater generality, however, the relevant $P$ and the conditions
under which satisfies an elliptic differential inequality are rather
implicitly given while in [6] and [7] are given explicitly. Nevertheless, one
advantage of being given implicitly rather than explicitly is that one can
extract many examples of $P-$functions that satisfy such conditions.

There are many other applications of $P-$functions that can be found in [18],
among others, such as lower bounds for eigenvalue problems. One additional
important application is in [1], where they showed that the monotonicity
assumption $u_{x_{n}}>0$, that is also stated in the De Giorgi’s conjecture,
does in fact imply the local minimality of $u$. Such implication is by no
means trivial and it is based on the construction of a so-called calibration
associated to the energy functional. Such notion is intimately connected to
the theory of null-Lagrangians, see [11], chapter 1 and chapter 4, section
2.4. In Theorem 4.4 in [1], then carry the construction of the appropriate
calibration for general integrands of the calculus of variations and such
construction relies explicitly on the $P-$ function.

Last but not least, there are applications such as gradient bounds similar to
(1.2) and Liouville-type properties for vector equations. To be more precise,
in Theorem 3.5 in [17], there is a gradient bound for the Ginzburg–Landau
system of equations.

In this paper, we introduce the notion of $P-$ function for general fully-
nonlinear differential equations or differential inequalities and we
incorporate the “$P-$ function technique” in a general setting. Assuming that
we have a $P-$ function with a particular form associated to a general type of
fully nonlinear equation that attains it’s supremum at a point, we can
determine the smooth entire solutions without excluding a priori some
potential singularities.

Moreover, we extract some implicit conditions for obtaining $P-$ functions for
a general class of quasi-linear equations. Additionally, we prove some
gradient bounds, Liouville-type properties and De Giorgi-type properties for
smooth entire solutions, utilizing techniques from [6] and we also obtain as a
special case an a priori gradient bound for the Allen-Cahn equation for
general potentials that is different from the Modica inequality. These bounds
hold for any $P-$function that satisfy some implicit conditions, so in fact,
for any explicit example we have a different gradient bound. This method allow
us to obtain many different types of gradient bounds, each one for every
explicit example of $P-$function that satisfy the implicit conditions
mentioned above. We also illustrate this claim more generally by proving
gradient bounds from the examples of $P-$ functions in [15]. One such bound,
generalizes the one in [6] for a more general class of quasi-linear equations.
Another consequence of determining many types of $P-$ functions is that we can
obtain a Liouville-type theorem for the equation in [6], when
$F^{\prime\prime}\geq 0$, in which case we have stability of solutions. For a
different class of equations, such Liouville-type properties imply non
existence of solutions and we give some examples.

We also provide a class of such quantities for the Monge-Ampere’s equation and
for higher order nonlinear equations together with some applications. For
instance, we establish a mean value-type theorem for the Monge-Ampere equation
and for nonlinear equations of order greater than two. One additional
application for higher order equations is an a priori bound for the Laplacian
and pointwise estimates through the mean value properties. Finally, some
Liouville-type properties can be extended for nonlinear equations of order
greater than two. In this setting, we believe that one can obtain many other
types of bounds for any order of derivatives, assuming a $C^{k,\alpha}$ a
priori estimate and that we have an appropriate $P-$ function related to the
respective equation.

##  2\. The notion of $P-$function and abstract consequences

We begin by defining the notion of P-function

######  Definition 2.1.

Let $u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ be a smooth
solution or subsolution of

(2.1) $F(u,\nabla u,...,\nabla^{m}u)=0$

where $F$ is a continuous function.

We say that $P=P(u,\nabla u,...,\nabla^{m-1}u)$ is a $P-$function of (2.1) if
there exists an elliptic operator $L$ and a non negative function
$\mu=\mu(x)\geq 0$

(2.2)
$\begin{gathered}L=-\sum_{i,j=1}^{n}a_{ij}\partial_{x_{i}x_{j}}+\sum_{i=1}^{n}b_{i}\partial_{x_{i}}+c\;\;,\;\textrm{with}\;\>c\geq
0\\\ \textrm{such that}\;\>\mu L\>P\leq
0\;\;,\;\textrm{in}\;\>\Omega.\end{gathered}$

an immediate corollary is that any $P-$function related to an equation or to a
differential inequality attains its maximum at the boundary $\partial\Omega$
or at a point $x\in\Omega$ such that $\mu(x)=0$.

The independence from the $x-$variables is needed in order the equation (2.1)
to be translation invariant. This ingredient is necessary for the gradient
bounds and similar applications. We initially state as a direct consequence a
strong maximum principle that holds in general (see Theorem 2.2 in [6] or
Theorem 4.7 in [7]).

######  Theorem 2.2.

Let $u$ be a smooth solution or subsolution of

(2.3) $\begin{gathered}F(u,\nabla
u,...,\nabla^{m}u)=0\;\;\;,\;\;u:\Omega\rightarrow\mathbb{R}^{d}\\\
\textrm{where}\;\>\Omega\;\>\textrm{is a connected, bounded subset
of}\;\>\mathbb{R}^{n}\end{gathered}$

such that $\inf_{\overline{\Omega}}g(\nabla^{k}u)>0$ for some
$g:\mathbb{R}^{n^{k}\times d}\rightarrow[0,+\infty)\;,\;k\in\\{1,...,m-1\\}$
and suppose that $P=P(u,\nabla u,...,\nabla^{m-1}u)$ is a $P-$function of
(2.3) with $\mu=\mu(g(\nabla^{k}u))\;,\>\mu(t)>0\;,\>\forall\>t>0$.

If there exists $x_{0}\in\Omega$ such that

(2.4)
$P(u(x_{0}),...,\nabla^{m-1}u(x_{0}))=\sup_{\Omega}P(u,...,\nabla^{m-1}u)$

then $P(u,\nabla u,...,\nabla^{m-1}u)$ is constant in $\Omega$.

###### Proof.

The proof is an immediate consequence of the strong maximum principle since
$\\\ \mu(g(\nabla^{k}u))>0$ in $\Omega$. ∎

The most common choice of $g$ in Theorem 2.2 above is the Euclidean norm. For
example, if $k=1$, $g(\nabla u)=|\nabla u|.$ If $\mu>0\;,\>\forall\>t\geq 0$,
then the assumption $\inf_{\overline{\Omega}}g(\nabla^{k}u)>0$ is dismissed.

######  Remark 2.3.

The constancy of $P-$functions with a particular form hides geometric
information on the level sets $\\{x\in\mathbb{R}^{n}\>|\>u(x)=t\\}$ of the
solution $u$, such as the property of being surfaces of zero mean curvature
(see Proposition 4.11 in [7]).

Next, we have a De Giorgi type result as in Theorem 5.1 in [6], without
excluding a priori potential singularities. We denote as $\mathcal{H}^{1}$ the
1-Hausdorff measure in $\mathbb{R}^{n}$.

######  Theorem 2.4.

Let $u$ be a smooth entire solution (or subsolution) of

(2.5) $F(u,\nabla u,\nabla^{2}u)=0\;\;\;,\;\;u:\mathbb{R}^{n}\setminus
S\rightarrow\mathbb{R}$

with $u_{x_{n}}>0$, except perhaps on a closed set $S$ of potential
singularities such that $\mathcal{H}^{1}(S)=0$ and $\mathbb{R}^{n}\setminus S$
is connected. Assume that $P=P(u,|\nabla u|)$ is a $P-$function of (2.5) with
$\mu=\mu(|\nabla u|)\;,\>\mu(t)>0\;,\>\forall\>t>0$, such that $P_{t}>0$ for
$t>0\;(P=P(s,t))$.

If there exists $x_{0}\in\mathbb{R}^{n}\setminus S$ such that

(2.6) $P(u(x_{0}),|\nabla u(x_{0})|)=\sup_{\mathbb{R}^{n}\setminus
S}P(u,|\nabla u|)<+\infty$

then there exists a function
$g:\mathbb{R}\rightarrow\mathbb{R}\;,\>a\in\mathbb{R}^{n}$ with $|a|=1$ and
$b\in\mathbb{R}$, such that

(2.7) $u(x)=g(a\cdot
x+b)\;\;,\;x\in\mathbb{R}^{n}\;\;\textrm{and}\;\>S=\emptyset.$

###### Proof.

Let $c_{0}=\sup_{\mathbb{R}^{n}\setminus S}P(u,|\nabla u|)$ and consider the
set

(2.8) $A=\\{x\in\mathbb{R}^{n}\setminus S\>:\>P(u,|\nabla u|)=c_{0}\\}$

$A$ is closed and by the assumption $A\neq\emptyset$. We are going to prove
that $A$ is open.

Let $x_{1}\in A$, we take $\delta>0$ such that
$B_{\delta}(x_{1})\subset\mathbb{R}^{n}\setminus S$. Since $u_{x_{n}}>0$, we
have $\inf_{\overline{B}_{\delta}(x_{1})}|\nabla u|>0$ and by Theorem 2.2 we
conclude that $P(u,|\nabla u|)\equiv c_{0}$ in $B_{\delta}(x_{1})$ and
therefore $A$ is open.

Since $\mathbb{R}^{n}\setminus S$ is connected, we have that
$A=\mathbb{R}^{n}\setminus S$, that is,

(2.9) $P(u,|\nabla u|)\equiv c_{0}\;\;,\;\forall\>x\in\mathbb{R}^{n}\setminus
S$

and $P_{t}>0$, thus

(2.10) $|\nabla u|=Q(u)\;\;\;,\;\;\textrm{in}\;\>\mathbb{R}^{n}\setminus
S\;\;,\;\textrm{for some function}\;\>Q:\mathbb{R}\rightarrow\mathbb{R}$

Now, if there exists $x_{2}\in\mathbb{R}^{n}\setminus S$ such that
$Q(u(x_{2}))=0$, so $|\nabla u(x_{2})|=0$, that contradicts the fact the
$u_{x_{n}}>0$.

So, $Q(u)>0\;\>,\;\forall\>x\in\mathbb{R}^{n}\setminus S$ and set

(2.11)
$\begin{gathered}v=G(u)\;\;\;,\;\;\textrm{where}\;\>G^{\prime}(s)=\frac{1}{Q(s)}\\\
\textrm{and}\;\>|\nabla v|^{2}=1\;\;\;\textrm{in}\;\>\mathbb{R}^{n}\setminus
S\end{gathered}$

Therefore, by the result in [5], we have that

(2.12) $\begin{gathered}\textrm{either}\;\>v(x)=a\cdot
x+b\;\>,\;a\in\mathbb{R}^{n}\;\>\textrm{with}\;\>|a|=1\;\>\textrm{and}\;\>b\in\mathbb{R}\\\
\textrm{or}\;\>v(x)=|x-z_{0}|+c\;\>,\;z_{0}\in\mathbb{R}^{n}\;\>\textrm{and}\;\>c\in\mathbb{R}\end{gathered}$

and we conclude

(2.13) $u(x)=g(a\cdot
x+b)\;\>,\;a\in\mathbb{R}^{n}\;\>\textrm{with}\;\>|a|=1\;,\;\>b\in\mathbb{R}\;\>\textrm{and}\;\>g(s)=G^{-1}(s)$

The radially symmetric solutions with respect to a point
$z_{0}\in\mathbb{R}^{n}$ are excluded since $u$ is monotone in $x_{n}$. ∎

##  3\. General Criterion for obtaining $P-$functions

We now provide a general criterion for obtaining $P-$functions for a class of
nonlinear elliptic equations of the form $\Delta u=F(u,|\nabla u|^{2})$ that
do not in general arise as variational problems. A special case for
$F(u,|\nabla u|^{2})=W^{\prime}(u)$, is the Allen-Cahn equation.

For any function $P=P(s,t)$ and consider $F=F(s,t)$, we define the quantity

(3.1)
$\begin{gathered}I(s,t)=P_{t}(s,t^{2})P_{s}(s,t^{2})F(s,t^{2})+\frac{P_{s}^{2}(s,t^{2})}{2}+2t^{2}P_{t}^{2}(s,t^{2})F_{s}(s,t^{2})\\\
-2t^{2}P_{t}(s,t^{2})P_{s}(s,t^{2})F_{t}(s,t^{2})\end{gathered}$

######  Theorem 3.1.

Let $u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth solution
of

(3.2) $\Delta u=F(u,|\nabla u|^{2})$

and $P(s,t)=P:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that $P_{t}>0$ for
$t>0$ and either

(3.3) $\begin{gathered}\begin{cases}Hes_{(s,t)}\>P\;\>\textrm{is positive
semidefinite and}\\\ I(s,t)\geq
0\;\;\;,\;\;\forall\;\>(s,t)\in\mathbb{R}\times[0,+\infty)\end{cases}\\\
\textrm{where}\;\>I=I(s,t)\;\>\textrm{is defined
in}\;\>\eqref{DefOfI(s,t)}\end{gathered}$

or

(3.4) $\begin{cases}P_{st}=0\;,\;P_{tt}\geq 0\;\>\textrm{and}\\\
t^{2}P_{ss}(s,t^{2})P_{t}(s,t^{2})+I(s,t)\geq
0\;\;\;,\;\;\forall\;\>(s,t)\in\mathbb{R}\times[0,+\infty)\end{cases}$

hold.

Then $P=P(u,|\nabla u|^{2})$ is a $P-$function of (3.2).

###### Proof.

(3.5) $P_{x_{i}}=P_{s}u_{x_{i}}+2P_{t}\sum_{j=1}^{n}u_{x_{j}}u_{x_{j}x_{i}}$
(3.6)
$\begin{gathered}\Rightarrow\sum_{i=1}^{n}(P_{x_{i}}-P_{s}u_{x_{i}})^{2}=\sum_{i=1}^{n}(2P_{t}\sum_{j=1}^{n}u_{x_{j}}u_{x_{j}x_{i}})^{2}\leq
4P_{t}^{2}|\nabla u|^{2}\sum_{i,j=1}^{n}u_{x_{i}x_{j}}^{2}\\\
\Rightarrow|\nabla P|^{2}-2P_{s}\nabla P\nabla u+P_{s}^{2}|\nabla u|^{2}\leq
4P_{t}^{2}|\nabla u|^{2}|Hes\>u|^{2}\end{gathered}$

and in addition,

(3.7)
$\begin{gathered}P_{x_{i}}u_{x_{i}}=P_{s}u_{x_{i}}^{2}+2P_{t}\sum_{j=1}^{n}u_{x_{i}}u_{x_{j}}u_{x_{i}x_{j}}\\\
\Rightarrow 2P_{t}\sum_{i,j=1}^{n}u_{x_{i}}u_{x_{j}}u_{x_{i}x_{j}}=\nabla
P\nabla u-P_{s}|\nabla u|^{2}\end{gathered}$

Also by (3.2) it holds that,

(3.8) $\Delta
u_{x_{j}}=F_{s}u_{x_{j}}+2F_{t}\sum_{k=1}^{n}u_{x_{k}}u_{x_{k}x_{j}}$

Now, by (3.5) we have

(3.9)
$\begin{gathered}P_{x_{i}x_{i}}=P_{ss}u_{x_{i}}^{2}+4P_{st}u_{x_{i}}\sum_{j=1}^{n}u_{x_{j}}u_{x_{j}x_{i}}+4P_{tt}(\sum_{j=1}^{n}u_{x_{j}}u_{x_{j}x_{i}})^{2}\\\
+P_{s}u_{x_{i}x_{i}}+2P_{t}[\sum_{j=1}^{n}(u_{x_{i}x_{j}}^{2}+u_{x_{j}}u_{x_{j}x_{i}x_{i}})]\end{gathered}$

So, if we assume (3.3) (since the hessian of $P$ is positive semidefinite),
utilizing (3.6), we have

(3.10) $\begin{gathered}\Delta P\geq P_{s}\Delta
u+2P_{t}|Hes\>u|^{2}+2P_{t}\sum_{j=1}^{n}u_{x_{j}}\Delta u_{x_{j}}\\\
P_{t}|\nabla u|^{2}\Delta P\geq P_{t}P_{s}|\nabla u|^{2}\Delta
u+\frac{1}{2}|\nabla P|^{2}-P_{s}\nabla P\nabla u+\frac{P_{s}^{2}}{2}|\nabla
u|^{2}\\\ +2P_{t}^{2}|\nabla u|^{2}\sum_{j=1}^{n}u_{x_{j}}\Delta
u_{x_{j}}\end{gathered}$

and by (3.2) and (3.8) we obtain

(3.11) $\begin{gathered}P_{t}|\nabla u|^{2}\Delta P\geq P_{t}P_{s}|\nabla
u|^{2}F(u,|\nabla u|^{2})+\frac{1}{2}|\nabla P|^{2}-P_{s}\nabla P\nabla
u+\frac{P_{s}^{2}}{2}|\nabla u|^{2}\\\ +2P_{t}^{2}|\nabla u|^{2}[F_{s}|\nabla
u|^{2}+2F_{t}\sum_{j,k=1}^{n}u_{x_{k}}u_{x_{j}}u_{x_{j}x_{k}}]\end{gathered}$

In addition, by (3.7) we have

(3.12) $\begin{gathered}P_{t}|\nabla u|^{2}\Delta P\geq P_{t}P_{s}|\nabla
u|^{2}F(u,|\nabla u|^{2})+\frac{1}{2}|\nabla P|^{2}-P_{s}\nabla P\nabla
u+\frac{P_{s}^{2}}{2}|\nabla u|^{2}\\\ +2P_{t}^{2}F_{s}|\nabla
u|^{4}+2P_{t}|\nabla u|^{2}F_{t}\nabla P\nabla u-2P_{t}P_{s}F_{t}|\nabla
u|^{4}\\\ \Rightarrow P_{t}|\nabla u|^{2}\Delta P\geq\frac{1}{2}|\nabla
P|^{2}+(2P_{t}|\nabla u|^{2}F_{t}-P_{s})\nabla P\nabla u+|\nabla
u|^{2}I(u,|\nabla u|)\end{gathered}$

Therefore, since by (3.1) it holds that $I(u,|\nabla u|)\geq 0$, we conclude

(3.13) $P_{t}|\nabla u|^{2}\Delta P-(2P_{t}|\nabla u|^{2}F_{t}-P_{s})\nabla
P\nabla u\geq\frac{1}{2}|\nabla P|^{2}\geq 0$

Finally, if we assume (3.4) instead of (3.3), equation (3.9) similarly becomes

(3.14) $\begin{gathered}P_{t}|\nabla u|^{2}\Delta P-(2P_{t}|\nabla
u|^{2}F_{t}-P_{s})\nabla P\nabla u\geq\frac{1}{2}|\nabla P|^{2}\\\
+P_{ss}P_{t}|\nabla u|^{4}+|\nabla u|^{2}I(u,|\nabla u|)\geq 0\end{gathered}$

∎

######  Corollary 3.2.

Let $u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth solution
of

(3.15) $\Delta u=f(u)$

and $P(s,t)=P:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is such that $P_{t}>0$ for
$t>0$ and either

(3.16) $\begin{gathered}\begin{cases}Hes_{(s,t)}\>P\;\>\textrm{is positive
semidefinite and}\\\
I(s,t):=P_{s}(s,t^{2})f(s)+\frac{P_{s}^{2}(s,t^{2})}{2P_{t}(s,t^{2})}+2P_{t}(s,t^{2})t^{2}f^{\prime}(s)\geq
0\;\;,\;\forall\>(s,t)\in\mathbb{R}\times[0,+\infty)\end{cases}\end{gathered}$

or

(3.17) $\begin{cases}P_{st}=0\;,\;P_{tt}\geq 0\;\>\textrm{and}\\\
t^{2}P_{ss}(s,t^{2})+P_{s}(s,t^{2})f(s)+\frac{P_{s}^{2}(s,t^{2})}{2P_{t}(s,t^{2})}+2P_{t}(s,t^{2})t^{2}f^{\prime}(s)\geq
0\end{cases}$

hold.

Then $P=P(u,|\nabla u|^{2})$ is a $P-$function of (3.15).

###  3.1. Examples of $P-$functions

(1) The well known $P-$function of (3.15) is

(3.18) $\begin{gathered}P(u,|\nabla u|^{2})=\frac{|\nabla u|^{2}}{2}-F(u)\\\
\textrm{where}\;\>F^{\prime}(u)=f(u)\end{gathered}$

(see [14] or Chapter 5 in [18]).

It is easy to see that (3.18) satisfies (3.17) in Corollary 3.2.

The major application is the well known gradient bound

(3.19) $|\nabla u|^{2}\leq 2F(u)$

that holds for every smooth and bounded entire solution of (3.15) and $F\geq
0$ (see [14]).

However we cannot obtain gradient bounds for $P-$functions in general.
Consider for example

$\displaystyle\Delta
u=e^{u}\;\>\textrm{and}\;\>P(s,t)=\frac{t}{2}-e^{s}+e^{-s}\;\>(\textrm{i.e.}\>\;P(u,|\nabla
u|^{2})=\frac{|\nabla u|^{2}}{2}-e^{u}+e^{-u})$

It is easy to see that $P=\frac{|\nabla u|^{2}}{2}-e^{u}+e^{-u}$ satisfies
(3.17), but the gradient bound $\\\ \frac{|\nabla u|^{2}}{2}\leq e^{u}-e^{-u}$
does not hold: if we take solutions of $\Delta u=e^{u}$ such that $|\nabla
u|^{2}=2e^{u}$ (i.e. that satisfy the equiparition of the energy), then we
have a contradition. (2) Another general example of $P-$function of (3.15) is

(3.20) $\begin{gathered}P(u,|\nabla u|^{2})=\frac{|\nabla
u|^{4}}{2}+2\int_{0}^{u}(\int_{0}^{y}\sqrt{f(z)f^{\prime}(z)}dz)^{2}dy\;\;\;\;,\;\;\textrm{if}\;\>f(t)f^{\prime}(t)\geq
0\;,\>\forall\>t\in\mathbb{R}\\\ P(u,|\nabla u|^{2})=\frac{|\nabla
u|^{4}}{2}-2\int_{0}^{u}(\int_{0}^{y}\sqrt{-f(z)f^{\prime}(z)}dz)^{2}dy\;\;\;\;,\;\;\textrm{if}\;\>f(t)f^{\prime}(t)\leq
0\;\;\;\end{gathered}$

and satisfies condition (3.17) of Corollary 3.2.

Note that the above example is not in the form $P=g(u)|\nabla u|^{2}+h(u)$
that we see in [18] as general form for $P$ related to equation (3.15). (3)
The next example can be found in [15]. Let $u$ be a solution of

(3.21) $\begin{gathered}\Delta u=u(k|\nabla u|^{2}+\lambda e^{-cu^{2}})\\\
\textrm{and
let}\;\>P(s,t)=\begin{cases}te^{-ks^{2}}+\frac{\lambda}{k+c}e^{-s^{2}(k+c)}\;\;,\;k\neq-c\\\
te^{cs^{2}}-\lambda
s^{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\;k=-c\end{cases}\end{gathered}$

Then $P=P(u,|\nabla u|^{2})$ is a $P-$function of (3.21).

(4) Let $u$ be a solution of

(3.22) $\Delta u=G(|\nabla
u|^{2}-u)\;\>,\;\textrm{where}\;\>G(z)\leq\frac{1}{2}\;,\>\forall\>z\in\mathbb{R}$

Then $P=P(u,|\nabla u|^{2})=|\nabla u|^{2}-u$ is a $P-$function of (3.22). It
is easy to see that $P$ satisfies condition (3.4) of Theorem 3.1.

(5) The following example is in [16] (see Theorem 1).

Let $u$ be a solution of

(3.23) $div(\Phi^{\prime}(|\nabla u|^{2})\nabla u)=\rho(|\nabla
u|^{2})F^{\prime}(u)$

with $\Phi^{\prime}(t),\rho(t)>0$ and
$\Phi^{\prime}(t)+2t\Phi^{\prime\prime}(t)>0\;,\>\forall\>t\geq 0$.

Consider the function

(3.24)
$P(s,t)=\int_{0}^{t}\frac{\Phi^{\prime}(y)+2y\Phi^{\prime\prime}(y)}{\rho(y)}dy-2F(s)$

Then $P=P(u,|\nabla u|^{2})$ is a $P-$function of (3.23).

Note that for $\rho\equiv 1$, the above example generalizes the one in [6].

##  4\. Gradient Bounds and properties of entire solutions of quasi-linear
equations

In this section we will see that utilizing the techniques of [6], we can
obtain gradient bounds for solutions of equations of the form (3.2) and for
more general class of quasi-linear equations. To be more precise, for any
explicit example of $P-$function with a specific property, we obtain a
particular gradient bound. Also, it holds an analogous of Theorem 2.4 that
gives De Giorgi-type results and additionally, if $P$ have a more specific
form, we can drop the assumption $u_{x_{n}}>0$.

Some of the regularity assumptions can be lessened in some cases, i.e. by
assuming a priori that $u\in W^{1,p}_{loc}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})$, as in assumption (i) in Theorem 1.6 in [6] and
utilizing regularity results in [19] afterwords. However, our main goal is not
the optimal regularity assumptions since we state the results in an abstract
form. Therefore, we will assume that the solutions are smooth and satisfy an
analog of assumption (ii) in Theorem 1.6 in [6].

Assumption.

(4.1) $\begin{gathered}u\in C^{2}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})\;,\>\nabla u\in
C^{\alpha}(\mathbb{R}^{n};\mathbb{R}^{n})\;\>\textrm{for
some}\;\>\alpha\in(0,1)\\\ \textrm{and}\;\>\textrm{there
exists}\;\>C=C(||u||_{L^{\infty}(\mathbb{R}^{n})})>0\;\>\textrm{such
that}\;\>|\nabla u(x)|\leq C\>,\;\textrm{for
any}\;\>x\in\mathbb{R}^{n}\end{gathered}$

The next theorem provides an a priori pointwise estimate for solutions of
(3.2). In contrast to the gradient bounds in [6] and [7], the theorem below
holds for any $P-$function that satisfies $P(u,0)\leq 0$. When $P$ is of the
form $P=P(u,\nabla u)$ we use the notation $P(u,0)$ instead of $P(u,0,...,0)$
and also we sometimes we write $P=P(u;x)$ for simplicity.

######  Theorem 4.1.

Let $u$ be an entire solution of

(4.2) $F(u,\nabla u,\nabla^{2}u)=0$

that satisfy assumption (4.1). If $P=P(u,\nabla u)$ is a $P-$function of
(4.2), with $\mu=\mu(|\nabla u|)$, $\mu(t)>0\;,\>\forall\>t>0$, such that
$P(s,0)\leq 0$,

Then

(4.3) $P(u(x),\nabla u(x))\leq 0\;\;,\;x\in\mathbb{R}^{n}$

###### Proof.

Let $u$ be a solution of (4.2) that satisfies assumption (4.1) and consider
the set

(4.4) $\mathcal{F}=\\{v\;\>\textrm{is a solution of
\eqref{ThmGradientBoundNonlinearEq} that satisfies
\eqref{AssumptionOnSolutionsSecondOrder}}\;:\>|v(x)|\leq||u||_{L^{\infty}(\mathbb{R}^{n})}\;\>,\;\forall\;x\in\mathbb{R}^{n}\\}$

$\mathcal{F}$ is non empty since $u\in\mathcal{F}$.

Let $P$ be a $P-$function of (4.2), with $\mu=\mu(|\nabla u|)$,
$\mu(t)>0\;,\>\forall t>0$, such that $P(u,0)\leq 0$. For simplicity, we
denote $P=P(u;x)$ instead of $P=P(u(x),\nabla u(x))$.

Consider now

(4.5) $P_{0}=\sup\\{P(v;x)\>|\>v\in\mathcal{F}\;,\;\>x\in\mathbb{R}^{n}\\}$

We claim that $P_{0}\leq 0$ and from this we conclude.

We argue by contradiction. Supposet that $P_{0}>0$, by (4.5) there exist two
sequences, $(v_{k})_{k\in\mathbb{N}}$ in $\mathcal{F}$ and
$(x_{k})_{k\in\mathbb{N}}$ in $\mathbb{R}^{n}$ such that

(4.6) $P_{0}-\frac{1}{k}\leq P(v_{k};x_{k})\leq
P_{0}\;\;\;\;,\;\;k\in\mathbb{N}$

Let $\tilde{v}_{k}(x)=v(x+x_{k})$. Since the equation (4.2) is translation
invariant, we have that $\tilde{v}_{k}\in\mathcal{F}$ and
$P(\tilde{v}_{k};0)=P(v_{k};x_{k})$, so that (4.6) can be rewritten as

(4.7) $P_{0}-\frac{1}{k}\leq P(\tilde{v}_{k};0)\leq
P_{0}\;\;\;\;,\;\;k\in\mathbb{N}$

Since $\nabla\tilde{v}_{k}\in C^{\alpha}(\mathbb{R}^{n})$, by the Ascoli-
Arzela theorem since we have equicontinuity of $\nabla\tilde{v}_{k}$ in any
compact subset of $\mathbb{R}^{n}$ together with a diagonal argument, we can
extract from $(\tilde{v}_{k})_{k\in\mathbb{N}}$ a subsequence, denoted by
$(\tilde{v}_{k}^{(k)})_{k\in\mathbb{N}}$ that converges with its first-order
derivatives, uniformly on compact subsets of $\mathbb{R}^{n}$. Denote by
$\tilde{v}$ the limit function. Then $\tilde{v}\in\mathcal{F}$ and
$P(\tilde{v}_{k}^{(k)};0)\rightarrow P(\tilde{v};0)$ as $k\rightarrow\infty$.
From (4.7) we have $P(\tilde{v};0)=P_{0}$.

Consider now the set

(4.8) $U=\\{x\in\mathbb{R}^{n}\>|\>P(\tilde{v};x)=P_{0}\\}$

$U$ is closed since $P$ is continuous on $\mathbb{R}^{n}$ and non empty since
$0\in U$. We will prove that $U$ is also open. Let $x_{0}\in U$, we observe
that $|\nabla\tilde{v}(x_{0})|\neq 0$, otherwise we would have

$\displaystyle
P_{0}=P(\tilde{v};x_{0})=P(\tilde{v}(x_{0}),\nabla\tilde{v}(x_{0}))=P(\tilde{v}(x_{0}),0)\leq
0$

against the fact that $P_{0}>0$.

By continuity, there exists $\delta>0$ such that

(4.9) $\inf_{\overline{B}_{\delta}(x_{0})}|\nabla\tilde{v}|>0\;\;\textrm{and
thus}\;\>\inf_{\overline{B}_{\delta}(x_{0})}\mu(|\nabla\tilde{v}|)>0$

and we conclude that $P(\tilde{v};x)\equiv P_{0}$ in $B_{\delta}(x_{0})$ by
Theorem 2.2. So $U$ is open and it follows that $U=\mathbb{R}^{n}$ by
connectedness.

On the other hand, since $|\tilde{v}|\leq||u||_{L^{\infty}(\mathbb{R}^{n})}$,
it holds that $\inf_{\mathbb{R}^{n}}|\nabla\tilde{v}|=0$. Let
$(y_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{n}$ such that
$|\nabla\tilde{v}(y_{j})|\rightarrow 0$ as $j\rightarrow+\infty$. By the
boundedness of $\tilde{v}$ we also have
$\tilde{v}(y_{j})=\tilde{v}_{j}\rightarrow v_{0}$ up to a subsequence that we
still denote as $y_{j}$, and so we obtain

$\displaystyle
0<P_{0}=\lim_{j\rightarrow\infty}P(\tilde{v}(y_{j}),\nabla\tilde{v}(y_{j}))=P(v_{0},0)$

which contradicts the assumption $P(s,0)\leq 0$. Therefore $P_{0}\leq 0$ and
we conclude. ∎

Next, we explore some additional consequences when $P$ related to (4.2) is of
the form

(4.10) $\begin{gathered}P(s,t)=B(t)-\Gamma(s)\\\ \textrm{such
that}\;\>B^{\prime}(t)>0\>,\;B^{\prime\prime}(t)\geq
0\;,\>\textrm{for}\;\>t>0\;,\>B(0)=0\;\>\textrm{and}\;\>\Gamma(s)\geq 0\\\
\textrm{and}\;\>\mu=\mu(|\nabla u|)\end{gathered}$

then, the condition (3.4) becomes

(4.11)
$\begin{gathered}-t^{2}\Gamma^{\prime\prime}(s)B^{\prime}(t^{2})-B^{\prime}(t^{2})\Gamma^{\prime}(s)F(s,t^{2})+\frac{(\Gamma^{\prime}(s))^{2}}{2}+2t^{2}(B^{\prime}(t^{2}))^{2}F_{s}(s,t^{2})\\\
+2t^{2}B^{\prime}(t^{2})\Gamma^{\prime}(s)F_{t}(s,t^{2})\geq 0\end{gathered}$

for $P$ as in (4.10), the gradient bound in Theorem 4.1 becomes

(4.12) $|\nabla
u|^{2}\leq\Psi(u)\;\;\;,\;\;\textrm{where}\;\>\Psi(u)=B^{-1}(\Gamma(u))$

and we observe that in Example (2) above, for solutions of (3.15),

(4.13) $\displaystyle P(u,|\nabla u|^{2})=\frac{|\nabla
u|^{4}}{2}-2\int_{0}^{u}(\int_{0}^{y}\sqrt{-f(z)f^{\prime}(z)}dz)^{2}dy\;\;\;\;,\;\;\textrm{if}\;\>f(t)f^{\prime}(t)\leq
0\;\;\;$

satisfies $P(u,0)\leq 0$.

Therefore, we have

######  Corollary 4.2.

Let $u$ be a smooth and bounded entire solution to

(4.14) $\begin{gathered}\Delta u=f(u)\\\ \textrm{where}\;\>f\in
C^{1,\alpha}(\mathbb{R}^{n})\;\>\textrm{and}\;\>f(t)f^{\prime}(t)\leq
0.\end{gathered}$

Then

(4.15) $\frac{|\nabla
u|^{4}}{4}\leq\int_{0}^{u}(\int_{0}^{y}\sqrt{-f(z)f^{\prime}(z)}dz)^{2}dy$

###### Proof.

By elliptic regularity theory we have that $u\in C^{2,\alpha}(\mathbb{R}^{n})$
and that $|\nabla u|$ is bounded in $\mathbb{R}^{n}$. It suffices to prove
that $P$ defined in (4.13) is a $P-$function of (4.14) and then the conclusion
is direct application of Theorem 4.1.

We have that $P$ satisfies (3.17),
$P(s,t)=\frac{t^{2}}{2}-2\int_{0}^{s}(\int_{0}^{y}\sqrt{-f(z)f^{\prime}(z)}dz)^{2}dy=\frac{t^{2}}{2}+q(s),\\\
$ so $P_{t}=t>0$ for $t>0$, $P_{tt}\geq 0\;,\;P_{st}=0$ and
$\mu=P_{t}(u,|\nabla u|^{2})|\nabla u|^{2}=\frac{1}{2}|\nabla u|^{4}$.

Finally,

$\displaystyle
t^{2}P_{ss}(s,t^{2})+P_{s}(s,t^{2})f(s)+2P_{t}(s,t^{2})t^{2}f^{\prime}(s)=2t^{4}f^{\prime}(s)+t^{2}q^{\prime\prime}(s)+q^{\prime}(s)f(s)\geq
0$

since the above polynomial has zero discriminant. ∎

######  Remark 4.3.

(1) Note that in Example (1) above, $P=\frac{|\nabla u|^{2}}{2}-e^{u}+e^{-u}$
which does not satisfy the gradient bound, fails to satisfy the condition
$P(s,0)\leq 0$ for $s<0.\\\ $ (2) We also observe that there are cases where
the bound (4.15) is more optimal than (1.2). Consider for example the
potential $W(u)=au^{k}\;,\>a>0$ and
$k\in(0,\frac{3-\sqrt{3}}{6})\cup(\frac{3+\sqrt{3}}{6},1)$ and for positive
solutions of (1.1), with $f(u)=W^{\prime}(u)$. Then we have

$\displaystyle\int_{0}^{u}(\int_{0}^{y}\sqrt{-f(z)f^{\prime}(z)}dz)^{2}dy=\frac{a^{2}k(1-k)}{2(k-\frac{1}{2})^{2}}u^{2k}<a^{2}u^{2k}=W^{2}(u)$

that is,

$\displaystyle\frac{1}{2}|\nabla
u|^{2}\leq\frac{a\sqrt{k(1-k)}}{\sqrt{2}\>|k-\frac{1}{2}|}u^{k}<au^{k}=W(u)\;\;\;,\;\;\textrm{for}\;\>k\in(0,\frac{3-\sqrt{3}}{6})\cup(\frac{3+\sqrt{3}}{6},1)$

In addition, we have the following gradient bounds for Examples (3) and (4)

######  Corollary 4.4.

Let $u$ be an entire solution of

(4.16) $\Delta u=u(k|\nabla u|^{2}+\lambda e^{-cu^{2}})$

that satisfy (4.1).

Then

(4.17) $|\nabla
u|^{2}\leq\begin{cases}-\frac{\lambda}{k+c}e^{-cu^{2}}\;\;\;,\;\textrm{if}\;\>\lambda(k+c)<0\\\
\lambda
u^{2}e^{-cu^{2}}\;\;\;\;\;\;,\;\textrm{if}\;\>k=-c\;\>\textrm{and}\;\>\lambda\geq
0\end{cases}$

###### Proof.

By [15], we have that

(4.18)
$P(s,t)=\begin{cases}te^{-ks^{2}}+\frac{\lambda}{k+c}e^{-s^{2}(k+c)}\;\;,\;k\neq-c\\\
te^{cs^{2}}-\lambda s^{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,\;k=-c\end{cases}$

is a $P-$function of (4.16) with $\mu(t)>0\;,\>\forall\>t\geq 0$ and
$P(s,0)\leq 0$ in both cases since either $\lambda(k+c)<0$ or $k=-c$ and
$\lambda\geq 0$. Therefore by Theorem 4.1 we conclude that $P(u,|\nabla
u|^{2})\leq 0\;\>\forall x\in\mathbb{R}^{n}$ and we obtain the gradient bound
(4.17). ∎

######  Remark 4.5.

For $\lambda=0$, Corollary 4.4 says that $|\nabla u|\equiv 0$ and thus $u$ is
a constant. That is a Liouville-type result and can also be obtained by
Liouville’s theorem by setting
$v=g(u)\;,\;\textrm{where}\;\>g(z)=\int_{0}^{y}e^{-kz^{2}}dz$ and then $\Delta
v=0$ and $v$ is bounded since $u$ is bounded.

######  Corollary 4.6.

Let $u$ be a non negative entire solution of

(4.19) $\begin{gathered}\Delta u=G(|\nabla u|^{2}-u)\\\
\textrm{where}\;\>G:\mathbb{R}\rightarrow\mathbb{R}\;\>\textrm{is such
that}\;\>G(z)\leq\frac{1}{2}\>,\;z\in\mathbb{R},\end{gathered}$

that satisfy (4.1).

Then

(4.20) $|\nabla u|^{2}\leq u$

###### Proof.

We have that the function

$\displaystyle P(s,t)=t-s$

satisfy the condition (3.4) with $\mu=P_{t}(u,|\nabla u|^{2})|\nabla
u|^{2}=|\nabla u|^{2}$ and also, $P(u,0)=-u\leq 0$ since $u$ is non negative
by assumption. Therefore we conclude by the Theorem 4.1. ∎

Another important application of Theorem 4.1, utilizing the Example (5), is
the following gradient bound

######  Corollary 4.7.

Let $u$ be an entire solution of

(4.21) $div(\Phi^{\prime}(|\nabla u|^{2})\nabla u)=\rho(|\nabla
u|^{2})F^{\prime}(u)\;\;,\;F\geq 0$

that satisfy assumption (4.1), with $\Phi^{\prime}(t),\rho(t)>0$ and
$\Phi^{\prime}(t)+2t\Phi^{\prime\prime}(t)>0\;,\>\forall\>t\geq 0$.

Then

(4.22) $\begin{gathered}|\nabla
u|^{2}\leq\Psi(u)\;\;,\;\textrm{where}\;\>\Psi(u)=Q^{-1}(2F(u))\\\
\textrm{and}\;\>Q(t)=\int_{0}^{t}\frac{\Phi^{\prime}(y)+2y\Phi^{\prime\prime}(y)}{\rho(y)}dy\end{gathered}$

###### Proof.

By Theorem 1 in [16], we have that $P(u,|\nabla u|^{2})=Q(|\nabla
u|^{2})-2F(u)$ is a $P-$function of (4.21) with $\mu(t)>0\;,\>\forall\>t\geq
0$ and satisfies $P(u,0)\leq 0$ since $F\geq 0$. Thus we apply Theorem 4.1 and
we conclude. ∎

######  Remark 4.8.

Note that the gradient bound in Corollary 4.7 is a quite more general form of
the gradient bound in [6].

For $P-$functions of the form (4.10), we have a Liouville-type result

######  Theorem 4.9.

Let $u$ be an entire solution of (4.2) that satisfies assumption (4.1) and $P$
is a $P-$function of the form (4.10). If there exists $x_{0}\in\mathbb{R}^{n}$
such that $\Gamma(u(x_{0}))=0$, then $u\equiv\textrm{const.}$ in
$\mathbb{R}^{n}$.

###### Proof.

We argue as in the proof of Theorem 1.8 in [6] with slight modifications. For
the convenience of the reader we provide the details.

Suppose that $\Gamma(u(x_{0}))=0$, let $u_{0}=u(x_{0})$ and consider the set

(4.23) $V=\\{x\in\mathbb{R}^{n}\>|\>u(x)=u_{0}\\}$

$V$ is a closed set and by the assumption, non empty. Let $x_{1}\in V$ and
consider the function $\phi(t)=u(x_{1}+t\omega)-u_{0}$, where $|\omega|=1$ is
arbitrarily fixed. We have $|\phi^{\prime}(t)|=|\nabla u(x_{1}+t\omega)|$. By
the gradient bound in Theorem 4.1 we have,

(4.24) $|\nabla
u|^{2}\leq\Psi(u)\;\;,\;\textrm{where}\;\>\Psi(s)=B^{-1}(\Gamma(s))$

Since $\Psi\in C^{2}(\mathbb{R})$ and $\Psi(u_{0})=0$, we have
$\Psi(u)=O(|u-u_{0}|^{2})$, as $|u-u_{0}|\rightarrow 0$. So, we conclude from
(4.24) that $|\phi^{\prime}(t)|\leq C|\phi(t)|$ for $t$ small enough. Since
$\phi(0)=0$, we must have $\phi\equiv 0$ on $[-\delta,\delta]$, for some
$\delta>0$. Thus $V$ is open, which gives that $V=\mathbb{R}^{n}$. ∎

Finally, we now prove a different version of Theorem 2.4 for the equation
(3.2), assuming that $P$ is of the form (4.10). In this case we can drop the
assumption that $u$ is monotone with respect to $x_{n}$, but we do not allow
any a priori singularities.

######  Theorem 4.10.

Let $u$ be an entire solution of

(4.25) $F(u,\nabla u,\nabla^{2}u)=0$

that satisfy assumption (4.1) and let $P=P(u,|\nabla u|^{2})$ be a
$P-$function of (4.25) of the form (4.10). If there exists
$x_{0}\in\mathbb{R}^{n}$ such that

(4.26) $P(u(x_{0}),|\nabla u(x_{0})|^{2})=0$

then there exists a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that

(4.27) $\begin{gathered}\textrm{either}\;\;\;\>u(x)=g(a\cdot
x+b)\;,\;a\in\mathbb{R}^{n}\;\>\textrm{with}\;\>|a|=1,\;,\>b\in\mathbb{R}\\\
\textrm{or}\;\;\;\>u(x)=g(|x-z_{0}|+c)\;\>,\;z_{0}\in\mathbb{R}^{n}\;\>\textrm{and}\;\>c\in\mathbb{R}\end{gathered}$

###### Proof.

By Theorem 4.1, we have that $P(u,|\nabla u|^{2})\leq 0$.

As in the proof of Theorem 2.4, we begin by considering the set

(4.28) $A=\\{x\in\mathbb{R}^{n}\>:\>P(u,|\nabla u|^{2})=0\\}$

$A$ is closed and by the assumption $A\neq\emptyset$. We are going to prove
that $A$ is open.

Let $x_{1}\in A$, If $\nabla u(x_{1})=0$, we obtain by the form
$P(s,t)=B(t)-\Gamma(s)$ that $P(u(x_{1}),0)=-\Gamma(u(x_{1}))=0$. By Theorem
4.9, we conclude that $u\equiv u(x_{1})$ and $\nabla u\equiv 0$ and hence
$P\equiv 0$.

On the other hand, if $\nabla u(x_{1})\neq 0$, we have
$\inf_{\overline{B}_{\delta_{1}}(x_{1})}|\nabla u|>0$ for some $\delta_{1}>0$
and by Theorem 2.2 we conclude that $P(u,|\nabla u|^{2})\equiv 0$ in
$B_{\delta_{1}}(x_{1})$ and therefore $A$ is open.

By connectedness, we have that $A=\mathbb{R}^{n}$, that is,

(4.29) $P(u,|\nabla u|^{2})\equiv 0\;\;,\;\forall\>x\in\mathbb{R}^{n}$

and $P_{t}=B^{\prime}(t)>0$, thus

(4.30) $|\nabla
u|^{2}=\Psi(u)\;\;\;,\;\;\textrm{in}\;\>\mathbb{R}^{n}\;\;,\;\textrm{where}\;\>\Psi(u)=B^{-1}(\Gamma(u))$

Now, if there exists $x_{2}\in\mathbb{R}^{n}$ such that $\Phi(u(x_{2}))=0$, so
$|\nabla u(x_{2})|=0$, again by Theorem 4.9 we have that $u\equiv u(x_{2})$.

If, on the other hand $\Psi(u(x))>0\;\>,\;\forall\>x\in\mathbb{R}^{n}$, we set

(4.31)
$\begin{gathered}v=G(u)\;\;\;,\;\;\textrm{where}\;\>G^{\prime}(s)=\frac{1}{\Psi(s)}\\\
\textrm{and}\;\>|\nabla
v|^{2}=1\;\;\;\textrm{in}\;\>\mathbb{R}^{n}\end{gathered}$

Therefore, by the result in [5], we have that

(4.32) $\begin{gathered}\textrm{either}\;\>v(x)=a\cdot
x+b\;\>,\;a\in\mathbb{R}^{n}\;\>\textrm{with}\;\>|a|=1\;\>\textrm{and}\;\>b\in\mathbb{R}\\\
\textrm{or}\;\>v(x)=|x-z_{0}|+c\;\>,\;z_{0}\in\mathbb{R}^{n}\;\>\textrm{and}\;\>c\in\mathbb{R}\end{gathered}$

So we conclude that

(4.33) $\begin{gathered}\textrm{either}\;\;\;\>u(x)=g(a\cdot
x+b)\;\>,\;a\in\mathbb{R}^{n}\;\>\textrm{with}\;\>|a|=1\;,\;\>b\in\mathbb{R}\;\>\textrm{where}\;\>g(s)=G^{-1}(s)\\\
\textrm{or}\;\;\;\>u(x)=g(|x-z_{0}|+c)\;\>,\;z_{0}\in\mathbb{R}^{n}\;\>\textrm{and}\;\>c\in\mathbb{R}\end{gathered}$

∎

##  5\. A Liouville-type property and nonexistence results

In this section we will see that if we are able to find a $P-$function,
related to any equation (4.2), of the form $P=g(\nabla u)$, where $g\geq 0$
and $g(t_{1},...,t_{n})=0$ iff $(t_{1},...,t_{n})=(0,...,0)$ then either the
solutions are constant or there are no solutions that satisfy assumption
(4.1). The most common example is when $P=|\nabla u|^{2}$.

In particular we have

######  Theorem 5.1.

Let $u$ be an entire solution of

(5.1) $F(u,\nabla u,\nabla^{2}u)=0$

that satisfy assumption (4.1). If $P=P(u,\nabla u)$ is a $P-$ function of
(5.1), with $\mu=\mu(|\nabla u|)\;,\;\mu(t)>0\;,\>\forall\>t>0$ and $P$ is
such that

(5.2) $P=g(\nabla
u)\;\;\;,\;\textrm{where}\;\>g:\mathbb{R}^{n}\rightarrow[0,+\infty)\;\>\textrm{and}\;\>\\{x\in\mathbb{R}^{n}\>:\>g(x)=0\\}=\\{(0,...,0)\\}$

Then $u$ is constant.

###### Proof.

The result of Theorem 5.1 is an immediate application of Theorem 4.1 since
$P(u,0)=0$ and thus $g(\nabla u)=0$ which gives that $u$ is constant. ∎

######  Corollary 5.2.

Let $u$ be an entire solution of

(5.3) $\Delta u=f(u)\;\;\;,\;\textrm{where}\;\>f^{\prime}(u)\geq 0$

that satisfies (4.1).

Then $u$ is constant.

###### Proof.

Consider $P(u,|\nabla u|^{2})=|\nabla u|^{2}$ (i.e. $P(s,t)=t$) and observe
that by the proof of Theorem 3.1, in particular by (3.14), we have

(5.4) $|\nabla u|^{2}\Delta P\geq\frac{1}{2}|\nabla P|^{2}+|\nabla
u|^{4}f^{\prime}(u)\geq 0$

since $I(u,|\nabla u|)=2|\nabla u|^{2}f^{\prime}(u)$ and so $\mu=|\nabla
u|^{2}$. Therefore by Theorem 5.1 we conclude that $u$ is constant. ∎

######  Remark 5.3.

The assumption $f^{\prime}(u)\geq 0$ implies stability for any solution, since

$\displaystyle\int_{\Omega}|\nabla\phi|^{2}+f^{\prime}(u)\phi^{2}\geq
0\;\>,\;\textrm{for any
open}\;\>\Omega\subset\mathbb{R}^{n}\;\>\textrm{and}\;\>\phi\in
C^{1}_{c}(\Omega).$

So this assumption makes the problem quite simple. When the condition
$f^{\prime}\geq 0$ is not satisfied the solutions of (5.3) are not necessarily
stable and the study of stable solutions in such cases is very important.
Stable solutions have been thoroughly studied for semilinear elliptic
equations in [3], [4] and [8] among others.

Note that if $u$ is solution of (5.3) in a domain $\Omega$, by (5.4) we have a
mean value inequality for the gradient of $u$ whenever the gradient does not
vanish in the domain.

Corollary 5.3 can be generalized to the equations of the form
$div(\Phi^{\prime}(|\nabla u|^{2}\nabla u)=F^{\prime}(u)$, studied in [6]
assuming that $F^{\prime\prime}(u)\geq 0$ and in fact even fewer assumptions
than (4.1) are needed. We can just assume (i) or (ii) from Theorem 1.6 in [6]
and the Liouville-type theorem below still holds.

In particular, let $\Phi$ as in [6], we have the following

######  Corollary 5.4.

Let $u$ be an entire solution of

(5.5) $div(\Phi^{\prime}(|\nabla u|^{2}\nabla
u)=F^{\prime}(u)\;\;\;,\;\textrm{where}\;\>F^{\prime\prime}(u)\geq 0$

that satisfies (4.1).

Then $u$ is constant.

###### Proof.

Consider $P=|\nabla u|^{2}$. Arguing as in the proof of Theorem 2.2 in [6], we
obtain

(5.6) $\sum_{i,j=1}^{n}(a_{ij}(\nabla u)P_{x_{i}})_{x_{j}}\geq
2F^{\prime\prime}(u)|\nabla u|^{2}+\frac{2|\nabla
u|^{2}\Phi^{\prime\prime}(|\nabla u|^{2})+\Phi^{\prime}(|\nabla
u|^{2})}{4|\nabla u|^{2}}|\nabla P|^{2}$

where
$a_{ij}(\sigma)=2\Phi^{\prime\prime}(|\sigma|^{2})\sigma_{i}\sigma_{j}+\Phi^{\prime}(|\sigma|^{2})\delta_{ij}$.

Thus, $P=|\nabla u|^{2}$ is a $P-$function of (5.5), since
$F^{\prime\prime}\geq 0$, and by Theorem 5.1 we conclude. ∎

In [6], it has been proved an analogous result to that of Corollary 5.4 using
a monotonicity formula for the energy for positive solutions of (5.5) that
vanish at infinity (see Theorem 4.5 in [6]).

We will now see a non existence result. If (5.1) do not admit constant
solutions then Theorem 5.1 states that the equation (5.1) do not admit any
entire solution that satisfy assumption (4.1).

######  Corollary 5.5.

Consider the equation

(5.7) $\begin{gathered}\Delta u=G(|\nabla
u|^{2})\;\;,\;\textrm{where}\;\>u:\mathbb{R}^{n}\rightarrow\mathbb{R}\\\
\textrm{and}\;\>G:\mathbb{R}\rightarrow\mathbb{R}\;\>\textrm{is such
that}\;\>G(0)\neq 0\end{gathered}$

Then there does not exist any solution of (5.7) that satisfies assumption
(4.1).

###### Proof.

Suppose that there exist a solution of (5.7) that satisfies assumption (4.1).
Let $P(u,|\nabla u|^{2})=|\nabla u|^{2}$ i.e. $P(s,t)=t$, then $P$ satisfies
condition (3.4) since $I(u,|\nabla u|)=0$. Therefore by Theorem 5.1 we
conclude that $u$ is a constant and we have contradiction since $G(0)\neq 0$.
∎

##  6\. $P-$functions for the Monge-Ampere equation

We will now prove that for any function such that the determinant of its
hessian is positive, we can obtain a class of functions that are $P-$functions
of this differential inequality. Then we obtain a Mean Value type property for
the Monge-Ampere equation.

######  Proposition 6.1.

Let $u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth function
that satisfies

(6.1) $det(\nabla^{2}u)>0\;\;,\;\textrm{in}\;\>\Omega$

and let $g=g(t_{1},...,t_{n})$ such that its hessian $Hes_{t}g$ is positive
semi definite.

Then $g=g(u_{x_{1}},...,u_{x_{n}})$ is a $P-function$ of (6.1).

###### Proof.

We have

(6.2)
$\begin{gathered}g_{x_{i}}=\sum_{j=1}^{n}g_{t_{j}}u_{x_{j}x_{i}}\;\;,\;i=1,...,n.\\\
\Leftrightarrow G_{x}=(\nabla^{2}u)G_{t}\\\
\textrm{where}\;\>G_{x}=(g_{x_{1}},...,g_{x_{n}})^{T}\;,\;G_{t}=(g_{t_{1}},...,g_{t_{n}})^{T}\;\;\textrm{and}\;\>\nabla^{2}u=Hes\>u\end{gathered}$

by (6.1), $(\nabla^{2}u)$ is invertible, and thus

(6.3) $\begin{gathered}G_{t}=(\nabla^{2}u)^{-1}G_{x}\\\
g_{t_{i}}=\sum_{k=1}^{n}A_{k}^{i}g_{x_{k}}\;\;,\;A_{k}^{i}=A_{k}^{i}(\nabla^{2}u)\end{gathered}$

Also,

(6.4)
$g_{x_{i}x_{i}}=\sum_{j=1}^{n}\sum_{k=1}^{n}g_{t_{j}t_{k}}u_{x_{j}x_{i}}u_{x_{k}x_{i}}+\sum_{j=1}^{n}g_{t_{j}}u_{x_{j}x_{i}x_{i}}\geq\sum_{j=1}^{n}g_{t_{j}}u_{x_{j}x_{i}x_{i}}$

since $Hes_{t}g$ is positive semidefinite.

So, by (6.3) we obtain

(6.5) $\begin{gathered}\Delta g\geq\sum_{j=1}^{n}g_{t_{j}}\Delta
u_{x_{j}}=\sum_{j=1}^{n}\sum_{k=1}^{n}A_{k}^{j}g_{x_{k}}\Delta
u_{x_{j}}=\sum_{k=1}^{n}g_{x_{k}}B_{k}\\\
\textrm{where}\;\>B_{k}=\sum_{j=1}^{n}A_{k}^{j}\Delta u_{x_{j}}\end{gathered}$

and thus,

(6.6) $L\>g\leq
0\;\;\;,\;\;\textrm{where}\;\>L=-\Delta+\sum_{i=1}^{n}B_{i}\partial_{x_{i}}$

∎

######  Corollary 6.2.

Let $u:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth solution
of

(6.7) $det(\nabla^{2}u)=f$

where $f:\mathbb{R}\rightarrow(0,+\infty)$ and let
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is such that its Hessian is positive
semi definite.

Then

(6.8) $\max_{\overline{\Omega}}g(|\nabla u|)\leq\max_{\partial\Omega}g(|\nabla
u|)$

In addition, for any $x_{0}\in\Omega$ there exists an increasing family
$D_{R}(x_{0})$ which satisfies

$\displaystyle B_{cR}(x_{0})\subset D_{R}(x_{0})\subset
B_{CR}(x_{0})\;,\;\textrm{with}\;\>c,C\;\>\textrm{depending only on}\;\>n$

and for $R<S$, we have

(6.9) $|\nabla
u(x_{0})|^{2}\geq\frac{1}{|D_{R}(x_{0})|}\int_{D_{R}(x_{0})}|\nabla
u|^{2}\geq\frac{1}{|D_{S}(x_{0})|}\int_{D_{S}(x_{0})}|\nabla u|^{2}$

###### Proof.

By Proposition 6.1 and the maximum principle we conclude. For the mean value
type inequality, we set $g(t)=t_{1}^{2}+...+t_{n}^{2}$ and apply Theorem 6.3
in [2]. ∎

######  Remark 6.3.

(1) Such mean value type inequality hold for any $P-$function in general
related to any equation. (2) If $\Omega=\mathbb{R}^{n}$ and consider
$g(t_{1},...,t_{n})=t_{1}^{2}+...+t_{n}^{2}$ in Proposition 6.1, then as
mentioned in the precious section, by Theorem 5.1 we have that that there is
no solution of (6.7) that satisfies the assumption (4.1). Indeed, if we assume
that $u$ is an entire solution of (6.7) that satisfies (4.1), then by Theorem
5.1 we have that $u$ is constant in $\mathbb{R}^{n}$, which contradicts the
fact that it’s Hessian has positive determinant. We can also see this as
follows, if $|\nabla u|$ is bounded in $\mathbb{R}^{n}$, then $\nabla u$ can
not be a global diffeomorphism and thus $det(\nabla^{2}u)$ can not be strictly
positive in $\mathbb{R}^{n}$. (3) Note that Theorem 2.2 holds for equation
(6.7) without assuming that $\\\ \inf_{\overline{\Omega}}|\nabla u|>0$. The
conclusion says that $u$ will be a solution of the Eikonal equation $\\\
|\nabla u|^{2}=c_{0}$. If in addition $u_{x_{n}}>0$ and
$F_{i}=\frac{u_{x_{i}}}{u_{x_{n}}}$, by Proposition 2.1 in [10],the function
$F=(F_{1},...,F_{n-1})$ will satisfy the Isobaric Euler equation.

##  7\. Higher order nonlinear equations

In this last section, we will provide examples of $P-$functions for higher
order nonlinear equations and their applications. In particular, an analogous
version of Theorems 4.1 and 5.1, allow us to obtain properties and pointwise
estimates of entire solutions even in this case. Moreover, we establish a
method of extracting pointwise estimates for nonlinear equations of order
greater than two, through the mean value properties of the $P-$functions or
utilizing an analogous bound to that of Theorem 4.1 for higher order
equations. This method can be applied to many other classes of higher order
nonlinear equations.

We begin by stating the analogous Theorem 4.1 for equations of general order.
Assumption

(7.1) $\begin{gathered}u\in C^{m}(\mathbb{R}^{n})\cap
L^{\infty}(\mathbb{R}^{n})\;\;,\;\nabla^{m-1}u\in
C^{\alpha}(\mathbb{R}^{n})\;\>\textrm{for some}\;\>\alpha\in(0,1)\\\
\textrm{and there exists}\;\>C>0\;\>\textrm{such that}\;\>|\nabla^{l}u|\leq
C\;\>,\;l=1,...,m-1.\end{gathered}$

######  Theorem 7.1.

Let $u$ be an entire solution or subsolution of

(7.2) $F(u,\nabla u,...,\nabla^{m}u)=0$

that satisfies assumption (7.1) and let $P=P(u,...,\nabla^{m-1}u)=P(u;x)$ be a
$P-$function of (7.2) such that one of the following holds: (i)
$\mu=\mu(g(\nabla^{k}u))$ for some
$g:\mathbb{R}^{n^{k}}\rightarrow\mathbb{R}\;,\>g(z)>0\;,\forall\>z\neq
0\;,\>g((0,...,0))=0\;,\\\ \mu(t)>0\;,\>\forall\>t>0$ and $P(u;x)\leq 0\;,$
when $\nabla^{k}u=(0,...,0)\;,\;k\in\\{1,...,m-1\\}\>,\\\ $ (ii)
$\mu=\mu(g(\nabla^{k}u))$ for some
$g:\mathbb{R}^{n^{k}}\rightarrow\mathbb{R}\;,\>g(z)>0\;,\forall\>z\neq
0\;,\>g((0,...,0))=0\;,\\\ \mu(t)>0\;,\>\forall\>t>0\;,\>P(u;x)\leq 0\;,$ when
$\nabla^{l}u=(0,...,0)\;,\>k\neq l\>,\;k,l\in\\{1,...,m-1\\}$ and
$g(\nabla^{k}u)>0\;,\>\forall\;x\in\mathbb{R}^{n}.$

Then $P(u,...,\nabla^{m-1}u)\leq 0\;\>\forall\>x\in\mathbb{R}^{n}.$

###### Proof.

The proof is similar to that of Theorem 4.1 with minor modifications. ∎

Note that Theorem 4.1 is a special case of Theorem 7.1 assuming (i),
$m=2\;,\>k=1$ and $g(z)=|z|$. The additional assumption
$g(\nabla^{k}u)>0\;,\>\forall\;x\in\mathbb{R}^{n}$ in (ii) is necessary in
order to utilize Theorem 2.2.

We now provide the analogous of Theorem 5.1 in the higher order case.

######  Theorem 7.2.

Let $u$ be an entire solution of

(7.3) $F(u,\nabla u,...,\nabla^{m}u)=0$

and let $P=P(u,...,\nabla^{m-1}u)=P(u;x)$ be a $P-$function of (7.2) such that
$\mu=\mu(g(\nabla^{k}u))$ for some
$g:\mathbb{R}^{n^{k}}\rightarrow\mathbb{R}\;,\>g(z)>0\;,\forall\>z\neq
0\;,\>g((0,...,0))=0\;,\>\mu(t)>0\;,\>\forall\>t>0$ and

(7.4)
$\begin{gathered}P=H(\nabla^{k}u)\;\;,\;\textrm{where}\;\>H\>:\mathbb{R}^{n^{k}}\rightarrow[0,+\infty)\\\
\textrm{and}\;\>\\{H=0\\}=\\{0\in\mathbb{R}^{n^{k}}\\}\;\>,\>k\in\\{1,...,m-1\\}\end{gathered}$

Then $\nabla^{k-1}u$ is a constant.

###### Proof.

The proof is direct consequence of Theorem 7.1 since $P=H(\nabla^{k}u)=0$ when
$\nabla^{k}u$ vanish which gives $P\equiv 0$ in $\mathbb{R}^{n}$. ∎

Furthermore, we give some examples of $P-$functions of the form $P=P(u,|\nabla
u|,\Delta u)$ related to forth order nonlinear equations.

######  Proposition 7.3.

Let $u$ be a smooth solution of

(7.5) $\begin{gathered}a(\Delta u)[|\nabla u|^{2}\Delta^{2}u-\Delta u(\nabla
u\cdot\nabla\Delta u)]=b(u)|\nabla u|^{4}\\\
\textrm{where}\;\>a,b\>:\mathbb{R}\rightarrow\mathbb{R}\;\>\textrm{and}\;\>a>0\;,\;a^{\prime}\geq
0\end{gathered}$

and set $P(s,t)=A(t)-B(s)$ such that $A^{\prime}=a$ and $B^{\prime\prime}=b$.

Then $P=P(u,\Delta u)=A(\Delta u)-B(u)$ is a $P-$function of (7.5).

In addition, if $u$ satisfies (7.1) with $m=4\;,\>B(u)\geq 0$ and
$u_{x_{n}}>0$, then

(7.6) $\Delta
u\leq\Gamma(u)\;\;\>\forall\>x\in\mathbb{R}^{n},\;\>\textrm{where}\;\>\Gamma(u)=A^{-1}(B(u)).$

###### Proof.

We have

(7.7) $P_{x_{i}}=P_{s}u_{x_{i}}+P_{t}\Delta u_{x_{i}}$

and so,

(7.8) $\begin{gathered}\Delta u(\nabla P\cdot\nabla u)=P_{s}|\nabla
u|^{2}\Delta u+P_{t}\Delta u\sum_{i=1}^{n}u_{x_{i}}\Delta u_{x_{i}}\\\
\Leftrightarrow-B^{\prime}(u)|\nabla u|^{2}\Delta u=\Delta u(\nabla
P\cdot\nabla u)-A^{\prime}(\Delta u)\Delta u\sum_{i=1}^{n}u_{x_{i}}\Delta
u_{x_{i}}\end{gathered}$

on the other hand we have

(7.9)
$\begin{gathered}P_{x_{i}x_{i}}=P_{ss}u_{x_{i}}^{2}+2P_{st}u_{x_{i}}\Delta
u_{x_{i}}+P_{tt}(\Delta u_{x_{i}})^{2}+P_{s}u_{x_{i}x_{i}}+P_{t}\Delta
u_{x_{i}x_{i}}\\\ \Rightarrow\Delta P=(-B^{\prime\prime}(u))|\nabla
u|^{2}+A^{\prime\prime}(\Delta u)\sum_{i=1}^{n}(\Delta
u_{x_{i}})^{2}-B^{\prime}(u)\Delta u+A^{\prime}(\Delta
u)\Delta^{2}u\end{gathered}$

and by (7.8) and the assumptions of $A$ and $B$, (7.9) becomes

(7.10) $|\nabla u|^{2}\Delta P-\Delta u(\nabla P\cdot\nabla u)\geq a(\Delta
u)[|\nabla u|^{2}\Delta^{2}u-\Delta u(\nabla u\cdot\nabla\Delta
u)]-b(u)|\nabla u|^{4}=0$

For the bound of the Laplacian, we have $P(u,0)=-B(u)\leq 0$ and $\mu=|\nabla
u|^{2}>0\;\>\forall\>x\in\mathbb{R}^{n}$ since $u_{x_{n}}>0$, so the
assumption (i) in Theorem 7.1 is satisfied and we conclude. ∎

######  Proposition 7.4.

Let $u$ be a smooth solution of

(7.11) $\begin{gathered}|Hes\>u|^{2}=F(u,|\nabla u|^{2},\Delta
u)+\frac{u}{2}\Delta^{2}u\\\
\textrm{where}\;\>F:\mathbb{R}^{3}\rightarrow\mathbb{R}\;\textrm{is such
that}\;\>F(s,t,w)\geq\frac{1}{2}w^{2}.\end{gathered}$

Then $P=P(u,|\nabla u|^{2},\Delta u)=|\nabla u|^{2}-u\Delta u$ is a
$P-$function of (7.13).

In addition, if $u$ is non negative, convex solution of (7.13) that satisfies
assumption (7.1), then

(7.12) $|\nabla u|^{2}\leq u\Delta u\;\;\;,\;\forall\;x\in\mathbb{R}^{n}.$

###### Proof.

We have that

$\displaystyle
P_{x_{i}}=2\sum_{j=1}^{n}u_{x_{j}}u_{x_{j}x_{i}}-u_{x_{i}}\Delta u-u\Delta
u_{x_{i}}$

and

$\displaystyle\Delta P=2|Hes\>u|^{2}+2\nabla u\nabla\Delta u-(\Delta
u)^{2}-2\nabla u\nabla\Delta u-u\Delta^{2}u$

so by (7.13),

$\displaystyle\Delta P=2F(u,|\nabla u|^{2},\Delta u)-(\Delta u)^{2}\geq 0$

For the gradient bound we see that $P(u,0,\Delta u)=-u\Delta u\leq 0$ since
$u$ is non negative and convex, so the assumption (i) of Theorem 7.1 is
satisfied and we conclude. ∎

As a result, we have the following pointwise estimate

######  Corollary 7.5.

Let $u:B_{2}\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth solution
of

(7.13) $\begin{gathered}|Hes\>u|^{2}=F(u,|\nabla u|^{2},\Delta
u)+\frac{u}{2}\Delta^{2}u\\\
\textrm{where}\;\>F:\mathbb{R}^{3}\rightarrow\mathbb{R}\;\textrm{is such
that}\;\>F(s,t,w)\geq\frac{1}{2}w^{2}.\end{gathered}$

Then

(7.14) $\begin{gathered}|\nabla u(x)|^{2}-u(x)\Delta u(x)\leq
C(||u||_{H^{1}(B_{2})}+||\Delta u||_{L^{2}(B_{2})})\;,\\\ \forall\>x\in
B_{1}=\\{y\in\mathbb{R}^{n}\>:\>|y|<1\\}\;,\>\textrm{and}\;\>C\;\>\textrm{depends
only on}\;\>n.\end{gathered}$

###### Proof.

By Proposition 7.4, we have that $P=|\nabla u|^{2}-u\Delta u=P(u;x)$ is
subharmonic. Therefore we have

(7.15)
$P(u;x)\leq\frac{1}{|B(x,r)|}\int_{B(x,r)}P(u;y)dy\;\;,\;\forall\;B(x,r)\subset
B_{2}$

Also, $P\leq|\nabla u|^{2}+\frac{1}{2}(u^{2}+(\Delta u)^{2})$.

So,

(7.16) $\int_{B(x,r)}P(u;y)dy\leq||u||_{H^{1}(B_{2})}+||\Delta
u||_{L^{2}(B_{2})}\;,\>\forall\;B(x,r)\subset B_{2}$

Thus, for any $x\in B_{1}$ (since $B(x,1)\subset B_{2}$), we have

(7.17) $P(u;x)\leq\frac{1}{|B_{1}|}(||u||_{H^{1}(B_{2})}+||\Delta
u||_{L^{2}(B_{2})})$

∎

######  Remark 7.6.

Note that if $F(u,|\nabla u|^{2},\Delta u)=\frac{1}{2}(\Delta u)^{2}$, we have
a reduction of order result, that is, if $u$ is a smooth and bounded entire
solution of

(7.18) $2|Hes\>u|^{2}=(\Delta u)^{2}+u\Delta^{2}u$

such that $\nabla u,\Delta u\in L^{\infty}(\mathbb{R}^{n})$, then $u$
satisfies $u\Delta u=|\nabla u|^{2}+c$ for some $c\in\mathbb{R}$. We can see
this from the proof of Proposition 7.4, where $P=|\nabla u|^{2}-u\Delta u$
will be harmonic for this particular equation. Also, $|P|\leq M$ for some
$M=M(||u||_{L^{\infty}(\mathbb{R}^{n})},||\nabla
u||_{L^{\infty}(\mathbb{R}^{n})},||\Delta u||_{L^{\infty}(\mathbb{R}^{n})})>0$
and thus $P\equiv$constant.

We also provide a consequence of Theorem 7.2

######  Corollary 7.7.

Let $u$ be a convex subsolution of

(7.19) $c|Hes\>u|^{2}-\Delta^{2}u=0\;\;,\;\textrm{with}\;\>c\geq 0$

that satisfies assumption (7.1) with $m=4$.

Then $u$ is constant.

###### Proof.

Consider $P=P(u,\nabla u,\Delta u)=(\Delta u)^{2}$, so

(7.20) $\begin{gathered}P_{x_{i}}=2\Delta u\Delta u_{x_{i}}\\\
P_{x_{i}x_{i}}=2(\Delta u_{x_{i}})^{2}+2\Delta u\Delta u_{x_{i}x_{i}}\\\
\Rightarrow\Delta P=2|\nabla\Delta u|^{2}+2\Delta u\Delta^{2}u\geq
2|\nabla\Delta u|^{2}+2c|Hes\>u|^{2}\Delta u\geq 0\end{gathered}$

and $P(u,\nabla u,0)=0$ with $\mu=1$, so by Theorem 7.2 we obtain $\Delta
u\equiv 0$ in $\mathbb{R}^{n}$ and $u$ is bounded by (7.1), so $u$ is
constant. ∎

Finally, we have the following De Giorgi-type property

######  Proposition 7.8.

Let $u:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth and bounded solution
of

(7.21) $F(u,\nabla u,\nabla^{2}u,\nabla^{3}u,\nabla^{4}u)=0$

such that $u_{y}>0$ and assume $P=P(u,\Delta u)$ is a $P-$function of (7.21),
such that $P_{t}>0\\\ (P=P(s,t))$ with $\mu=\mu(|\nabla
u|)\;,\>\mu(t)>0\;,\>\forall\>t>0$.

If there exists $x_{0}\in\mathbb{R}^{2}$ such that

(7.22) $P(u(x_{0}),\Delta u(x_{0}))=\sup_{\mathbb{R}^{n}}P(u,\Delta
u)<+\infty$

then there exists a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that

(7.23) $u(x)=g(ax+by)\;\;\;,\;\>\textrm{for}\;\>a,b\in\mathbb{R}$

###### Proof.

Arguing as in the proof of Theorem 2.4 we obtain that

(7.24) $P(u,\Delta u)\equiv
c_{0}\;\;\;,\;\;\textrm{where}\;\>c_{0}=\sup_{\mathbb{R}^{n}}P(u,\Delta u)$

since $P_{t}>0$ we have

(7.25) $\Delta u=f(u)\;\;\;,\;\;\textrm{for
some}\;\>f:\mathbb{R}\rightarrow\mathbb{R}$

and $u$ is bounded entire solution of (7.25) such that $u_{y}>0$.

Therefore, by Theorem 1.1 in [12], we conclude that

(7.26) $u(x)=g(ax+by)\;\;\;,\;\>\textrm{for
some}\;\>g:\mathbb{R}\rightarrow\mathbb{R}$

∎

Acknowledgments: I would like to thank my advisor professors N. Alikakos and
C. Makridakis for their support. Also, I would like to thank professor N.
Alikakos for his useful suggestions that improved this work.

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