\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 137, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/137\hfil Minimization of energy integrals]
{Minimization of energy integrals associated with the $p$-Laplacian
in $\mathbb{R}^N$ for rearrangements}

\author[S. Ji, J. Yin, R. Huang \hfil EJDE-2014/137\hfilneg]
{Shanming Ji, Jingxue Yin, Rui Huang}  % in alphabetical order

\address{Shanming Ji \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou 510631, China}
\email{sam@m.scnu.edu.cn}

\address{Jingxue Yin \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou 510631, China}
\email{yjx@scnu.edu.cn}

\address{Rui Huang \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou 510631. \newline
Department of Mathematics, South China University of Technology,
Guangzhou 510640, China}
\email{huang@scnu.edu.cn}

\thanks{Submitted April 11, 2014. Published June 11, 2014.}
\subjclass[2000]{35A01 35J15 35Q99}
\keywords{Optimization problem; rearrangements; energy integral; penalty;
\hfill\break\indent  $p$-Laplacian}

\begin{abstract}
 In this article, we establish the existence of minimizers for energy
 integrals associated with the $p$-Laplacian in $\mathbb{R}^N$
 with the admissible set being a rearrangement class of a given function.
 Some representation formulae of the minimizers are also stated.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\def\pd#1#2{\frac{\partial#1}{\partial#2}}


\section{Introduction}

In this article, we study the optimization problems of minimizing the energy
integrals associated with the  $p$-Laplacian equation
\begin{gather}\label{eq1-1}
-\Delta_p u=f-h,\quad x\in\mathbb{R}^N,\\ \label{eq1-2}
\lim_{|x|\to+\infty}u(x)=0.
\end{gather}
Here $1<p<N$, $\Delta_p$ stands for the usual $p$-Laplacian; i.e.,
 $\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$.
Let $f_0, h\in L^\infty(\mathbb{R}^N)$ be fixed nonnegative functions with
compact supports, and let $\mathcal{R}$ be the class of rearrangements of
$f_0$ with compact support; that is,
$\mathcal{R}=\{f\in L^\infty(\mathbb{R}^N); |\{x;f(x)\ge\alpha\}|
=|\{x;f_0(x)\ge\alpha\}|, \forall \alpha\in\mathbb{R},
\operatorname{supp}f\text{ is  bounded}\}$, where
$|\cdot|$ is the Lebesgue measure.
For $\lambda\ge0$ and $f$ varying in $\mathcal{R}$, we define the energy
functional with ($\lambda>0$) or without ($\lambda=0$)
penalty as
\begin{equation}\label{eq1-3}
\Psi_\lambda(f)=\int_{\mathbb{R}^N}|\nabla u_f|^p \,dx+\lambda\int_{\mathbb{R}^N}gf
\,dx,
\end{equation}
where $u_f$ is the solution of problem \eqref{eq1-1}--\eqref{eq1-2},
$g\in C^2(\mathbb{R}^N)$ is the penalty function,
$\lim_{|x|\to+\infty}g(x)=+\infty$ and
$\Delta_p g\ge\sigma^{p-1}$ for some constant $\sigma>0$.
The optimization problem \eqref{P-lambda} is to find the minimizer for
energy integral $\Psi_\lambda(f)$, namely,
\begin{equation} \label{P-lambda}  %\tag{$\mathrm P_\lambda$}
\min_{f\in\mathcal{R}}\Psi_\lambda(f).
\end{equation}


The optimization problems of maximizing or minimizing convex functionals over
the set of rearrangements of a given function
have been investigated by many authors.
In such problems, the theory of rearrangements and functionals on rearrangements
established by Burton \cite{B1, B2} has proved to be a crucial tool
in addressing questions such as existence, uniqueness and symmetry of optimal
solutions. This theory has already been applied to shape optimization problems
of membranes, solid and fluid mechanics,
eigenvalue optimization problems of some differential operators and so on,
see \cite{E3} and references therein.

In recent years, a great deal of attention has been devoted to optimization problems
where the cost functionals are the energy integrals associated with elliptic
equations.
For problems in bounded domains, numerous elliptic operators have been studied,
including the Laplacian \cite{B1, C2, E1},
$p$-Laplacian \cite{C1, Ma} and some semilinear operators \cite{E2}.
For example, Marras \cite{Ma} studied the minimization problem of energy
integral $\Psi_0(f)$ associated with
the $p$-Laplacian on bounded domain
\begin{gather*}
-\Delta_pu=f, \quad x\in\Omega, \\
u=0, \quad x\in\partial\Omega,
\end{gather*}
where $p>1$, $f\in \mathcal{R}$.
There are also some works dealing with elliptic operators in unbounded domains.
Bahrami and Fazli \cite{Ba} considered the minimization problem of energy integral
$$
\Phi_\lambda(f)=\frac{1}{2}\int_{\mathbb{R}^3}fu_f\,dx
+\lambda\int_{\mathbb{R}^3}gf\,dx,
$$
where $u_f$ is the solution of Poisson's equation
\begin{equation} \label{eq1-4}
\begin{gathered}
-\Delta u=f-2h, \quad x\in \mathbb{R}^3, \\
\lim_{|x|\to+\infty}u(x)=0,
\end{gathered}
\end{equation}
where $f\in\mathcal{R}$, $h\in L^\infty(\mathbb{R}^3)$, $g\in C^2(\mathbb{R}^3)$,
$\lim_{|x|\to+\infty}g(x)=+\infty$, $\Delta g\ge c>0$ and $\lambda\ge0$.

We mention here some details of the previous works.
The weakly sequentially continuity of the functional $\Psi_\lambda(f)$ on space
$L^q(\Omega)$ for $q\ge1$ and bounded domain $\Omega$,
is essential in the proof of \cite{Ma} and other works when applying Burton's
theory of rearrangements.
However, the continuity is generally not true on unbounded domains due to the
general loss of compact imbedding of Sobolev spaces
on unbounded domains, especially on the whole space.
Thus the authors in \cite{Ba} investigate the problem on bounded domains
 to solve the optimization problem on unbounded domains.

We are interested in the extension of the work of Bahrami and Fazli \cite{Ba}
to the nonlinear diffusion case.
As a matter of fact, the $p$-Laplacian arises in various physical contexts:
non-Newtonian fluids, reaction diffusion problems,
nonlinear elasticity, electrochemical machining, elastic-plastic torsional
creep, etc., see \cite{C1} and references therein.

We state here our main results of existence and representation formulae
of minimizers for problem \eqref{P-lambda} in the case
$\lambda>0$ and $\lambda=0$ respectively.

\begin{theorem}\label{th1}
The optimization problem \eqref{P-lambda} has a solution for
\[
\lambda>\lambda_0\equiv\frac{p'}{\sigma}\|f_0\|_\infty^{\frac{1}{p-1}}.
\]
Moreover, if $f_\lambda\in\mathcal{R}$ is a solution of \eqref{P-lambda}
and $u_{f_\lambda}$ is the solution of
problem \eqref{eq1-1}--\eqref{eq1-2} corresponding to $f_\lambda$,
then there exists a decreasing function $\varphi_\lambda$ such that
$$
f_\lambda=\varphi_\lambda\circ(p'u_{f_\lambda}+\lambda g),
$$
almost everywhere in $\mathbb{R}^N$.
\end{theorem}

\begin{theorem}\label{th2}
Let $f_0$ and $h$ be as introduced above.
There exists a constant $\kappa=\kappa(N,p)\in(0,\frac{1}{2}]$ depending only
on $N$ and $p$, such that
if $\|f_0\|_{\infty}<\|h\|_{-\infty;\operatorname{supp} h}$,
$\operatorname{supp} h\subset B_{r_h}$ with $r_h>0$
appropriately large, and
\begin{equation}
|\operatorname{supp} f_0|\le\kappa
\Big(\frac{\|h\|_{-\infty;\operatorname{supp} h}}{\|h\|_{\infty}}
\Big)^{\frac{p}{p-1}}
\Big(\frac{|\operatorname{supp} h|}{|B_{r_h}|}\Big)^\frac{N-p}{N(p-1)}
|\operatorname{supp} h|,
\end{equation}
then the optimization problem \eqref{P-lambda} with $\lambda=0$ has a solution.
Moreover, if $\hat f\in\mathcal{R}$ is a solution of 
\eqref{P-lambda} with $\lambda=0$  and $u_{\hat f}$
is the solution of problem \eqref{eq1-1}--\eqref{eq1-2} corresponding to $\hat f$,
then there exists a decreasing function $\hat\varphi$ such that
$$
\hat f=\hat\varphi\circ u_{\hat f},
$$
almost everywhere in $\mathbb{R}^N$.
\end{theorem}

The crucial point of the proofs, compared with the linear diffusion case,
lies in the estimates on the different contributions
of the two opposed-signed functions $f$ and $-h$ to the solution.
In the previous work \cite{Ba}, the classical theory of linear elliptic equations
was applied, namely,
the explicit expression of solutions based on the superposition principle is
feasible and effective in deriving the estimates
on solutions of linear elliptic equations.
However, such a method is inapplicable in the current problem due to the
nonlinearity of the $p$-Laplacian.
It turns out to be more difficult as we attempt to estimate the different
contributions of the two opposed-signed functions.
To overcome those difficulties, we use the De Giorgi and Moser iteration
techniques to derive estimates in quasilinear case
and we take advantage of the representation formulas of the problem on bounded
domains since they provide additional correlation between the solution and
the free term.

The organization of this paper is as follows.
Section 2 is devoted to the basic notations and some preliminary results,
especially some fundamental estimates.
Then we will discuss the case with ($\lambda>0$) and without ($\lambda=0$)
penalty in Section 3 and Section 4 respectively.


\section{Definitions and preliminary results}
Throughout this paper, we assume that $1<p<N$, where $N$ is the spatial dimension,
$p'=\frac{p}{p-1}$ the conjugate exponent of $p$, $p_*=\frac{Np}{N-p}$
the Sobolev conjugate exponent of $p$ and $p_*'=\frac{p_*}{p_*-1}$.
The measure of a Lebesgue measurable set $A\subset\mathbb{R}^N$ is denoted by $|A|$.
By $B_r(x)$ we denote the ball centered at $x\in\mathbb{R}^N$ with radius $r$;
if the center is the origin, we write $B_r$ for simplicity.
Constant $\omega_N$ stands for the measure of the unit ball in $\mathbb{R}^N$.

Let us begin with the usual concept of rearrangement.
Denote by $L_\alpha(f)$ the level set of a measurable function $f$ at height
$\alpha$; that is $L_\alpha(f)=\{x\in\mathbb{R}^N;f(x)=\alpha\}$.
The strong support of a nonnegative function $f$ is defined as
$\operatorname{supp} f=\{x\in\mathbb{R}^N;f(x)>0\}$.
Furthermore, we define
$$
\|f\|_{-\infty;\operatorname{supp} f}
=\sup\{M\ge0;f(x)\ge M, {\rm ~a.e.~in~supp} f\}.
$$
When $f$ and $g$ are nonnegative measurable functions that vanish outside
sets of finite measure in $\mathbb{R}^N$,
we say $f$ is a rearrangement of $g$ whenever
$$
|\{x\in\mathbb{R}^N;f(x)\ge\alpha\}|=|\{x\in\mathbb{R}^N;g(x)\ge\alpha\}|,
\quad\forall\alpha\ge0.
$$

Now fix $f_0\in L^{\infty}(\mathbb{R}^N)$ being a measurable nonnegative
function vanishing outside a set of finite measure.
As defined in Section 1, $\mathcal{R}$ denotes the set of all rearrangements
on $\mathbb{R}^N$ of $f_0$ with bounded support.
The subset of $\mathcal{R}$ containing functions vanishing outside the ball
$B_r$ is denoted by $\mathcal{R}(r)$;
here we assume $\omega_N r^N\ge|\operatorname{supp} f_0|$ in order that
$\mathcal{R}(r)$ is nonempty.
The weak closure in $L^{p_*'}(B_r)$ of $\mathcal{R}(r)$ is denoted by
$\overline{\mathcal{R}(r)^w}$.

Henceforth we may regard a function $f\in L^q(\mathbb{R}^N)$ as a function in
$L^q(B_r)$ by restricting its domain;
we can also regard a function $f\in L^q(B_r)$ as its zero extension in
$L^q(\mathbb{R}^N)$ when necessary for $1\le q\le+\infty$.

To solve the optimization problems \eqref{P-lambda}, we first need to consider the similar problems
whose admissible sets are nested subsets of $\mathcal{R}$.
We define minimizing problems \eqref{P-lambda-r} as follows:
\begin{equation} \label{P-lambda-r} %\tag{$\mathrm P_\lambda(r)$}
\min_{f\in\mathcal{R}(r)}\Psi_\lambda(f).
\end{equation}
The sets of solutions of \eqref{P-lambda} and \eqref{P-lambda-r} are
denoted by $S_\lambda$ and $S_\lambda(r)$ respectively.

In the following part of this section we state and prove some lemmas
which are essential in our analysis.
We begin with some results about properties of rearrangement classes
and the relative variational problems proved by Burton in \cite{B2}.

\begin{lemma}\label{le2-1}
For $r>0$ satisfying $\omega_N r^N\ge|\operatorname{supp} f_0|$ and $q\ge1$, we have
\begin{itemize}
\item[(i)] $\|f\|_q=\|f_0\|_q$, for $f\in\mathcal{R}(r)$;
\item[(ii)] $\overline{\mathcal{R}(r)^w}$ is weakly sequentially compact
in $L^q(B_r)$;
\item[(iii)] $\overline{\mathcal{R}(r)^w}
=\{f\in L^1(B_r);\int_0^s f^\triangle dt\le\int_0^s f_0^\triangle dt,
0<s\le\omega_Nr^N, \int_{B_r}fdx=\int_{B_r}f_0dx\}$,
where $f^\triangle$ is a decreasing function on the interval $(0,\omega_Nr^N)$
satisfying
$$
|\{s\in(0,\omega_Nr^N);f^\triangle(s)\ge\alpha\}|
=|\{x\in B_r;f(x)\ge\alpha\}|,\quad\forall\alpha>0.
$$
\end{itemize}
\end{lemma}

{\noindent\bf Remark} From the representation of $\overline{\mathcal{R}(r)^w}$
in (iii), we find that the weak closure
of $\mathcal{R}(r)$ in $L^{p_*'}(B_r)$ is actually the weak closure in $L^q(B_r)$
for $1\le q<+\infty$.
Combining (i) and the property of weak closure we have
\begin{equation}\label{2-1}
\|f\|_q\le\|f_0\|_q, \quad \forall f\in\overline{\mathcal{R}(r)^w},\;
 1\le q\le+\infty.
\end{equation}

The following two lemmas are simple variations of
\cite[Lemma 2.15 and Theorem 3.3]{B2}.

\begin{lemma}[{\cite[Lemma 2.15]{B2}}] \label{le2-2}
Let $T:L^{p'}(B_r)\to\mathbb{R}$ be the linear functional defined as
 $T(f)=\int_{B_r}fv dx$ for $r>0$,
$\omega_N r^N\ge|\operatorname{supp} f_0|$ and $v\in L^{p}(B_r)$.
If $\hat f$ is a minimizer of $T$ relative to $\overline{\mathcal{R}(r)^w}$ and
$$
|L_\alpha(v)\cap \operatorname{supp}\hat f|=0, \quad\forall\alpha\in\mathbb{R},
$$
we have $\hat f\in\mathcal{R}(r)$ and
$\hat f=\varphi\circ v$ a.e. i $B_r$,
for a decreasing function $\varphi$.
\end{lemma}

\begin{lemma}[{\cite[Theorem 3.3]{B2}}] \label{le2-3}
Let $\Psi:L^{p'}(B_r)\to\mathbb{R}$ be a weakly sequentially continuous
and G\^ateaux differentiable functional.
\begin{itemize}
\item[(i)] There exists a minimizer for $\Psi$ relative to
$\overline{\mathcal{R}(r)^w}$.
\item[(ii)] If $f^*$ is a minimizer for $\Psi$ relative to
$\overline{\mathcal{R}(r)^w}$ and the G\^ateaux differential of
$\Psi$ at $f^*$ is
${\Psi}'(f^*)\in L^{p}(B_r)$, $f^*$ is a minimizer for the functional
$\langle\cdot,{\Psi}'(f^*)\rangle$ relative to $\overline{\mathcal{R}(r)^w}$.
\end{itemize}
\end{lemma}

The following Sobolev's inequality plays an important role in our estimates.
For more details, see \cite{Gi}.

\begin{lemma}[{\cite[Theorem 7.10]{Gi}}] \label{le2-4}
For $1<p<N$ and $p_*=\frac{Np}{N-p}$, we have
\begin{equation}\label{2-2}
\|u\|_{p_*}\le C_0\|\nabla u\|_p, \quad \forall u\in W^{1,p}(\mathbb{R}^N),
\end{equation}
where $C_0=C_0(N,p)$ is a constant depending only on $N$ and $p$.
\end{lemma}

{\noindent\bf Remark}
Invoking usual approximations, we see that this estimate is also valid provided
$u\in L^{p_*}(\mathbb{R}^N)$,
$\nabla u\in L^p(\mathbb{R}^N;\mathbb{R}^N)$.

Henceforth, we assume $r>0$, $\omega_N r^N\ge|\operatorname{supp} f_0|$ and
$\lambda\ge0$. For $f\in L^{p_*'}(\mathbb{R}^N)$, we consider the problem
 \eqref{eq1-1}--\eqref{eq1-2}.
It is a classical result of variational theory that such a problem has a unique
solution
$u\in W\equiv\{w\in W_{\rm loc}^{1,p}(\mathbb{R}^N);
w\in L^{p_*}(\mathbb{R}^N),\nabla w\in L^p(\mathbb{R}^N;\mathbb{R}^N)\}$
satisfying
\begin{gather}\label{2-3}
\begin{aligned}
\sup_{v\in W}\int_{\mathbb{R}^N}\big(p(f-h)v-|\nabla v|^p\big)dx
&=\int_{\mathbb{R}^N}\big(p(f-h)u-|\nabla u|^p\big)dx\\
&=(p-1)\int_{\mathbb{R}^N}|\nabla u|^pdx,
\end{aligned} \\ \label{2-4}
\int_{\mathbb{R}^N}(f-h)v dx=\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla
u\cdot\nabla v dx, \quad \forall v\in W.
\end{gather}

\begin{lemma}\label{le2-5}
Let $u$ be the solution of problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f\in L^{p_*'}(\mathbb{R}^N)$.
We have
\begin{gather}\label{2-5}
\|\nabla u\|_p \le C_0^{\frac{1}{p-1}}(\|f_0\|_{p_*'}
 +\|h\|_{p_*'})^{\frac{1}{p-1}},\\ \label{2-6}
\|u\|_{p_*} \le C_0^{\frac{p}{p-1}}(\|f_0\|_{p_*'}+\|h\|_{p_*'})^{\frac{1}{p-1}},
\end{gather}
where $C_0$ is the constant in \eqref{2-2}.
\end{lemma}

\begin{proof}
 From \eqref{2-4}, we apply H\"{o}lder's inequality to find
$$\int_{\mathbb{R}^N}|\nabla u|^p dx
=\int_{\mathbb{R}^N}(f-h)u dx\le\|f-h\|_{p_*'}\|u\|_{p_*}.$$
Combining this inequality with Sobolev's inequality \eqref{2-2},
 we get the results.
\end{proof}

\begin{lemma}\label{le2-6}
The functional $\Psi_\lambda$ defined in \eqref{eq1-3} is weakly
sequentially continuous and G\^ateaux differentiable in $L^{p'}(B_r)$
with derivative $p'u_f+\lambda g\in L^{p}(B_r)$ at $f\in L^{p'}(B_r)$,
where $u_f$ is the solution of problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f$.
\end{lemma}

\begin{proof}
It suffices to prove that the functional
$I(f)\equiv\int_{\mathbb{R}^N}|\nabla u_f|^p dx$ is weakly sequentially
continuous and G\^ateaux differentiable in $L^{p'}(B_r)$ with derivative
$p'u_f\in L^{p}(B_r)$ at $f\in L^{p'}(B_r)$.
Let $f_n\rightharpoonup f$ in $L^{p'}(B_r)$ and $u_{f_n}$, $u_{f}$ be
the solutions of the problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f_n$, $f$ respectively.
Using \eqref{2-3}, we have
\begin{equation}
\begin{aligned}
&(p-1)I(f)+\int_{\mathbb{R}^N}p(f_n-f)u_fdx\\
&=\int_{\mathbb{R}^N}\big(p(f_n-h)u_f-|\nabla u_f|^p\big)dx\le(p-1)I(f_n)\\
&=\int_{\mathbb{R}^N}\big(p(f-h)u_{f_n}-|\nabla u_{f_n}|^p\big)dx
 +\int_{\mathbb{R}^N}p(f_n-f)u_{f_n}dx\\
&\le(p-1)I(f)+\int_{\mathbb{R}^N}p(f_n-f)u_{f_n}dx.
\end{aligned} \label{2-7}
\end{equation}
By assumption, we have
\begin{equation}\label{2-8}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(f_n-f)u_fdx
=\lim_{n\to\infty}\int_{B_r}(f_n-f)u_fdx=0.
\end{equation}
Let us prove that
\begin{equation}\label{2-9}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(f_n-f)u_{f_n}dx
=\lim_{n\to\infty}\int_{B_r}(f_n-f)u_{f_n}dx=0.
\end{equation}
From \eqref{2-5}, \eqref{2-6} and
$\|f\|_{p_*';\mathbb{R}^N}=\|f\|_{p_*';B_r}\le|B_r|^{\frac{1}{N}}\|f\|_{p';B_r}$
for $f\in L^{p'}(B_r)$, we see that the norms $\|\nabla u_{f_n}\|_{p;\mathbb{R}^N}$,
$\|u_{f_n}\|_{p_*;\mathbb{R}^N}$
and $\|u_{f_n}\|_{1,p;B_r}$ are bounded by constants independent of $n$.
Therefore, we can choose a subsequence of $\{u_{f_n}\}$ denoted by
$\{u_{f_{n_k}}\}$ and a function $w\in W$,
such that $\{u_{f_{n_k}}\}$ converges weakly in $L^{p_*}(\mathbb{R}^N)$ and
strongly in $L^p(B_r)$ to $w$,
$\{\nabla u_{f_{n_k}}\}$ converges weakly in
$L^p(\mathbb{R}^N;\mathbb{R}^N)$ to $\nabla w$.
From
$$
\int_{\mathbb{R}^N}(f_{n_k}-f)u_{f_{n_k}}dx
=\int_{\mathbb{R}^N}(f_{n_k}-f)wdx+\int_{\mathbb{R}^N}(f_{n_k}-f)(u_{f_{n_k}}-w)dx,
$$
and
$$
\big|\int_{\mathbb{R}^N}(f_{n_k}-f)(u_{f_{n_k}}-w)dx\big|
\le\|f_{n_k}-f\|_{p';B_r}\|u_{f_{n_k}}-w\|_{p;B_r},
$$
the limit \eqref{2-9} is valid for a subsequence $\{n_k\}$.
Combining this with \eqref{2-7}--\eqref{2-8}, we deduce
\begin{equation}\label{2-10}
\lim_{k\to\infty}I(f_{n_k})=I(f).
\end{equation}
We claim that the function $w$ is actually $u_f$ , which is a fixed function
independent of the choice of subsequence $\{n_k\}$,
to show that the sequence $\{u_{f_n}\}$ itself converges and equality
\eqref{2-9} is valid.
Indeed, from
\begin{gather*}
(p-1)I(f_{n_k})=\int_{\mathbb{R}^N}\big(p(f_{n_k}-h)u_{f_{n_k}}
 -|\nabla u_{f_{n_k}}|^p\big)dx,\\
\lim_{k\to\infty}\int_{\mathbb{R}^N}(f_{n_k}-h)u_{f_{n_k}}dx
=\int_{\mathbb{R}^N}(f-h)w\,dx,
\end{gather*}
and the classical result
\begin{equation}
\liminf_{k\to\infty}\int_{\mathbb{R}^N}|\nabla u_{f_{n_k}}|^pdx
\ge\int_{\mathbb{R}^N}|\nabla w|^pdx,
\end{equation}
using \eqref{2-10} and \eqref{2-3}, we get
\begin{equation}
(p-1)I(f)\le\int_{\mathbb{R}^N}\big(p(f-h)w-|\nabla w|^p\big)dx\le(p-1)I(f).
\end{equation}
By the uniqueness of the maximizer of
$\int_{\mathbb{R}^N}\big(p(f-h)v-|\nabla v|^p\big)dx$ in $W$, we have $w=u_f$.
Thus \eqref{2-7}--\eqref{2-9} yield the weak continuity.

Let $z\in L^{p'}(B_r)$ and $\{t_n\}$ be a positive sequence such that
$\lim_{n\to\infty}t_n=0$.
Taking $f_n=f+t_nz$ in the inequality \eqref{2-7},
we find
$$
\int_{\mathbb{R}^N}p'u_fzdx\le\frac{I(f+t_nz)-I(f)}{t_n}
\le\int_{\mathbb{R}^N}p'u_{f_n}z\,dx.
$$
As already observed, $\{u_{f_n}\}$ converges to $u_f$ strongly in $L^p(B_r)$.
Therefore,
$$
\lim_{n\to\infty}\frac{I(f+t_nz)-I(f)}{t_n}=\int_{\mathbb{R}^N}p'u_fz\,dx.
$$
Since the sequence $\{t_n\}$ and the function $z$ are arbitrary,
it follows that $I(f)$ is G\^ateaux differentiable
with derivative $p'u_f$.
\end{proof}

Note that the functional $\Psi_\lambda$ is not weakly
sequentially continuous in $L^{p'}(\mathbb{R}^N)$.

\begin{lemma}\label{le2-7}
Let $u$ be the solution of the problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f\in\overline{\mathcal{R}(r)^w}$.
We have
\begin{equation}\label{2-11}
\|u\|_{\infty;\mathbb{R}^N}\le C_1(N,p)
\|f-h\|_{\infty}^{\frac{N-p}{Np-N+p}}\|f-h\|_{p_*'}^{\frac{p^2}{(p-1)(Np-N+p)}},
\end{equation}
where $C_1(N,p)$ is a constant depending only on $N$ and $p$.
\end{lemma}

\begin{proof}
 For any $k>0$, take $v=(u-k)_+\in W$ in \eqref{2-4}.
We deduce
$$
\int_{\mathbb{R}^N}|\nabla v|^pdx\le\int_{\mathbb{R}^N}|f-h||v|dx.
$$
By Sobolev's inequality and H\"{o}lder's inequality, we have
$$
\|v\|_{p_*;A(k)}^p \le C_0^p\int_{A(k)} |f-h| |v|\,dx
\le C_0^p\|v\|_{p_*;A(k)} \|f-h\|_{p_*';A(k)},
$$
where $A(k)=\{x\in\mathbb{R}^N;u(x)>k\}$.
Therefore,
$$
\|v\|_{p_*;A(k)}^{p-1}\le C_0^p\|f-h\|_{p_*';A(k)}
\le C_0^p\|f-h\|_{\infty}|A(k)|^{1/{p_*'}}.
$$
Combining this with
$$
\|v\|_{p_*;A(k)}\ge\|v\|_{p_*;A(h)}\ge(h-k)|A(h)|^{1/p_*},
\quad\forall h>k>0,
$$
we have
\begin{equation}
|A(h)|\le\Big(\frac{C_0^{p'}\|f-h\|_{\infty}^{\frac{1}{p-1}}}{h-k}
\Big)^{p_*}|A(k)|^{\frac{p_*-1}{p-1}},
\quad \forall h>k>0.
\end{equation}
By iteration, we see that $|A(k_0+d)|=0$ for $k_0>0$,
\[
d=C_0^{p'}\|f-h\|_{\infty}^{\frac{1}{p-1}}
2^{\frac{(p_*-1)(N-p)}{p^2}}|A(k_0)|^{\frac{p'}{N}}.
\]
From  estimate \eqref{2-6}, we see that
$$
k_0|A(k_0)|^{\frac{1}{p_*}}\le \|u\|_{p_*}
\le C_0^{p'}\|f-h\|_{p_*'}^{\frac{1}{p-1}}.
$$
Hence
$$
u\le k_0+d\le k_0+2^NC_0^{p'+\frac{{p'}^2p_*}{N}}\|f-h\|_\infty^{\frac{1}{p-1}}
\frac{\|f-h\|_{p_*'}^{\frac{p'p_*}{N(p-1)}}}{k_0^\frac{p'p_*}{N}}.
$$
Let $\alpha=\frac{p'p_*}{N}=\frac{p^2}{(N-p)(p-1)}$,
$A=2^NC_0^{{p'}^2p_*}\|f-h\|_\infty^{\frac{1}{p-1}}
\|f-h\|_{p_*'}^{\frac{p'p_*}{N(p-1)}}$
and $k_0=(\alpha A)^{\frac{1}{\alpha+1}}$.
We get $u\le(\alpha^{\frac{1}{\alpha+1}}
+\alpha^{-\frac{\alpha}{\alpha+1}})A^\frac{1}{\alpha+1}$.
By considering $-u$ instead of $u$, we complete the proof.
\end{proof}

In Section 4, the case $\lambda=0$, more precise estimates are required
to demonstrate our result.
We begin with an estimate on the lower bound of the energy functional
$\int_{\mathbb{R}^N}|\nabla u|^p dx$.

\begin{lemma}\label{le2-8}
Let $u$ be the solution of the problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f\in\overline{\mathcal{R}(r)^w}$
and $\operatorname{supp}h\subset B_{r_h}$, $\|f_0\|_1\le\|h\|_1$.
We have
\begin{equation}\label{2-12}
\|\nabla u\|_p\ge C_2(N,p)r_h^{-\frac{N-p}{p(p-1)}}(\|h\|_1-\|f_0\|_1)^\frac{1}{p-1},
\end{equation}
where $C_2(N,p)$ is a constant depending only on $N$ and $p$.
\end{lemma}

\begin{proof}
From \eqref{2-3}, it suffices to prove that there exists $v\in W$ such that
$$
\int_{\mathbb{R}^N}\big(p(f-h)v-|\nabla v|^p\big)dx
\ge C(N,p)r_h^{-\frac{N-p}{p-1}}(\|h\|_1-\|f_0\|_1)^\frac{p}{p-1},
$$
for a constant $C(N,p)$ depending only on $N$ and $p$.
We verify that the function $v(x)\equiv-k\min\{(\frac{r_h+a-|x|}{a})_+,1\}\in W$
fulfills the conditions
for some specially selected positive constants $k$ and $a$.
Indeed, noticing the signs of $f$, $h$ and $v$, we have
\begin{align*}
\int_{\mathbb{R}^N}\big(p(f-h)v-|\nabla v|^p\big)dx
&\ge kp(\|h\|_1-\|f_0\|_1)-\omega_N\big(\frac{k}{a}\big)^p(r_h+a)^N\\
&=kp(\|h\|_1-\|f_0\|_1)-\omega_Nk^p\frac{N^Nr_h^{N-p}}{p^p(N-p)^{N-p}}\\
&=\frac{(p-1)p^{p'}(N-p)^\frac{N-p}{p-1}}{\omega_N^\frac{1}{p-1}N^\frac{N}{p-1}}
r_h^{-\frac{N-p}{p-1}}(\|h\|_1-\|f_0\|_1)^\frac{p}{p-1},
\end{align*}
for $a=\frac{p}{N-p}r_h$ and
$k^{p-1}=\frac{p^p(N-p)^{N-p}(\|h\|_1-\|f_0\|_1)}{\omega_NN^Nr_h^{N-p}}$.
\end{proof}

Next we deduce the local boundedness of solutions by the Moser iteration technique.

\begin{lemma}\label{le2-9}
Let $u$ be the solution of the problem \eqref{eq1-1}--\eqref{eq1-2}
 corresponding to nonnegative function $f\in L^{p_*'}(B_r)$,
$v=(-u)_+$ and $\operatorname{supp}h\subset B_{r_h}$.
There holds
\begin{equation}\label{2-13}
\|v\|_{\infty;B_{R/2}(x_0)}\le C_3(N,p)\Big(\frac{1}{R^N}
\int_{B_R(x_0)}|v|^{p_*}dx\Big)^{1/p_*},
\end{equation}
for any $x_0\in\mathbb{R}^N$ and  $R>0$ provided $B_R(x_0)\cap B_{r_h}=\emptyset$,
where $C_3(N,p)$ is a constant depending only on $N$ and $p$.
\end{lemma}

\begin{proof}
For $0<\rho<{\rho}'\le R$, let $\eta(x)$ be a cut-off function
$\eta\in C_0^\infty(B_{{\rho}'}(x_0))$,  satisfying $0\le\eta\le1$, $\eta(x)=1$ on
$B_\rho(x_0)$, $\eta(x)=0$ on $\mathbb{R}^N\backslash B_{{\rho}'}(x_0)$ and
$|\nabla\eta(x)|\le\frac{2}{{\rho}'-\rho}$.
We write $B_R=B_R(x_0)$ in this proof for the sake of convenience.

Choose $\eta^pv^s$ as a test function in \eqref{2-4} for $s\ge1$ and set
$q=s+p-1$.
We have
$$
\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla u\cdot\nabla(\eta^pv^s)dx
=\int_{\mathbb{R}^N}(f-h)\eta^pv^sdx
=\int_{B_R}f\eta^pv^sdx\ge0,
$$
or
$$
-\int_{B_R}\eta^p|\nabla v|^{p-2}\nabla v\cdot\nabla(v^s)dx
-\int_{B_R}v^s|\nabla v|^{p-2}\nabla v\cdot\nabla(\eta^p)dx\ge0.
$$
Therefore, using Young's inequality, we deduce
\begin{align*}
&\int_{B_R}\eta^p|\nabla(v^\frac{q}{p})|^pdx\\
&=\frac{q^p}{sp^p}\int_{B_R}\eta^p|\nabla v|^{p-2}\nabla v\cdot\nabla(v^s)dx
\le\frac{q^p}{sp^p}\int_{B_R}|\nabla v|^{p-1}|\nabla(\eta^p)|v^sdx\\
&\le\frac{q^p}{sp^{p-1}}\int_{B_R}\eta^{p-1}|\nabla\eta||\nabla v|^{p-1}v^sdx
=\frac{q}{s}\int_{B_R}\eta^{p-1}|\nabla(v^\frac{q}{p})|^{p-1}|\nabla\eta|v^\frac{q}{p}dx\\
&\le\frac{p-1}{p}\int_{B_R}\eta^p|\nabla(v^\frac{q}{p})|^pdx+\frac{q^p}{ps^p}
\int_{B_R}|\nabla\eta|^pv^qdx.
\end{align*}
Hence we obtain
$$
\int_{B_R}\eta^p|\nabla(v^\frac{q}{p})|^pdx
\le\frac{q^p}{s^p}\int_{B_R}|\nabla\eta|^pv^qdx
\le p^p\int_{B_R}|\nabla\eta|^pv^qdx,\quad\forall q\ge p.
$$
Combining this with Sobolev's inequality \eqref{2-2}, we have
$$
\Big(\int_{B_R}\eta^\frac{Np}{N-p}v^\frac{Nq}{N-p}dx\Big)^\frac{N-p}{N}
\le C_0^p\int_{B_R}|\nabla(\eta v^\frac{q}{p})|^pdx
\le(4pC_0)^p\int_{B_R}|\nabla\eta|^pv^qdx.
$$
It follows that
$$
\Big(\frac{1}{R^N}\int_{B_\rho}v^{\frac{Nq}{N-p}}dx\Big)^\frac{N-p}{N}
\le(8pC_0)^p
\Big(\frac{1}{R^{N-p}({\rho}'-\rho)^p}\int_{B_{{\rho}'}}v^qdx\Big).
$$
Denote $\rho_k=\frac{R}{2}(1+\frac{1}{2^k})$, $k=0,1,\dots$ and choose
$q=p_*(\frac{N}{N-p})^k$, $\rho=\rho_{k+1}$, ${\rho}'=\rho_k$.
Since $\frac{N}{N-p}>1$, invoking iterations we see that \eqref{2-13}
is valid.
\end{proof}

There are difficulties in carrying out an estimate independent of $r$
on the corresponding solution of \eqref{eq1-1}--\eqref{eq1-2}
due to the fact that $f$ varies in $\overline{\mathcal{R}(r)^w}$.
Hence we introduce the following comparison principle.

\begin{lemma}\label{le2-10}
Let $u_f$ and $u_0$ be the solutions of the problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f\in\overline{\mathcal{R}(r)^w}$
and $f=0$ respectively.
There holds
$$
u_f(x)\ge u_0(x),\quad {\rm a.e.~in~}\mathbb{R}^N.
$$
\end{lemma}

\begin{proof}
From \eqref{2-4}, we see that
$$
\int_{\mathbb{R}^N}(|\nabla u_f|^{p-2}\nabla u_f-|\nabla u_0|^{p-2}\nabla u_0)
\cdot\nabla\varphi dx
=\int_{\mathbb{R}^N}f\varphi dx,\quad\forall\varphi\in W.
$$
Choosing $\varphi=(u_0-u_f)_+\in W$, we obtain
$$
\int_A(|\nabla u_f|^{p-2}\nabla u_f-|\nabla u_0|^{p-2}\nabla u_0)
\cdot(\nabla u_f-\nabla u_0)dx=-\int_Af\varphi dx\le0,
$$
where $A=\{x\in\mathbb{R}^N;u_f(x)\le u_0(x)\}$.
Thus $\nabla\varphi\equiv0$ and $\varphi\equiv0$ from Sobolev's inequality.
 \end{proof}

Now we could give a locally lower bound of the solution independent of $f$ and $r$.

\begin{lemma}\label{le2-11}
Let $u$ be the solution of the problem \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f\in\overline{\mathcal{R}(r)^w}$.
For any $\varepsilon>0$, there exists $r_\varepsilon>0$ depending only on
$N$, $p$, $h$ and $\varepsilon$,
such that
$$
u(x)\ge-\varepsilon,\quad\forall x\in\mathbb{R}^N\backslash B_{r_\varepsilon}.
$$
\end{lemma}

\begin{proof}
Let $u_0$ be as defined in Lemma \ref{le2-10}, which is independent of $f$ and $r$.
Using the similar method in the proof of Lemma \ref{le2-10},
 we demonstrate $u_0\le0$ in $\mathbb{R}^N$.
Utilizing Lemma \ref{le2-9}, we find
$$
\|u_0\|_{\infty;B_{1/2}(x_0)}\le C_3(N,p)\|u_0\|_{p_*;B_1(x_0)},
$$
provided $B_1(x_0)\cap\operatorname{supp}h=\emptyset$.
Let $r\ge r_h+1$ and $|x_0|\ge r$, where $r_h>0$ satisfies
$\operatorname{supp}h\subset B_{r_h}$.
From \eqref{2-6}, we see that
$\|u_0\|_{p_*;\mathbb{R}^N}\le C_0^{p'}\|h\|_{p_*'}^\frac{1}{p-1}$.
It follows $\|u_0\|_{\infty;B_{1/2}(x_0)}\le\varepsilon$ provided
$|x_0|$ is large enough.
By applying Lemma \ref{2-10}, we complete the proof.
\end{proof}

\section{The case $\lambda>0$}

First we are concerned with the existence of minimizers for the energy
functional in bounded domains,
then we will show that a solution valid in a sufficiently large bounded domain
is in fact valid in the whole space.

\begin{lemma}\label{le3-1}
Let $\lambda\ge0$, $r>0$ and $\omega_Nr^N\ge|\operatorname{supp}f_0|$.
\begin{itemize}
\item[(i)] The functional $\Psi_\lambda$ attains its minimum relative to
$\overline{\mathcal{R}(r)^w}$.

\item[(ii)] If $f_{r,\lambda}$ is a minimizer for $\Psi_\lambda$ relative to
$\overline{\mathcal{R}(r)^w}$,
$f_{r,\lambda}$ is a solution of the  variational problem
$$
\min_{f\in\overline{\mathcal{R}(r)^w}}\int_{\mathbb{R}^N}
f(p'u_{f_{r,\lambda}}+\lambda g)dx,
$$
where $u_{f_{r,\lambda}}$ is the solution of \eqref{eq1-1}--\eqref{eq1-2}
corresponding to $f_{r,\lambda}$.
\end{itemize}
\end{lemma}

The above lemma is a simple consequence of Lemma \ref{le2-3} and
Lemma \ref{le2-6}.

\begin{lemma}\label{le3-2}
Let $\lambda>\lambda_0\equiv\frac{p'}{\sigma}\|f_0\|_\infty^{\frac{1}{p-1}}$.
If $f_{r,\lambda}$ is a minimizer of $\Psi_\lambda$ relative to
 $\overline{\mathcal{R}(r)^w}$ and
$\psi_{r,\lambda}=p'u_{f_{r,\lambda}}+\lambda g$, we have
$$
|L_\alpha(\psi_{r,\lambda})\cap\operatorname{supp}f_{r,\lambda}|=0,\quad
 \forall\alpha\in\mathbb{R}.
$$
\end{lemma}

\begin{proof} We argue by contradiction.
Suppose there exists $\hat\alpha\in\mathbb{R}$ such that $|S_{\hat\alpha}|>0$,
$S_{\hat\alpha}=L_{\hat\alpha}(\psi_{r,\lambda})\cap\operatorname{supp}
f_{r,\lambda}\subset B_r$.
We have $\psi_{r,\lambda}=p'u_{f_{r,\lambda}}+\lambda g=\hat\alpha$, a.e. in
$S_{\hat\alpha}$.
Therefore,
$$
\|f\|_\infty\ge f-h=-\Delta_p u_{f_{r,\lambda}}=\Delta_p
\big(\frac{\lambda}{p'}g\big)
\ge\big(\frac{\lambda}{p'}\big)^{p-1}\sigma^{p-1}>\|f_0\|_\infty,
\quad \text{a.e. in }S_{\hat\alpha},
$$
in the sense of distribution, which contradicts to \eqref{2-1}.
This completes the proof.
 \end{proof}

\begin{lemma}\label{le3-3}
Let $\lambda_0$ be as defined in the lemma above and
$\lambda>\lambda_0$, $\omega_Nr^N\ge|\operatorname{supp}f_0|$.
The set of solutions of the variational problem \eqref{P-lambda-r}
denoted by $S_\lambda(r)$ is nonempty.
If $f_{r,\lambda}\in S_\lambda(r)$, we have
\begin{equation}\label{3-1}
f_{r,\lambda}=\varphi_{r,\lambda}\circ(p'u_{f_{r,\lambda}}+\lambda g),
\end{equation}
almost everywhere in $B_r$ for a decreasing function $\varphi_{r,\lambda}$.
\end{lemma}

\begin{proof}
From Lemma \ref{le3-1}, there exists $f_{r,\lambda}\in\overline{\mathcal{R}(r)^w}$,
which is a minimizer of $\Psi_\lambda$ relative to $\overline{\mathcal{R}(r)^w}$.
By Lemma \ref{le3-2}, the level sets of
$\psi_{r,\lambda}=p'u_{f_{r,\lambda}}+\lambda g$
on $\operatorname{supp} f_{r,\lambda}$ have zero measure.
Utilizing Lemma \ref{le3-1} (ii) and Lemma \ref{le2-2}, we see that
$f_{r,\lambda}\in\mathcal{R}(r)$ solves
the variational problem \eqref{P-lambda-r} and has the representation \eqref{3-1}.
As already shown in the proof, the minimum for $\Psi_\lambda$ relative to
$\overline{\mathcal{R}(r)^w}$ actually equals
the minimum relative to $\mathcal{R}(r)$ under the assumption of this lemma.
Thus for any $f_{r,\lambda}\in S_\lambda(r)$, $f_{r,\lambda}$ has a
representation as \eqref{3-1} for some $\varphi_{r,\lambda}$.
\end{proof}

We have proved that the variational problem \eqref{P-lambda-r} has a solution
for $\lambda>\lambda_0$ and $\omega_Nr^N\ge|\operatorname{supp}f_0|$.
Now we will show that if $r$ is chosen large enough, it ceases to have any
influence on the variational problem \eqref{P-lambda-r}.

\begin{lemma}\label{le3-4}
Let $\lambda>\lambda_0$.
There exists $r_\lambda>0$ satisfying
$\omega_Nr_\lambda^N\ge|\operatorname{supp}f_0|$ such that for any
$r\ge r_\lambda$ and
$f_{r,\lambda}\in S_\lambda(r)$, we have
$\operatorname{supp}f_{r,\lambda}\subset B_{r_\lambda}$.
\end{lemma}

\begin{proof}
Let $r_a>0$, $\omega_Nr_a^N>|\operatorname{supp}f_0|
=|\operatorname{supp}f_{r,\lambda}|$.
From estimates \eqref{2-1} and \eqref{2-11}, we see that
$\|u_{f_{r,\lambda}}\|_{\infty;\mathbb{R}^N}$ is bounded by a constant
depending on $N$, $p$, $\|f_0\|_{\infty}$,
$\|h\|_{\infty}$, $|\operatorname{supp}f_0|$, $|\operatorname{supp}h|$ and
independent of $r$, $\lambda$.
Since $\lambda>\lambda_0$, $g\in C^2(\mathbb{R}^N)$ and
$\lim_{|x|\to+\infty}g(x)=+\infty$,
we can find a constant $r_\lambda\ge r_a$ such that
\begin{equation}\label{3-2}
p'u_{f_{r,\lambda}}(x)+\lambda g(x)\ge p'u_{f_{r,\lambda}}(z)+\lambda g(z),
\quad\forall x\in\mathbb{R}^N\backslash B_{r_\lambda}, \; z\in\overline{B}_{r_a}.
\end{equation}
Using the representation \eqref{3-1} of $f_{r,\lambda}$ in $B_r$,
the decreasing property of $\varphi_{r,\lambda}$ and the fact
$\operatorname{supp}f_{r,\lambda}\subset B_r$, we deduce
$$
0\le f_{r,\lambda}(x)\le\inf_{|z|\le r_a}f_{r,\lambda}(z),
\quad\forall x\in\mathbb{R}^N\backslash B_{r_\lambda}.
$$
By the assumption of $r_a$, we get $\inf_{|z|\le r_a}f_{r,\lambda}(z)=0$
since $|B_{r_a}\backslash\operatorname{supp}f_{r,\lambda}|>0$.
It follows $\operatorname{supp}f_{r,\lambda}\subset B_{r_\lambda}$.
\end{proof}

Now we are ready to prove Theorem \ref{th1}.

\begin{proof}[Proof of Theorem \ref{th1}]
Let $\lambda>\lambda_0$, $r\ge r_\lambda$ and $f_{r,\lambda}\in S_\lambda(r)$.
From Lemma \ref{le3-4}, we have
$\operatorname{supp}f_{r,\lambda}\subset B_{r_\lambda}$.
Therefore, $f_{r,\lambda}\in\mathcal{R}(r_\lambda)\subset\mathcal{R}(r)$.
It shows that the minimum of $\Psi_\lambda$
relative to $\mathcal{R}(r)$ is attained at and only at some points in subset
$\mathcal{R}(r_\lambda)$ for $r\ge r_\lambda$.
Since $\mathcal{R}=\bigcup_{r\ge r_\lambda}\mathcal{R}(r)$, we obtain
$S_\lambda=S_\lambda(r_\lambda)=S_\lambda(r)$
for $r\ge r_\lambda$.
It follows \eqref{P-lambda} has a solution.
To prove the last part of this theorem, for any $r\ge r_\lambda$ and
$f_\lambda\in S_\lambda=S_\lambda(r)$,
we have by applying Lemma \ref{le3-3}
$$
f_\lambda=\varphi_{r,\lambda}\circ(p'u_{f_\lambda}+\lambda g),
\quad\text{a.e. in }B_r,
$$
for a decreasing function $\varphi_{r,\lambda}$.
We can use the similar method in the proof of \eqref{3-2} to choose
$r\ge r_\lambda$ and $C_\lambda\in\mathbb{R}$ such that
$$
p'u_{f_\lambda}(x)+\lambda g(x)\ge C_\lambda=\sup_{z\in\overline{B}_{r_\lambda}}
(p'u_{f_\lambda}(z)+\lambda g(z)), \quad\forall x\in\mathbb{R}^N\backslash B_{r}.
$$
Noticing that $\operatorname{supp}f_\lambda\subset B_{r_\lambda}$, we have
that $\varphi_{r,\lambda}(t)=0$ for $t\in[C_\lambda,C_\lambda ']$, and
$C_\lambda '=\sup_{z\in\overline{B}_r}(p'u_{f_\lambda}(z)+\lambda g(z))\ge C_\lambda$.
Now define
$$
\varphi_\lambda(t)=
\begin{cases}
\varphi_{r,\lambda}(t), &t\le C_\lambda,\\
0, &t>C_\lambda.
\end{cases}
$$
Clearly $\varphi_\lambda$ is a decreasing function and
$f_\lambda=\varphi_\lambda\circ(p'u_{f_\lambda}+\lambda g)$
a.e. in $\mathbb{R}^N$.
\end{proof}

\section{The case $\lambda=0$}

To derive the existence result in this case, we need some additional
conditions on $f$ and $h$.
Similarly, we first deduce the following lemma in bounded domains.

\begin{lemma}\label{le4-1}
Suppose $\|f_0\|_\infty<\|h\|_{-\infty;\operatorname{supp}h}$, $r>0$ and
$\omega_Nr^N\ge|\operatorname{supp}f_0|$.
Let $f_r$ be a minimizer of $\Psi_0$ relative to $\overline{\mathcal{R}(r)^w}$
and $u_{f_r}$ be the solution of
the problem \eqref{eq1-1}--\eqref{eq1-2} corresponding to $f_r$.
We have
$$
|L_\alpha(u_{f_r})\cap\operatorname{supp}f_r|=0,\quad\forall\alpha\in\mathbb{R}.
$$
\end{lemma}

\begin{proof}
We argue by contradiction.
Suppose there exists $\hat\alpha\in\mathbb{R}$ such that $|A_{\hat\alpha}|>0$,
$A_{\hat\alpha}=L_{\hat\alpha}(u_{f_r})\cap\operatorname{supp}f_r\subset B_r$.
We have $u_{f_r}=\hat\alpha$ a.e. in $A_{\hat\alpha}$.
Hence $-\Delta_pu_{f_r}=f-h=0$ a.e. in $A_{\hat\alpha}$, in the sense of
distributions.
Since $A_{\hat\alpha}\subset\operatorname{supp}f_r$, we find $f>0$ in
$A_{\hat\alpha}$,
which follows $h>0$ a.e. in $A_{\hat\alpha}$ and
$A_{\hat\alpha}\subset\operatorname{supp}h$.
Thus $\|h\|_{-\infty;\operatorname{supp}h}\le\|f\|_\infty\le\|f_0\|_\infty$
from \eqref{2-1}, which is a contradiction to our assumption.
\end{proof}

\begin{lemma} \label{le4-2}
The set of solutions of the variational problem 
\eqref{P-lambda-r} with $\lambda=0$ denoted by $S_0(r)$
is nonempty under the assumption of the lemma above.
Moreover, if $f_r\in S_0(r)$, we have
\begin{equation}\label{4-1}
f_r=\varphi_r\circ u_{f_r},\quad{\rm a.e.~in~}B_r,
\end{equation}
for a decreasing function $\varphi_r$.
\end{lemma}

\begin{proof}
Utilizing Lemma \ref{le3-1}, Lemma \ref{le4-1} and Lemma \ref{le2-2},
we obtain the required results by using the similar method in the proof
 of Lemma \ref{le3-3}.
\end{proof}

\begin{lemma}\label{le4-3}
There exists a constant $\kappa=\kappa(N,p)\in(0,\frac{1}{2}]$
 depending only on $N$ and $p$, such that
if $\|f_0\|_{\infty}<\|h\|_{-\infty;\operatorname{supp} h}$,
$\operatorname{supp} h\subset B_{r_h}$ and
\begin{equation}\label{4-2}
|\operatorname{supp} f_0|\le\kappa
\Big(\frac{\|h\|_{-\infty;\operatorname{supp} h}}{\|h\|_{\infty}}
\Big)^{\frac{p}{p-1}}
\Big(\frac{|\operatorname{supp} h|}{|B_{r_h}|}\Big)^\frac{N-p}{N(p-1)}
|\operatorname{supp} h|,
\end{equation}
we have $\operatorname{supp}f_r\subset B_{r_0}$ for any $r\ge r_0$ and
$f_r\in S_0(r)$, where $r_0\ge r_h$ is a constant independent of $r$ and $f_r$.
\end{lemma}

\begin{proof}
From the representation of $f_r$ in \eqref{4-1} and the decreasing property of
$\varphi_r$, we see that
\begin{equation}\label{4-3}
\sup_{x\in\operatorname{supp}f_r}u_{f_r}(x)=s_0
\le\inf_{z\in B_r\backslash\operatorname{supp}f_r}u_{f_r}(z).
\end{equation}
Using \eqref{2-4}, we calculate
\begin{equation} \label{4-4}
\begin{aligned}
&\int_{\mathbb{R}^N}|\nabla u_{f_r}|^pdx\\
&=\int_{\mathbb{R}^N}(f_r-h)u_{f_r}dx\\
&=\int_{\operatorname{supp}f_r\backslash\operatorname{supp}h}
f_ru_{f_r}dx-\int_{\operatorname{supp}h\backslash\operatorname{supp}f_r}hu_{f_r}dx
+\int_{\operatorname{supp}f_r\cap\operatorname{supp}h}(f_r-h)u_{f_r}dx\\
&\le s_0\|f_r\|_{1;\operatorname{supp}f_r\backslash\operatorname{supp}h}
-s_0\|h\|_{1;\operatorname{supp}h\backslash\operatorname{supp}f_r}
+\|u_{f_r}\|_\infty\|f_r-h\|_\infty|\operatorname{supp}f_r|.
\end{aligned}
\end{equation}
By assumption, for any $\kappa\le\frac{1}{2}$, utilizing \eqref{2-1}
and \eqref{2-11}, we have
\begin{equation}
\begin{aligned}
\|f_r\|_{1;\operatorname{supp}f_r\backslash\operatorname{supp}h}
&\le\|f_r\|_1\le\|f_0\|_1\le\|f_0\|_{\infty}|\operatorname{supp}f_0|\\
&<\|h\|_{-\infty;\operatorname{supp}h}(|\operatorname{supp}h|
 -|\operatorname{supp}f_r|)
 \le\|h\|_{1;\operatorname{supp}h\backslash\operatorname{supp}f_r},
\end{aligned}\label{4-5}
\end{equation}
\begin{equation}
\|f_0\|_1
\le\|f_0\|_{\infty}|\operatorname{supp}f_0|<\frac{1}{2}
\|h\|_{-\infty;\operatorname{supp}h}|\operatorname{supp}h|
\le\frac{1}{2}\|h\|_1,\label{4-6}
\end{equation}
\begin{equation}
\begin{aligned}
&\|u_{f_r}\|_\infty\|f_r-h\|_\infty|\operatorname{supp}f_r|\\
&\le C_1\|f_r-h\|_\infty^\frac{Np}{Np-N+p}\|f_r-h\|_{p_*'}
 ^\frac{p^2}{(p-1)(Np-N+p)}|\operatorname{supp}f_r|\\
&\le C_12^\frac{p^2}{(p-1)(Np-N+p)}\|h\|_\infty^\frac{Np}{Np-N+p}
 \|h\|_{p_*'}^\frac{p^2}{(p-1)(Np-N+p)}|\operatorname{supp}f_0|\\
&\le C_12^\frac{p^2}{(p-1)(Np-N+p)}\|h\|_\infty^\frac{p}{p-1}
 |\operatorname{supp}h|^\frac{p}{N(p-1)}|\operatorname{supp}f_0|,
\end{aligned}\label{4-7}
\end{equation}
where $C_1=C_1(N,p)$ is the constant in \eqref{2-11}.
From the assumption and the estimates \eqref{2-12}, \eqref{4-6}, we deduce
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^N}|\nabla u_{f_r}|^pdx
&\ge C_2^pr_h^{-\frac{N-p}{p-1}}(\|h\|_1-\|f_0\|_1)^\frac{p}{p-1}
\ge C_2^pr_h^{-\frac{N-p}{p-1}}\big(\frac{1}{2}\|h\|_1\big)^\frac{p}{p-1}\\
&\ge 2^{-p'}C_2^pr_h^{-\frac{N-p}{p-1}}(\|h\|_{-\infty;\operatorname{supp}h}
|\operatorname{supp}h|)^\frac{p}{p-1},
\end{aligned}\label{4-8}
\end{equation}
where $C_2=C_2(N,p)$ is the constant in \eqref{2-12}.
Let 
\[
\kappa=\min\{\frac{1}{2},\frac{C_2^p\omega_N^\frac{N-p}{N(p-1)}}{2\cdot2^{p'}
\cdot2^\frac{p^2}{(p-1)(Np-N+p)}C_1}\}.
\]
Combining \eqref{4-2}, \eqref{4-4}--\eqref{4-5}, \eqref{4-7}--\eqref{4-8},
 we obtain
\begin{align*}
s_0&\le-\frac{C_2^p\|h\|_1^{p'}r_h^{-\frac{N-p}{p-1}}}
{2\cdot2^{p'}(\|h\|_{1;\operatorname{supp}h\backslash\operatorname{supp}f_r}
-\|f_r\|_{1;\operatorname{supp}f_r\backslash\operatorname{supp}h})}\\
&\le-\frac{C_2^p}{2\cdot2^{p'}}\|h\|_1^\frac{1}{p-1}
r_h^{-\frac{N-p}{p-1}}\equiv-\delta,
\end{align*}
where $\delta>0$ is a constant independent of $r$ and $f_r$.

For $\varepsilon=\frac{1}{2}\delta$, applying Lemma \ref{2-11},
we find that there exists $r_0\ge r_h$ independent of $r$ and $f_r$ such that
$$
u_{f_r}(x)\ge-\frac{1}{2}\delta>-\delta\ge s_0
=\sup_{x\in\operatorname{supp}f_r}u_{f_r}(x),\quad\forall
x\in\mathbb{R}^N\backslash B_{r_0}.
$$
It follows $\operatorname{supp}f_r\subset B_{r_0}$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{th2}]
The first part of this theorem can be proved
by using the similar method in the proof of Theorem \ref{th1}.
It follows $S_0=S_0(r_0)=S_0(r)$ for $r\ge r_0$.
To prove the last part of this theorem, for any
$\hat f\in S_0=S_0(r_0)$, we have from \eqref{4-1}
$$
\hat f=\varphi_{r_0}\circ u_{\hat f},\quad \text{a.e. in }B_{r_0},
$$
for a decreasing function $\varphi_{r_0}$.
Combining this with \eqref{4-3}, we obtain $\varphi_{r_0}(t)\ge0$
for $t\le s_0$ and $\varphi_{r_0}(t)=0$ for $t\in[s_0,s_0']$,
$s_0'=\sup_{x\in B_{r_0}}u_{\hat f}(x)$.
Now define
$$
\hat\varphi(t)= \begin{cases}
\varphi_{r_0}(t), &t\le s_0,\\
0, &t>s_0.
\end{cases}
$$
Clearly, $\hat\varphi$ is a decreasing function and
$\hat f=\hat\varphi\circ u_{\hat f}$ a.e. in $\mathbb{R}^N$.
 \end{proof}


\subsection*{Acknowledgments}
The first author was supported in part by the Scientific Research Foundation of
Graduate School of South China Normal University.
The second author and the third author were supported in part by National
Natural Scientific Foundation of China and Specialized Research Fund for the
Doctoral Program of Higher Education.

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\end{document}