\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 60, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/60\hfil Optimization problems] {Optimization problems involving Poisson's equation in $\mathbb{R}^3$} \author[F. Bahrami, H. Fazli\hfil EJDE-2011/60\hfilneg] {Fariba Bahrami, Hossain Fazli} % in alphabetical order \address{Fariba Bahrami \newline Department of Applied Mathematics, University of Tabriz, Tabriz, Iran} \email{fbahram@tabrizu.ac.ir} \address{Hossain Fazli \newline Department of Applied Mathematics, University of Tabriz, Tabriz, Iran} \email{fazli64@gmail.com} \thanks{Submitted February 16, 2011. Published May 10, 2011.} \subjclass[2000]{49J20, 49K20} \keywords{Rearrangements class; variational problem; Poisson's equation; \hfill\break\indent energy functional; minimization} \begin{abstract} In this article, we prove the existence of minimizers for integrals associated with a second-order elliptic problem. For this three-dimensional optimization problem, the admissible set is a rearrangement class of a given function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} We consider the Poisson's equation $$\label{poson} \begin{gathered} -\Delta u=f-2h \quad \text{in } \mathbb{R}^3\\ \lim_{|x|\to +\infty} u(x)=0, \quad f\in L^p_b(\mathbb{R}^3), \end{gathered}$$ where $L^p_b(\mathbb{R}^3)=\{f\in L^p(\mathbb{R}^3): f \text{ has bounded support}\}$ and $p>3$. Here $h$ is a given non-negative function in $h\in L^\infty(\mathbb{R}^3)$ with bounded support. For the sake of convenience in the discussions, we have $2h$ instead of $h$, but it can be replaced by $h$. By standard results on elliptic equations, problem \eqref{poson} has a unique solution $u\in W^{2,p}_{\rm loc}(\mathbb{R}^3)$; see \cite{gilbarg}. Let $u_{f}$ be the solution of \eqref{poson}, we define energy functional corresponding to \eqref{poson} as $$\Psi_\lambda(f)=\frac{1}{2}\int_{\mathbb{R}^3}f u_{f} +\lambda \int_{\mathbb{R}^3}gf,$$ for $f \in L^p_b(\mathbb{R}^3)$ where $g\in C^2(\mathbb{R}^3)$, $\lim_{|x|\to+\infty}g=+\infty$ and $\Delta g>c$ for some $c>0$ and $\lambda\geq 0$. In this paper we minimize the functional $\Psi_\lambda$ on rearrangement class of a fixed function. We separate the investigation of the particular case $\lambda=0$, since the discussion in the case $\lambda>0$ does not carry over the case $\lambda=0$. The same optimization problems have been investigated in bounded domains for the Laplacian operator in \cite{Burton1, cuccu, ema}, for the p-Laplacian operator in \cite{fabriz, marras}, for semilinear operators in \cite{zivari}. For the current problem we face two mathematical difficulties: firstly the awkward nature of rearrangements class, and secondly a loss of compactness which is caused by the unboundness of the domain $\mathbb{R}^3$. To overcome these difficulties we first investigate the problem in a bounded domain. Then using Burton's theory on rearrangements class, we show that a solution valid in a sufficiently large bounded domain is in fact valid in the whole space. \section{Notation, definitions and statement of the main result} Henceforth we assume $p\in(3,\infty)$ and $p'$ is the conjugate exponent of $p$; that is, $\frac{1}{p}+\frac{1}{p'}=1$. Points in $\mathbb{R}^3$ are denoted by $x = (x_1, x_2,x_3)$, $y=(y_1,y_2,y_3)$, and so on. By $B_r (x)$ we denote the ball centered at $x\in \mathbb{R}^3$ with radius $r$; if the center is the origin, we write $B_r$. Measure will refer to Lebesgue measure on $\mathbb{R}^3$, and if $A\subseteq \mathbb{R}^3$ is measurable then $|A|$ will denote the measure of $A$. If $A \subset \mathbb{R}^3$ is a measurable set, then we say $x\in A$ is a density point of $A$ whenever $$|B_\varepsilon(x) \cap A|>0,$$ for all positive $\varepsilon$. For a measurable function $f:\mathbb{R}^3 \to \mathbb{R}^+$, the strong support or simply the support of $f$ is denoted $\operatorname{supp}( f)$ and is defined by $$\operatorname{supp}( f) = \{x :f(x)>0\}.$$ For a measurable function $f : \mathbb{R}^3\to \mathbb{R}^+$ we define $$\|f\|_{-\infty}=\operatorname{ess\,inf}(f)=\sup\{M\geq 0 : f(x)\geq M, \text{ for almost all }x\}.$$ When $f$ and $g$ are non-negative measurable functions that vanish outside sets of finite measure in $\mathbb{R}^3$, we say $f$ is a rearrangement of $g$ whenever $$|\{x \in\mathbb{R}^3 :f(x)\geq\alpha\}| = |\{x \in\mathbb{R}^3 :g(x)\geq\alpha\}|,$$ for every positive $\alpha$. For any real integrable and non-negative function $f$ vanishing outside a bounded set $\Omega\subset\mathbb{R}^3$ of measure $m$, we can define a decreasing rearrangement $f^\Delta$ which is a decreasing function on the interval $(0,m)$ satisfying $$|\{s \in(0,m) : f^\Delta(s)\geq\alpha\}| =|\{x \in\Omega : f(x)\geq\alpha\}|,$$ for every positive $\alpha$. Also there exists a Schwarz rearrangement $f^*$ for $f$, that is a rearrangemet of $f$ as a radial decreasing function on a ball. Let us fix $f_0 \in L^p(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$ to be a measurable and non-negative function vanishing outside a set of measure $4\pi a^3/3$, for some positive $a\in \mathbb{R}$. The set of all rearrangements on $\mathbb{R}^3$ of $f_0$ with bounded support is denoted by $\mathcal{R}$. The subset of $\mathcal{R}$ containing functions vanishing outside the ball $B_r$, where $r\geq a$, is denoted by $\mathcal{R}(r)$; henceforth we assume $r\geq a$ in order that $\mathcal{R}(r)$ is non-empty. The weak closure in $L^p(B_r )$ of $\mathcal{R}(r)$ is denoted by $\overline{\mathcal{R}(r)^w}$. Now we are ready to introduce our minimizing problems $P_\lambda$ as follows: $$\label{variation} \min_{f\in\mathcal{R}}\ \Psi_\lambda (f).$$ The set of solutions of $P_\lambda$ is denoted by $S_\lambda$. Similarly, for $r\geq a$ we define $P_\lambda(r)$ as follows: $$\label{bvariation} \min_{f\in\mathcal{R}(r)}\ \Psi_\lambda (f),$$ and the set of solutions is denoted by $S_\lambda(r)$. Our main results are the following: \begin{theorem} \label{thm1} There exists $\lambda_0>0$ such that for every $\lambda>\lambda_0$, the optimization problem $P_\lambda$ has a solution. Moreover, if ${f_{\lambda}}\in S_{\lambda}$ and $u_{ f_{\lambda}}$ be the solution of \eqref{poson} corresponding with energy minimizer, then there exists a decreasing function $\varphi_\lambda$ such that $$\label{partial differential} f_{\lambda}=\varphi_\lambda \circ ( u_{ f_{\lambda}}+\eta+\lambda g),$$ almost everywhere in $\mathbb{R}^3$ where $\eta$ will be presented later. \end{theorem} \begin{theorem}\label{thm2} Let $f_0$ and $h$ be as introduced above. Let $|\operatorname{supp}(f_0)|=4\pi a^3/3$ and $|\operatorname{supp}(h)|=4\pi b^3/3$ for some $a, b$ positive real numbers. We assume $$\label{necess-cond-supp} b>\sqrt{3}a,\quad \| f_0\|_{\infty} < \| h\|_{-\infty}.$$ Then the optimization problem $P_0$ has a solution. \end{theorem} \section{Preliminary results} In this section we state and/or prove some lemmas which are essential in our analysis. We begin with a result proved by Burton in \cite{Burton2}. \begin{lemma}\label{rearangement} For $r\geq a$ and $q\geq1$, 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)] $\overline{\mathcal{R}(r)^w}=\{f\in L^1(B_r): \int_{0}^sf^\Delta(t)dt\leq\int_0^sf_0^\Delta(t)dt,\; 0|x|/2$ for $y\in \operatorname{supp}(f)\cup \operatorname{supp}(h)$. Thus, $K(f-2h)$ is dominated by $2\|f-2h\|_1/|x|$. To prove (ii), let $f$ be as in the lemma, we have $$|Kf(x)|\leq\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{|f(y)|}{|x-y|}dy \leq\frac{1}{4\pi}\int_{B_{r^*}(x)}\frac{{f^*}(y)}{|x-y|}dy,$$ where ${f^*}$ is the Schwarz rearrangement of $f$ with respect to $x$ and $r^*=\big(\frac{3|\operatorname{supp}(f)|}{4\pi}\big) ^{1/3}.$ The inequality is a consequence of Hardy-Littlewood inequality \cite{Hardy}. Now by H\"older's inequality, we obtain the assertion where $$\label{fix c} C=\frac{1}{4\pi}\Big(\int_{B_{r^*}(x)}\frac{1}{|x-y|^{p'}}dy\Big) ^{1/p'}=\frac{(3|\operatorname{supp}(f)|)^{\frac{1}{p'}-\frac{1}{3}}} {(4\pi)^{2/3}(3-p')^{1/p'}},$$ and $p'$ is the conjugate exponent of $p$. \end{proof} \begin{lemma}\label{Laplasian} Let $K$ be as the above lemma. \begin{itemize} \item[(i)] If $U$ is a bounded open subset in $\mathbb{R}^3$, $K:L^p(U)\to L^{p'}(U)$ is a linear compact operator. \item[(ii)]For $f\in L^p(U)$, $Kf\in W^{2,p}(U)$ and $-\Delta Kf=f$, almost everywhere in $U$. \end{itemize} \end{lemma} \begin{proof} Since $W^{1,2}(U)$ is compactly embedded into $L^{p'}(U)$ for $p>3$, in order to show the compactness of $K$ it is sufficient to prove the bondedness of $K$ as a map from $L^p(U)$ into $W^{1,2}(U)$. To do this, let $f\in L^p(U)$ we have $$|\nabla Kf(x)| \leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{|f(y)|}{ |x-y|^2}dy, \quad x\in U.$$ Similar to the proof of the lemma above and the fact that $p'<\frac{3}{2}$, we deduce $\|\nabla Kf\|_{2}\leq C\|f\|_{p},$ where $C$ depends on $|U|$ and $p$. This completes the proof. For a proof of part (ii) see \cite{Em2000}. \end{proof} The following lemma is a simple variation of \cite[Lemma 2.15]{Burton2}. \begin{lemma}\label{2.15} Let $r\geq a$ and $\upsilon\in L^{p'}(B_r)$. Denote by $L_{\alpha}(\upsilon)$ the level set of $\upsilon$ at height $\alpha$; that is, $$L_{\alpha}(\upsilon)=\{x\in B_r : \upsilon(x)=\alpha\}.$$ Let ${ T} : L^p(B_r)\to \mathbb{R}$ be the linear functional defined by $${T}(f)=\int_{B_r}f \upsilon.$$ If $\hat{f}$ is a minimizer of $T$ relative to $\overline{\mathcal{R}(r)^w}$ and if $$\left|L_\alpha(\upsilon)\cap \operatorname{supp}(\hat{f})\right|=0,$$ for every $\alpha\in\mathbb{R}$, then $\hat{f}\in\mathcal{R}(r)$ and $$\hat{f}=\varphi o \upsilon,$$ almost everywhere in $B_r$, for some decreasing function $\varphi$. \end{lemma} \section{Investigation in the case: $\lambda>0$} In this section we consider the case in which $\lambda>0$. First we are concerned with the existence of minimizers for the energy functional in a bounded domain, then we will demonstrate the problem in the unbounded domain. \subsection{Bounded domains} We begin with the following lemma. \begin{lemma}\label{burton-lemma} \begin{itemize} \item[(i)] The energy functional $\Psi_{\lambda}$ attains its minimum relative to $\overline{\mathcal{R}(r)^w}$ for $r\geq a$. \item[(ii)] If $f_{r,\lambda}$ is any minimizer for $\Psi_{\lambda}$ relative to $\overline{\mathcal{R}(r)^w}$, then $f_{r,\lambda}$ is a solution of the following variational problem $$\label{minimum} \inf_{f\in\overline{\mathcal{R}(r)^w}}\int_{\mathbb{R}^3} f(u_{f_{r,\lambda}}+\eta+\lambda g),$$ where $u_{f_{r,\lambda}}$ is the solution of \eqref{poson} corresponding to $f_{r,\lambda}$ and $\eta=Kh$. \end{itemize} \end{lemma} \begin{proof} From Lemma \ref{bound}, the optimization problem \eqref{bvariation} is equivalent to $\inf_{f\in \mathcal{R}(r)}\big\{\frac{1}{2}\int_{\mathbb{R}^3}f Kf-\int_{\mathbb{R}^3}\eta f+\lambda\int_{\mathbb{R}^3}g f\big\}.$ By Lemma \ref{Laplasian}, $K$ is compact and symmetric, then $\Psi_{\lambda}$ is a weakly sequentially continuous and G\^ateaux differentiable functional. From Lemma \ref{rearangement}, $\overline{\mathcal{R}(r)^w}$ is weakly sequentially compact, hence $\Psi_{\lambda}$ attains its minimum on it. If $f_{r,\lambda}$ is a minimizer of $\Psi_{\lambda}$ on $\overline{\mathcal{R}(r)^w}$, since the G\^ateaux differential of $\Psi_{\lambda}$ at $f_{r,\lambda}$ is $Kf_{r,\lambda}-\eta+{\lambda}g$, then by \cite[Theorem 3.3]{Burton2}, $f_{r,\lambda}$ is a solution of the variational problem \eqref{minimum}. \end{proof} \begin{lemma}\label{level set1} Let $r\geq a$ and $f_{r,\lambda}$ be a minimizer of $\Psi_{\lambda}$ relative to $\overline{\mathcal{R}(r)^w}$. Let $\psi_{r,\lambda}=Kf_{r,\lambda}-\eta+\lambda g$ and denote by $L_\alpha(\psi_{r,\lambda})$ the level set of $\psi_{r,\lambda}$ at height $\alpha$. Then there exists $\lambda_0>0$ such that for every $\lambda>\lambda_0$, $$\left|L_{\alpha}(\psi_{r,\lambda})\cap \operatorname{supp} (f_{r,\lambda})\right|=0,\quad \forall\alpha\in\mathbb{R}.$$ \end{lemma} \begin{proof} Let $r\geq a$. From Lemma \ref{burton-lemma}, for every $\lambda>0$, the minimizer $f_{r,\lambda}$ of $\Psi_\lambda$ on $\overline{\mathcal{R}(r)^w}$ exists. Suppose there exists $\hat{\alpha}\in\mathbb{R}$ such that $\left|L_{\hat{\alpha}}(\psi_{r,\lambda})\cap \operatorname{supp}(f_{r,\lambda})\right|>0$. Let $S_{\hat{\alpha}}= L_{\hat{\alpha}}(\psi_{r,\lambda})\cap \operatorname{supp}(f_{r,\lambda})$. Since $\psi_{r,\lambda}=u_{f_{r,\lambda}}+\eta+\lambda g$, using \cite[Theorem 7.7]{gilbarg}, lemma \ref{Laplasian} and equation \eqref{poson}, we have $$\label{1} -\Delta\psi_{r,\lambda}=f_{r,\lambda}-h-\lambda\Delta g=0, \quad \text{ a.e. in }\ S_{\hat{\alpha}}.$$ On the other hand, by Lemma \ref{rearangement}, $$\label{2} \int_{0}^sf_{r,\lambda}^\Delta(t)dt\leq\int_0^sf_0^\Delta(t)dt, \quad s>0.$$ Then we deduce $$\label{3} \| f_{r,\lambda}^\Delta\|_{\infty} \leq \| f_0^\Delta\|_{\infty}.$$ Since $f_{r,\lambda}^\Delta$ is a rearrangement of $f_{r,\lambda}\in \overline{\mathcal{R}(r)^w}$ and $f_0^\Delta$ is a rearrangement of $f_0$, from equation \eqref{3}, we conclude $$\label{4} \| f_{r,\lambda}\|_{\infty} \leq \| f_0\|_{\infty}\,.$$ If we assume that $\lambda_0=\| f_0\|_\infty / \| \Delta g \|_{-\infty}$, then for every $\lambda>\lambda_0$, we have $$\label{4'} \| f_0\|_{\infty} < \| h+\lambda \Delta g \|_{-\infty}$$ Finally, from \eqref{4} and \eqref{4'} for every $\lambda> \lambda_0$, we deduce $$\label{5} \| f_{r,\lambda}\|_{\infty} < \| h+\lambda \Delta g\|_{-\infty}$$ which is a contradiction to \eqref{1}. This completes the proof. \end{proof} \begin{lemma}\label{increasing function} Let $\lambda_0$ be as in the lemma above. Then for every $\lambda>\lambda_0$, the variational problem $P_\lambda(r)$ has a solution for $r\geq a$. If $f_{r,\lambda}$ is any solution of $P_\lambda(r)$, then $$\label{e6} f_{r,\lambda}=\varphi_\lambda\circ (u_{f_{r,\lambda}}+\eta+\lambda g),$$ almost everywhere in $B_r$, for a decreasing unknown function $\varphi_\lambda$. \end{lemma} \begin{proof} Let $r\geq a$. From Lemma \ref{burton-lemma}, there exists $f_{r,\lambda}\in\overline{\mathcal{R}(r)^w}$ such that $f_{r,\lambda}$ is a minimizer of $\Psi_\lambda$ relative to $\overline{\mathcal{R}(r)^w}$ and a solution of \eqref{minimum}. By Lemma \ref{level set1}, for every $\lambda>\lambda_0$, the level sets of $\psi_{r,\lambda}=Kf_{r,\lambda}-\eta+\lambda g$ on $\operatorname{supp}(f_{r,\lambda})$ have zero measure. We can use Lemma \ref{2.15} to deduce equation \eqref{e6}. \end{proof} \subsection{Unbounded domain} We proved that the variational problem $P_\lambda(r)$ has a solution for $\lambda>\lambda_0$ and $r\geq a$. Now we will show that if $r$ is chosen large enough, it ceases to have any influence whatever on the variational problem, $P_\lambda(r)$. To do this, we now perform some calculations to deduce the following result. \begin{lemma}\label{exist-unb-domain} Let $\lambda>\lambda_0$. Then, there exists $r_0>a$ such that for $r\geq r_0$ and $f_{r,\lambda}\in S_\lambda(r)$ we have $\operatorname{supp}(f_{r,\lambda})\subset B_{r_0}.$ \end{lemma} \begin{proof} To prove this lemma, it is sufficient to show that the support of $f_{r,\lambda}$ does not have any dense point on the boundary of $B_r$ when $r$ is chosen large enough. Let $r_h>a$ be the smallest positive number for which $\operatorname{supp}(h)\subset B_{r_h}$. We consider $r>r_h+1$ and $f_{r,\lambda}\in S_\lambda(r)$. From Lemma \ref{increasing function} we have $$\label{7} f_{r,\lambda}=\varphi_\lambda\circ (u_{f_{r,\lambda}}+\eta+\lambda g),$$ almost everywhere in $B_r$, for a decreasing unknown function $\varphi_\lambda$ where $u_{f_{r,\lambda}}$ is the solution of \eqref{poson} corresponding with $f_{r,\lambda}$. To seek a contradiction suppose the assertion is false. Then there exists $x_0\in den(\operatorname{supp}(f_{r,\lambda} ))$(set of dense points of support) such that $|x_0|=r$. Let $A=\operatorname{supp}(f_{r,\lambda})\cap B_1(x_0)$, then $|A|>0$. For $x\in A$ $$\label{8} Kf_{r,\lambda}(x) =\frac{1}{4\pi} \int_{B_r}\frac{1}{|x-y|}f_{r,\lambda}(y)dy\\ \geq \frac{1}{4\pi}\frac{\| f_0\|_1}{2r}$$ and $$\label{9} \eta(x)=\frac{1}{4\pi} \int_{B_{r_h}}\frac{1}{|x-y|}h(y)dy \leq \frac{1}{4\pi}\frac{\| h\|_1}{r-r_h-1}$$ From \eqref{8}, \eqref{9} and relation $u_{f_{r,\lambda}}= Kf_{r,\lambda}-2\eta$, we obtain $$\label{lower bound} u_{f_{r,\lambda}}(x)+\eta(x)+\lambda g(x)\geq \frac{1}{4\pi}\Big(\frac{1}{2r}\| f_0\|_1-\frac{1}{r-r_h-1}\| h\|_1\Big)+\lambda g(x).$$ Since $|\operatorname{supp}(f_{r,\lambda} )|=4\pi a^3/3$ and $r_h>a$, there exists $D\subset B_{r_h}$ such that $D\cap \operatorname{supp}(f_{r,\lambda})$ is empty and $|D|>0$. For $z\in D$ from Lemma \ref{bound} we have $$\label{11} Kf_{r,\lambda}(z)=\frac{1}{4\pi} \int_{B_r}\frac{1}{|z-y|} f_{r,\lambda}(y)dy \leq C\| f_0 \|_p,$$ where $C$ depends on $p$ and $|\operatorname{supp}(f_{r,\lambda})|$. Also $$\eta(z)=\frac{1}{4\pi} \int_{B_{r_h}}\frac{1}{|z-y|}h(y)dy \geq\frac{1}{4\pi}\frac{1}{2r_h} \| h\|_1. \label{12}$$ Then, from \eqref{11} and \eqref{12} we derive $$u_{f_{r,\lambda}}(z)+\eta(z)+\lambda g(z)\leq \lambda g(z)-C_1. \label{12'}$$ Now, since $|z|\leq r_h$ for $z\in D$ and \eqref{12'}, we deduce $$\label{upper bound} u_{f_{r,\lambda}}(z)+\eta(z)+\lambda g(z)\leq C_2, \quad z\in D,$$ where $C_2$ is a constant independent of $r$. If we make $r$ large we derive from \eqref{lower bound} and \eqref{upper bound} $$(u_{f_{r,\lambda}}(x)+\eta(x)+\lambda g(x)) -(u_{f_{r,\lambda}}(z)+\eta(z)+\lambda g(z))>0,$$ for $x\in A$ and $z\in D$. Since $|A|>0$, $|D|>0$, this is a contradiction to \eqref{7}. \end{proof} \subsection{Proof of Theorem \ref{thm1}} Let $r_0$ be as in Lemma \ref{exist-unb-domain}. Assume $f_{r,\lambda}$ to be a solution of $P_\lambda(r)$ for $r\geq r_0$ and $\lambda>\lambda_0$. From Lemma \ref{exist-unb-domain}, $\operatorname{supp}(f_{r,\lambda})\subset B_{r_0}$ for $r>r_0$, therefore we obtain the inclusion $S_\lambda(r_0)\subset S_\lambda$ that it means $P_{\lambda}$ has a solution. Let $f_{\lambda}\in S_{\lambda}$ for $\lambda>\lambda_0$. To prove the last part of theorem, if ${f}_\lambda\in S_\lambda$ we have by applying Lemma \ref{increasing function} $$\label{14} {f}_\lambda=\varphi_\lambda\circ (u_{f_{\lambda}}+\eta+\lambda g),$$ almost everywhere in $B_r$ for $r>r_0$ and a decreasing unknown function $\varphi_\lambda$. Notice that we can suppose $\varphi_\lambda\geq0$. Since $u_{f_{\lambda}}+\eta+\lambda g$ is a continuous function on the compact set $B_{r_0}$, and $\operatorname{supp}(f_{\lambda})\subset B_{r_0}$, there exists $k\in\mathbb{R}$ such that $$\label{15} u_{f_{\lambda}}+\eta+\lambda g< k\quad \text{a.e } \operatorname{supp}(f_{\lambda}).$$ On the other hand, by applying condition \eqref{lower bound} we have $u_{f_{\lambda}}+\eta+\lambda g\to +\infty$, as $|x|\to\infty$. Then we can find $r > r_0$ such that $$\label{16} u_{f_{\lambda}}+\eta+\lambda g\geq k\quad\text{a.e outside } B_r.$$ Now define $$\hat{\varphi}_\lambda(t)=\begin{cases} {\varphi}_\lambda(t) &t0. Let A_{\hat{\alpha}}= L_{\hat{\alpha}}(\psi_r)\cap \operatorname{supp}(f_r). Then from equation \eqref{poson}, we have $$\label{e1} -\Delta \psi_r=f_r-h=0, \quad \text{a.e. in } A_{\hat{\alpha}}.$$ So A_{\hat{\alpha}}\subset \operatorname{supp}(h). On the other hand, by Lemma \ref{rearangement}, we have \begin{equation*} \int_{0}^sf_r^\Delta(t)dt\leq\int_0^sf_0^\Delta(t)dt, \quad s>0. \end{equation*} Then $$\label{e3} \| f_r^\Delta\|_{\infty} \leq \|f_0^\Delta\|_{\infty}.$$ Since f_r^\Delta is a rearrangement of f_r and f_0^\Delta is a rearrangement of f_0, from \eqref{e3} we obtain $$\label{e4} \| f_r\|_{\infty} \leq \|f_0\|_{\infty}.$$ Finally, from \eqref{e4} and condition \eqref{necess-cond-supp}, we deduce $$\label{e5} \| f_r\|_{\infty} < \| h\|_{-\infty}.$$ which is a contradiction to \eqref{e1}. \end{proof} \subsection{Proof of Theorem \ref{thm2}} Since the level sets of \psi_r have zero measure, similar to the proof of Lemma \ref{increasing function} we can claim that there exists minimizer f_r for P_r such that $$\label{6} f_r=\varphi o (u_{f_r}+\eta),$$ almost everywhere in B_r, for a decreasing unknown function \varphi. To prove the existence in unbounded domain, it is enough to show that the support of f_r does not have any dense point at the boundary of B_r when r is chosen large enough. Let r_h>a be the smallest positive number for which \operatorname{supp}(h)\subset B_{r_h}. Since b>\sqrt{3}a, then similar to presented trend in the proof of Lemma \ref{exist-unb-domain} there exits A\subset \operatorname{supp}(f_r) with positive measure and D\subset \operatorname{supp}(h)\cap(\operatorname{supp}(f_r))^c such that |D|>0 and |z-y|r_h+1 we have \begin{gather}\label{10} u_{f_r}(x)+\eta(x)\geq\frac{1}{4\pi}\Big(\frac{1}{2r}\| f_0\|_1-\frac{1}{r-r_h-1}\| h\|_1\Big), \quad \text{a.e. in } A, \\ u_{f_r}(z)+\eta(z)\leq \frac{a^2}{2}\| f_0\|_\infty-\frac{1}{8\pi b} \| h\|_{-\infty}|\operatorname{supp}(h)|,\quad\text{a.e. in } D. \end{gather} Utilizing conditions mentioned in \eqref{necess-cond-supp}, there exists C<0 such that $$\label{13} u_{f_r}(z)+\eta(z)\leq C \quad\text{a.e. in } D.$$ If we make r large enough we derive from \eqref{10} and \eqref{13},$$ \big(u_{f_r}(x)+\eta(x)\big)-\big(u_{f_r}(z)+\eta(z)\big)>0,  for $x\in A$ and $z\in D$. Since $|A|>0$ and $|D|>0$, this is a contradiction to \eqref{6}. Let $r_0$ be such that $r>r_0$, support of $f_r$ does not touch the boundary of $B_r$ where $f_r$ is a solution of $P(r)$ for $r\geq r_0$. Then, $\operatorname{supp}(f_r)$ does not have any density point on the boundary of $B_r$ for $r>r_0$. This means that $\operatorname{supp}(f_r)$ has a positive distance from the boundary of $B_r$. Hence $\operatorname{supp}(f_r)\subset B(r_0)$. Therefore we obtain the inclusion $S(r_0)\subset S$. It yields that $P$ has a solution. \subsection*{Acknowledgement} The first author wants to thank Dr. Behrouz Emamizadeh for his useful suggestions. \begin{thebibliography}{00} \bibitem{Burton1} G. R. Burton; {Rearrangements of functions, maximization of convex functionals and vortex rings,} \emph{Math. Ann}. 276, 225-253(1987). \bibitem{Burton2} G. R. Burton; Variational problems on classes of rearrangements and multiple configurations for steady vorticies. \emph{Ann. Inst. H. Poincar\'{e} -- Anal. Non Lin\'{e}aire}, 6, 295-319(1989). \bibitem{fabriz} F. 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