\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Existence of solutions to an elliptic equation } { Marie-Fran\c{c}oise Bidaut-V\'{e}ron } \begin{document} \setcounter{page}{23} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems,\newline Electronic Journal of Differential Equations, Conference 08, 2002, pp 23--34. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Necessary conditions of existence for an elliptic equation with source term and measure data involving $p$-Laplacian % \thanks{ {\em Mathematics Subject Classifications:} 35J70, 35B45, 35D05. \hfil\break\indent {\em Key words:} Degenerate quasilinear equations, measure data, capacities, a priori estimates. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002.} } \date{} \author{Marie-Fran\c{c}oise Bidaut-V\'{e}ron} \maketitle \begin{abstract} We study the nonnegative solutions to equation $$-\Delta_{p}u=u^{q}+\lambda\nu,$$ in a bounded domain $\Omega$ of $\mathbb{R}^{N}$, where $1p-1$, $\nu$ is a nonnegative Radon measure on $\Omega$, and $\lambda>0$ is a parameter. We give necessary conditions on $\nu$ for existence, with $\lambda$ small enough, in terms of capacity. We also give a priori estimates of the solutions. \end{abstract} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction}\label{intro} Let $\Omega$ be a bounded regular domain in $\mathbb{R}^{N}$. We denote by $\mathcal{M}(\Omega)$ the set of Radon measures on $\Omega$, $\mathcal{M}^{+}(\Omega)$ the set of nonnegative ones, and by $\mathcal{M}_{b}(\Omega),\mathcal{M}_{b}^{+}(\Omega)$ the subsets of bounded ones. We consider the quasilinear elliptic problem with a source term: $$\begin{gathered} -\Delta_{p}u=-\mathop{\rm div}(| \nabla u| ^{p-2}\nabla u)=| u| ^{q-1}u+\mu,\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega, \end{gathered} \label{un}$$ with $1p-1$, and $\mu\in\mathcal{M}_{b}^{+}(\Omega)$. We look for conditions on the measure $\mu$ ensuring that the problem admits a nonnegative solution, and essentially in terms of capacity. In order to take account of the size of the measure, we will study the problem with $\mu=\lambda\nu,\quad\text{ }\lambda\geq0,$ where $\nu\in\mathcal{M}_{b}^{+}(\Omega)$ is fixed and $\lambda$ is a parameter. Recall a result of \cite{BaPi} in case $p=2$ , $N\geq3$, which gives a necessary and sufficient condition for existence: \begin{theorem}[\cite{BaPi}]\label{bapi}The following problem: $$\begin{gathered} -\Delta u=u^{q}+\lambda\nu,\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega, \end{gathered} \label{2la}%$$ where $\nu\in\mathcal{M}_{b}^{+}(\Omega)$, $\nu\neq0$, has a nonnegative solution (in the integral sense) if and only if $$\lambda\int_{\Omega}\varphi d\nu\leq\frac{q-1}{q^{q'}}\int_{\Omega }\varphi^{1-q'}(-\Delta\varphi)^{q'}dx,\label{star}$$ for any $\varphi\in W_{0}^{1,\infty}(\Omega)\cap W^{2,\infty}(\Omega)$ such that $-\Delta\varphi\geq0$, with compact support in $\Omega$. \end{theorem} Thus if $q$ is subcritical, that means $q1$, $\mathop{\rm cap}{}_{m,r}$ is the capacity associated to the Sobolev space $W_{0}% ^{m,r}({\Omega})$, defined by $\mathop{\rm cap}{}_{m,r}(K,{\Omega})=\inf\left\{ \left\| \psi\right\| _{W_{0}% ^{m,r}({\Omega})}^{r}:\psi\in\mathcal{D}({\Omega}),0\leq\psi \leq1,\psi=1\text{ on }K\text{ }\right\} .$ In fact it was proved in \cite{AdPi} that (\ref{cap2}) is also sufficient: \begin{theorem} [\cite{AdPi}] Assume that $\nu$ has a compact support in $\Omega$. Then problem (\ref{2la}) has a solution for any $\lambda\geq0$ small enough if and only if there exists $C>0$ such that (\ref{cap2}) holds. \end{theorem} Condition (\ref{cap2}) implies that $\mu$ does not charge the sets with $2,q'$- capacity zero. But it is stronger: if $q>N/(N-2)$ (resp. $q=N/(N-2)$), there exists a function $\nu\in L^{s}(\Omega)$ with $1\leq s0$. $\medskip$ Concerning problem (\ref{un}) with $p\neq2$, the question is much harder, because the full duality argument used in \cite{BaPi} cannot be used for the $p$-Laplacian. The first thing is to define a notion of solution, as it is the case for the problem without reaction term. In Section \ref{ens} we recall the usual notions of entropy solutions, which suppose that the measure is bounded; this leads to assume that $u^{q}\in L^{1}(\Omega)$. We denote by $\overline{P}=\frac{N(p-1)}{N-p}%$ the critical exponent linked to the $p$-Laplacian, and we set $q^{\ast}=q/(q-p+1),$ (hence $q^{\ast}=q'$ if $p=2$). In Section \ref{nec} we prove our main result: \begin{theorem} \label{T1} Let $\nu\in\mathcal{M}_{b}^{+}(\Omega)$ and $\lambda\geq0$. Assume that problem% $$\begin{gathered} -\Delta_{p}u=u^{q}+\lambda\nu,\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega, \end{gathered} \label{pla}$$ has a nonnegative entropy solution (hence $u^{q}\in L^{1}(\Omega)$). Then for any $R>pq^{\ast}$, there exists $C=C(N,p,q,R,\Omega)>0$ such that% $$\lambda\int_{\Omega}\varphi d\nu+\int_{\Omega}u^{q}\varphi\,dx \leq C\Big(\int_{\Omega}\varphi^{1-R}| \nabla\varphi| ^{R}dx\Big) ^{pq^{\ast}/R},\label{map}$$ for any $\varphi\in W_{0}^{1,p}(\Omega)\cap W^{1,s}(\Omega)$ $(s>N)$ such that $0\leq\varphi\leq1$ in $\Omega$. And for any $\alpha<0$, there exists $C=C(\alpha,N,p,q,R,\Omega)>0$ such that $$\int_{\Omega}(u+1)^{\alpha-1}|\nabla u|^{p}\varphi\, dx \leq C\Big(1+\int_{\Omega}u^{q}\varphi\,dx\Big) \Big( \int_{\Omega}\varphi^{1-R}| \nabla\varphi| ^{R}dx\Big) ^{p/R}.\label{mip}%$$ \end{theorem} This Theorem gives a priori estimate not only of the size of the measure, but also of the integral $\int_{\Omega}u^{q}\varphi\,dx$, independently on\textit{ }$u$. In the case $p=2$, this was first remarked by \cite{DFLN} when $\mu=0$ ; it was the starting point for proving $L^{\infty}$ universal estimates. It was also used in \cite{BiVi} and \cite{BiY} for obtaining a priori estimates with a general measure $\mu$. As a consequence we deduce the following: \begin{theorem} \label{T2} If problem (\ref{pla}) has a solution, then, for any $R>pq^{\ast}$, there exists $C=C(N,p,q,R,\Omega)>0$ such that% $$\lambda\int_{K}d\nu\leq C\;(\mathop{\rm cap}{}_{1,R} (K,{\Omega}))^{pq^{\ast}/R}, \quad\text{for every compact set K\subset\Omega}.\label{lup}$$ and if $\nu$ has a compact support in $\Omega$, there exists $C=C(N,p,q,R,\mu )>0$ such that $$\lambda\int_{K}d\nu\leq C\;(\mathop{\rm cap}{}_{1,R}(K,\mathbb{R}^{N}))^{pq^{\ast}/R}% ,\quad\text{for every compact set K\subset\Omega}.\label{lip}$$ In particular, if $q>\overline{P}$, then $\nu$ does not charge the point sets. Moreover for any $1\leq s0$, problem (\ref{pla}) admits no solution.\medskip \end{theorem} In Section \ref{open}, we mention some partially or fully open problems linked to this study. We refer to \cite{Bi2} for more complete results for problem (\ref{un}) with possible signed measure $\mu$, and for the problem with an absorption term% $$\begin{gathered} -\Delta_{p}u+| u| ^{q-1}u=\mu,\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega. \end{gathered} \label{abs}$$ \section{Entropy solutions}\label{ens} First recall some well-known results concerning the problem $$\begin{gathered} -\Delta_{p}u=\mu,\quad\text{in }\Omega,\\ u=0,\quad\text{on }\partial\Omega, \end{gathered} \label{basic}$$ with $\mu\in\mathcal{M}_{b}(\Omega)$. We set $P_{0}=\frac{2N}{N+1},\quad P_{1}=2-\frac{1}{N},$ so that $1P_{0}$ $\Longleftrightarrow\overline{P}>1$. When $p>P_{1}$, problem (\ref{basic}) admits at least a solution $u$ in the sense of distributions, such that $u\in W_{0}^{1,r}(\Omega)$ for any $1\leq r<\overline{P}$ . In the general case, one can define a notion of entropy or renormalized solutions in four equivalent ways, see \cite{DMOP}, which allow to give a sense to the gradient in any case: they are solutions such that $\nabla T_{k}(u)\in L_{loc}^{1}(\Omega)$ for any $k>0$, where $$T_{k}(s)=\begin{cases} s, &\text{if } |s| \leq k,\\ k \mathop{\rm sign}(s),& \text{if }|s| >k, \end{cases} \label{tk}$$ and the gradient of $u$, denoted by $y=\nabla u$ is defined by $$\nabla(T_{k}(u))=y\times1_{\left\{ | u| \leq k\right\} }% \quad\text{a.e. in }\Omega.\label{yg}%$$ For any $p>1$ there exists at least an entropy solution of (\ref{basic}), and it is unique if $\mu\in L^{1}(\Omega)$. Moreover any entropy solution satisfies the equation in the sense of distributions. The role of $P_{0}$ and $P_{1}$ is shown by the estimates \begin{gather*} u^{p-1} \in L^{s}(\Omega),\quad\text{for any }1\leq sP_{1}$and$u$itself is in$L^{1}(\Omega)$if and only if$p>P_{0}$. \medskip Recall that any measure$\mu\in\mathcal{M}_{b}(\Omega)$can be decomposed as $\mu=\mu_{0}+\mu_{s}^{+}-\mu_{s},%$ where$\mu_{0}\in\mathcal{M}_{0,b}(\Omega)$, set of bounded measures such that $$\mu_{0}(B)=0\quad\text{for any Borel set }B\subset\Omega\text{ such that }\mathop{\rm cap}{}_{1,p}(B,\Omega)=0;\label{mu0}%$$ and$\mu_{s}^{+},\mu_{s}^{-}$are nonnegative and concentrated on a set$E$with$\mathop{\rm cap}{}_{1,p}(E,\Omega)=0$. If$\mu\in\mathcal{M}_{b}^{+}(\Omega)$, then$\mu_{0}$is nonnegative, and$\mu=\mu_{0}+\mu_{s}^{+}$. \medskip We will use one of the four equivalent definitions of solution:$u$is an entropy solution if$u$is measurable and finite$a.e$. in$\Omega$, and $$T_{k}(u)\in W_{0}^{1,p}(\Omega)\quad\text{for every }k>0,\label{gtk}%$$ and the gradient defined by (\ref{yg}) satisfies | \nabla u| ^{p-1}\in L^{r}(\Omega),\quad\text{for any }1\leq rN$, such that $h(u)\varphi\in W_{0}^{1,p}(\Omega).\medskip$ In the same way, for given $\mu=\mu_{0}+\mu_{s}^{+}\in\mathcal{M}_{b}% ^{+}(\Omega)$, a nonnegative entropy solution $u$ of problem (\ref{un}) will be a measurable function $u$ such that $u^{q}\in L^{1}(\Omega)$ and $u$ is an entropy solution of problem $\begin{gathered} -\Delta_{p}u=\mu-u^{q}\quad\text{in }\Omega,\\ u=0\quad\text{on }\partial\Omega. \end{gathered}$ In particular \begin{equation*} \int_{\Omega}| \nabla u| ^{p-2}\nabla u.\nabla(h(u)\varphi )dx+\int_{\Omega}u^{q}h(u)\varphi\,dx=\int_{\Omega}h(u)\varphi d\mu _{0}+h(+\infty)\int_{\Omega}\varphi d\mu_{s}^{+}, \label{ploc}% \end{equation*} for any $h$ and $\varphi$ as above. \section{Proofs and comments}\label{nec} \paragraph{Proof of Theorem \ref{T1}} Let $\mu=\lambda\nu=\mu_{0}+\mu_{s}^{+}$, where $\mu_{0}\in\mathcal{M}_{0,b}(\Omega)$ and $\mu_{s}^{+}$ is singular, and let $\alpha\in\left( 1-p,0\right)$ be a parameter. For any $k>0$, we set $u_{k}=T_{k}(u)$, and, for any $\varepsilon\in\left( 0,k\right)$, $h_{\alpha,k,\varepsilon}(r)=(T_{k}(r^{+})+\varepsilon)^{\alpha} =\begin{cases} \varepsilon^{\alpha}, &\text{if }r\leq0,\\ (r+\varepsilon)^{\alpha}, &\text{if }0\leq r\leq k,\\ (k+\varepsilon)^{\alpha}, &\text{if }r\geq k. \end{cases}$ We choose in (\ref{ploc}) the test functions $h=h_{\alpha,k,\varepsilon}$, and $\varphi\in W_{0}^{1,p}(\Omega)\cap W^{1,s}(\Omega)$, with $s>N$ and $\varphi\geq0$ in $\Omega$, and obtain% \begin{align*} \int_{\Omega}(u_{k}&+\varepsilon)^{\alpha}\varphi d\mu_{0}+(k+\varepsilon )^{\alpha}\int_{\Omega}\varphi d\mu_{s}^{+}+\int_{\Omega}(u_{k}+\varepsilon )^{\alpha}u^{q}\varphi\,dx\\ +| \alpha|& \int_{\Omega}\int_{\Omega}(u_{k}+\varepsilon )^{\alpha-1}|\nabla u_{k}|^{p}\varphi\,dx\\ =&\int_{\Omega}(u_{k}+\varepsilon )^{\alpha}| \nabla u| ^{p-2}\nabla u.\nabla\varphi\,dx\\ \leq&\int_{\Omega}(u_{k}+\varepsilon)^{\alpha}| \nabla u_{k}| ^{p-1}| \nabla\varphi| dx+\int_{\left\{ u\geq k\right\} }% (u_{k}+\varepsilon)^{\alpha}| \nabla u| ^{p-1}| \nabla\varphi| dx\\ \leq&\frac{| \alpha| }{2}\int_{\Omega}(u_{k}+\varepsilon )^{\alpha-1}|\nabla u_{k}|^{p}\text{ }\varphi\,dx+C\int_{\Omega}(u_{k}% +\varepsilon)^{\alpha+p-1}\varphi^{1-p}| \nabla\varphi| ^{p}\text{ }dx\\ & +(k+\varepsilon)^{\alpha}\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx, \end{align*} where $C=C(\alpha)>0$. Now from H\"{o}lder inequality, setting $\theta=q/(p-1+\alpha)>1$, \begin{multline*} \int_{\Omega}(u_{k}+\varepsilon)^{\alpha+p-1}\varphi^{1-p}| \nabla\varphi| ^{p}\text{ }dx \\ \leq \Big( \int_{\Omega}(u_{k}+\varepsilon)^{q} \varphi\,dx\Big)^{1/\theta}\Big( \int_{\Omega} \varphi^{1-p\theta'}| \nabla\varphi| ^{p\theta' }dx\Big) ^{1/\theta'}. \end{multline*} In particular for any $k>1$, \begin{multline} \frac{| \alpha| }{2}\int_{\Omega}\int_{\Omega}(u_{k} +\varepsilon)^{\alpha-1}|\nabla u_{k}|^{p}\varphi\,dx\\ \leq C\Big( \int_{\Omega}(u_{k}+\varepsilon)^{q}\varphi\,dx\Big) ^{1/\theta}\Big( \int_{\Omega}\varphi^{1-p\theta'}| \nabla\varphi| ^{p\theta'}dx\Big) ^{1/\theta'} +\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx. \label{mir} \end{multline} Letting $\varepsilon$ tend to $0$, we get \begin{align} \frac{| \alpha| }{2}\int_{\Omega}u_{k}^{\alpha-1}|\nabla u_{k}|^{p}\varphi\,dx \leq &C\Big( \int_{\Omega}u_{k}^{q}\varphi\,dx\Big) ^{1/\theta}\Big( \int_{\Omega}\varphi^{1-p\theta'}| \nabla\varphi| ^{p\theta'}dx\Big) ^{1/\theta' }\nonumber\\ & +\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx. \label{mil} \end{align} Choosing now $h(u)=1$ in (\ref{ploc}), with the same $\varphi$, we find \begin{align} \int_{\Omega}\varphi d\mu_{0}&+\int_{\Omega}\varphi d\mu_{s}^{+}+\int_{\Omega }u^{q}\varphi\,dx=\int_{\Omega}| \nabla u| ^{p-2}\nabla u.\nabla\varphi\,dx \nonumber\\ \leq&\int_{\Omega}u_{k}^{(\alpha-1)/p'}| \nabla u| ^{p-1}u_{k}^{(1-\alpha)/p'}| \nabla\varphi| dx+\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx\nonumber\\ \leq&\Big( \int_{\Omega}u_{k}^{\alpha-1}| \nabla u_{k}| ^{p}\text{ }\varphi\,dx\Big) ^{1/p'}\Big( \int_{\Omega} u_{k}^{(1-\alpha)(p-1)}\varphi^{1-p}| \nabla\varphi| ^{p}dx\Big) ^{1/p}\nonumber\\ & +\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx.\label{zer}% \end{align} Since $q>p-1$, we can fix $\alpha\in(1-p,0)$ such that $\tau=q/(1-\alpha)(p-1)>1$. From (\ref{mil}) and (\ref{zer}), we derive% \begin{align*} \int_{\Omega}&\varphi d\mu+\int_{\Omega}u^{q}\varphi\,dx\\ \leq&\Big( \int_{\Omega}u_{k}^{\alpha-1}| \nabla u_{k}| ^{p}\text{ }\varphi\,dx\Big) ^{1/p'}\Big( \int_{\Omega}u_{k}% ^{q}\varphi\,dx\Big) ^{1/\tau p}\Big( \int_{\Omega}\varphi^{1-\tau 'p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{1/\tau'p}\\ & +\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx\\ \leq&\Big( C\Big( \int_{\Omega}u_{k}^{q}\varphi\,dx\Big) ^{1/\theta }\Big( \int_{\Omega}\varphi^{1-p\theta'}| \nabla \varphi| ^{p\theta'}dx\Big) ^{1/\theta'} +\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx\Big) ^{1/p'}\\ & \times\Big( \int_{\Omega}u_{k}^{q}\varphi\,dx\Big) ^{1/\tau p}\left( \int_{\Omega}\varphi^{1-\tau'p}| \nabla\varphi| ^{\tau'p}dx\right) ^{1/\tau'p}+\int_{\left\{ u\geq k\right\} }| \nabla u| ^{p-1}| \nabla\varphi| dx. \end{align*} Now we can let $k$ tend to $\infty$, since $u^{q}+| \nabla u| ^{p-1}\in L^{1}(\Omega)$. It follows that \begin{align} \label{cst} \int_{\Omega}\varphi d\mu+\int_{\Omega}u^{q}\varphi\,dx &\leq C\Big( \int_{\Omega}u^{q}\varphi\,dx\Big) ^{1/p'\theta+1/\tau p} \\ &\times \Big( \int_{\Omega}\varphi^{1-p\theta'}| \nabla\varphi| ^{p\theta'}dx\Big) ^{1/p'\theta'} \Big( \int_{\Omega}\varphi^{1-\tau'p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{1/\tau'p},\nonumber \end{align} with a new $C=C(\alpha,N,p,q)$. Since $1/\theta'p'+1/\tau'p=1/q^{\ast}=1-(1/\theta p'+1/\tau p)$, we find in particular \begin{align*} \Big( \int_{\Omega}&u^{q}\varphi\,dx\Big) ^{1-(p-1)/q} \\ =&\Big(\int_{\Omega}u^{q}\varphi\,dx\Big) ^{(1/p'\theta'+1/\tau'p)}\\ \leq& C\Big( \int_{\Omega}\varphi^{1-p\theta'}| \nabla\varphi| ^{p\theta'}dx\Big) ^{1/p'% \theta'}\Big( \int_{\Omega}\varphi^{1-\tau'p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{1/\tau'p}. \end{align*} Consequently \begin{align*} \int_{\Omega}&u^{q}\varphi \,dx\\ \leq &C\Big( \int_{\Omega}\varphi^{1-p\theta '}| \nabla\varphi| ^{p\theta'}dx\Big) ^{\tau'p/(\tau'p+p'\theta')}\Big( \int_{\Omega}\varphi^{1-\tau'p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{p'\theta'/(\tau'p+p'\theta')}. \end{align*} Notice that $\tau0$. Moreover, from (\ref{cst}) and (\ref{aut}), $\int_{\Omega}\varphi d\mu\leq C\Big( \int_{\Omega}\varphi^{1-\tau'% p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{(q^{\ast }-1+1/p'+1/p)/\tau'},%$ then $\int_{\Omega}\varphi d\mu+\int_{\Omega}u^{q}\varphi \,dx\leq C\Big( \int_{\Omega}\varphi^{1-\tau'p}| \nabla\varphi| ^{\tau'p}dx\Big) ^{q^{\ast}/\tau'}.$ We can choose $| \alpha|$ sufficiently small, such that $pq^{\ast}2q' has also a 2,q'- capacity zero, see \cite{AdHe}. The capacity involved in Theorem \ref{T1} is of order 1 instead of 2, because we cannot use the full duality argument of the linear case. However, observe that a point set has a 1,2q'- capacity zero if and only if q>N/(N-2), that means if and only if it has a 2,q'- capacity zero. \paragraph{Proof of Theorem \ref{T2}} Let \psi_{n}\in \mathcal{D}({\Omega}) such that 0\leq\psi_{n}\leq1 and \psi_{n}\geq\chi_{K} and \left\| \psi_{n}\right\| _{W^{1,R}({\Omega})}^{R} tends to \mathop{\rm cap}{}_{1,R}% (K,\Omega) as n tends to \infty. Choosing \varphi=\psi_{n}^{R} in (\ref{map}), we deduce that \[ \lambda\int_{K}d\nu\leq C\Big( \int_{\Omega}| \nabla\psi_{n}| ^{R}dx\Big) ^{pq^{\ast}/R}\leq C\left\| \psi_{n}\right\| _{W^{1,R}% ({\Omega})}^{R},$ with new constants $C=C(N,p,q,R,\Omega)$, and (\ref{lup}) follows. If $\nu$ has a compact support $X$ in $\Omega$, then (\ref{lip}) holds after localization on a neighborhood of $X$. Assume moreover that $q>\overline{P}$, then we can choose $R$ such that $pq^{\ast}0$ small enough, we derive $$\lambda\int_{B(x_{0},r)}d\nu\leq Cr^{N-R},\label{tru}$$ with $C=C(N,p,q,R,x_{0},\Omega)$. For any $1\leq s0$, $\lambda\nu$ does not satisfy (\ref{tru}) for $pq^{\ast}P_{0}$, the existence of solutions of problem (\ref{un}), with possibly signed measure $\mu$, is shown in \cite{Gr}. In the supercritical case, the problem is entirely open, even for $L^{s}$ functions. In particular it would be interesting to extend to the case $p\neq2$ a consequence of Theorem \ref{bapi}: \begin{theorem} [\cite{BaPi}]Assume that $N\geq3$, and $\nu\in L^{s}(\Omega)$, for some $s\geq1$. If $q>N/(N-2)$ and $s\geq N/2q'$, or $q=N/(N-2)$ and $s>N/2q'$, then problem (\ref{2la}) has a solution for $\lambda$ small enough.\medskip \end{theorem} \noindent\textbf{Problem 2:} Can we solve problems (\ref{basic}) and (\ref{pla}) if $\mu$ is not bounded?\\ Let us begin by the case without reaction term. For any $x\in\Omega$, denote by $\rho(x)$ the distance from $x$ to $\partial\Omega$. When $p=2$, problem (\ref{basic}) is well posed in fact for any measure $\mu$, possibly unbounded, such that $\int_{\Omega}\rho d| \mu| <\infty:$ it admits a unique integral solution $$u(x)=G(\mu)=\int_{\Omega}\mathcal{G}(x,y)d\mu(y),\label{int}$$ where $\mathcal{G}$ is the Green kernel. And $u$ is also the weak solution of the problem in the sense that $u\in L^{1}(\Omega)$ and $$\int_{\Omega}u(-\Delta\xi)dx=\int_{\Omega}\xi d\mu,\label{weak}%$$ for any $\xi\in C^{1}(\overline{\Omega})$ vanishing on $\partial\Omega$ with $\nabla\xi$ is Lipschitz continuous, see \cite{BiVi}. The case where $\mu$ is a function $f$, such that $\int_{\Omega}\rho fdx<\infty$, was first considered by Br\'{e}zis, see \cite{Ve}. The problem is open when $p\neq2:$ up to now we have no existence results concerning equation (\ref{basic}) when $\mu$ may be unbounded, even in the case $p>P_{1}$, where the definition of the gradient does not cause any problem. Now let us consider the problem with source term. When $p=2$, it was studied in \cite{KaVb} and specified in \cite{BrCa}: \begin{theorem}[\cite{KaVb}] Let $\nu\in\mathcal{M}^{+}(\Omega)$, $\nu\neq0$ such that $\int_{\Omega}\rho d\nu<\infty$. Then problem (\ref{2la}) has a solution such that $G(u^{q})<\infty$, $a.e$. in $\Omega$, for any $\lambda\geq0$ small enough, if and only if there exists $C>0$ such that $$G(G^{q}(\nu))\leq CG(\nu),\quad a.e.\text{ in }\Omega.\label{gns}%$$ \end{theorem} Notice that condition $G(u^{q})<\infty$ $a.e$. in $\Omega$, is satisfied as soon as $\int_{\Omega}\rho fu^{q}dx<\infty$, and the solutions are taken in the integral sense. More recently new existence results and a priori estimates were given in \cite{BiY}, covering the case of measures $\mu$ such that $\int_{\Omega}\rho^{\gamma}d\mu<\infty$ for some $0\leq\gamma\leq1$. Condition (\ref{gns}) allows to obtain a supersolution, and then a solution by using an iterative scheme. It is available for much more general linear operators, see \cite{KaVb} and \cite{Vb}. It seems to be difficult to extend to nonlinear ones, since it is based on a representation formula. However Kalton and Verbitski \cite{KaVb} also gave necessary and sufficient in terms of capacity with weights, extending the result of \cite{AdPi} to general measures: \begin{theorem}[\cite{KaVb}] Let $\nu\neq0$ be a nonnegative Radon measure on $\Omega$. Then problem (\ref{2la}) has a solution for any $\lambda\geq0$ small enough if and only if there exists $C>0$ such that $\int_{K}d\nu\leq C\mathop{\rm cap}{}_{2,q',\rho}(K), \quad\text{ for every compact set K\subset\Omega},$ where $\mathop{\rm cap}{}_{2,q',\rho}(K)=\inf\Big\{ \int_{\Omega}w^{q'}% \rho^{1-q'}dx:w\geq0,\quad Gw\geq\rho\chi_{K}\quad a.e.\text{ in }\Omega\Big\} .$ \end{theorem} One can ask if results of this type can be obtained for the $p$-Laplacian, using capacities of order 1 with suitable weights. \begin{thebibliography}{99} \bibitem{AdHe}D. R. Adams and L. I. Hedberg, Functions spaces and potential theory, Grundlehren der Mathematischen wissenchaften, Springer Verlag, Berlin, 314 (1996). \bibitem{AdPi}D. 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