Quasilinear and Hessian Lane-Emden type systems with measure data

TL;DR

This paper investigates nonlinear PDE systems involving p-Laplacian and Hessian operators with measure data, establishing conditions for solutions based on capacity theory in bounded domains and Euclidean space.

Contribution

It provides necessary and sufficient existence conditions for complex nonlinear systems with measure data using capacity criteria, extending previous results to new operators.

Findings

Existence conditions characterized by Riesz or Bessel capacities.

Results apply to systems with measure data in bounded domains and ^N.

Includes systems involving p-Laplacian and Hessian operators.

Abstract

We study nonlinear systems of the form $-\Delta\_pu=v^{q\_1}+\mu,\;-\Delta\_pv=u^{q\_2}+\eta$ and $F\_k[-u]=v^{s\_1}+\mu,\;F\_k[-v]=u^{s\_2}+\eta$ in a bounded domain $\Omega$ or in $\mathbb{R}^N$ where $\mu$ and $\eta$ are nonnegative Radon measures, $\Delta\_p$ and $F\_k$ are respectively the $p$-Laplacian and the $k$-Hessian operators and $q\_1$, $q\_2$, $s\_1$ and $s\_2$ positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.

Equations

$${\Delta }_{p}u:=div\left(|\nabla u{|}^{p-2}\nabla u\right)$$

$$F_k[u]=\sum _{1\le j_1<j_2<...<j_k\le N}{\lambda }_{j_1}{\lambda }_{j_2}...{\lambda }_{j_k}$$

$$\displaystyle\begin{array}[]{ll}-\Delta_{p}u=u^{q}+\mu&\text{in }\Omega\\ \phantom{-\Delta_{p}}u=0&\text{on }\partial\Omega,\end{array}$$

$$\displaystyle\begin{array}[]{ll}F_{k}[-u]=u^{q}+\mu&\text{in }\Omega\\ \phantom{F_{k}[-]}u=0&\text{on }\partial\Omega.\end{array}$$

$$\begin{array}[]{ll}\mu(E)\leq c_{1}\text{Cap}_{\mathbf{G}_{p},\frac{q}{q+1-p}}(E)\qquad\text{for all compact set }E\subset\Omega,\end{array}$$

$$\begin{array}[]{ll}{\displaystyle\int_{B}}\left({\bf W}^{R}_{1,p}[\mu_{B}](x)\right)^{q}dx\leq c_{2}\mu(B)\qquad\text{for all ball }B\text{ s.t. }B\cap\text{supp}\mu\not=\emptyset,\end{array}$$

$$\mu (E)\le c_3{\text{Cap}}_{{\mathbf{G}}_{2k},\frac{q}{q-k}}(E)\text{for all compact set }E\subset \Omega ,$$

$${\int }_B{\left[{\mathbf{W}}_{\frac{2k}{k+1},k+1}^R[{\mu }_B(x)]\right]}^{q}dx\le c_4\mu (B)\text{for all ball }B\text{ s.t. }B\cap \text{supp}\mu \ne \varnothing ,$$

$$\displaystyle\begin{array}[]{ll}-\Delta_{p}u=v^{q_{1}}+\mu&\text{in }\Omega\\ -\Delta_{p}v=u^{q_{2}}+\eta&\text{in }\Omega\\ \phantom{-\Delta_{p}}u=v=0&\text{on }\partial\Omega,\end{array}$$

$$\displaystyle\begin{array}[]{ll}F_{k}[-u]=v^{s_{1}}+\mu&\text{in }\Omega\\ F_{k}[-v]=u^{s_{2}}+\eta&\text{in }\Omega\\ \phantom{F_{k}[-]}u=v=0&\text{on }\partial\Omega,\end{array}$$

$$\left.\begin{array}[]{ll}-\Delta_{p}u=v^{q_{1}}+\mu&\text{in }\;\mathbb{R}^{N}\\ -\Delta_{p}v=u^{q_{2}}+\eta&\text{in }\;\mathbb{R}^{N},\\ \end{array}\right.$$

$$\eta (E)+{\int }_E{\left({\mathbf{W}}_{1,p}[\mu ](x)\right)}^{q_2}dx\le c_5{\mathrm{Cap}}_{{\mathbf{I}}_{\frac{p(q_1+p-1)}{q_1},\frac{q_1{q}_2}{q_1{q}_2-(p-1{)}^2}}}(E)\text{ for all Borel sets }E.$$

$$v\le c_8{\mathbf{W}}_{1,p}[\omega ],u\le c_9{\mathbf{W}}_{1,p}[{\left({\mathbf{W}}_{1,p}[\omega ]\right)}^{q_1}]+c_7{\mathbf{W}}_{1,p}[\mu ]$$

$$\mu (E)+{\int }_E{\left({\mathbf{W}}_{1,p}[\eta ](x)\right)}^{q_1}dx\le c_5^{\prime }{\mathrm{Cap}}_{{\mathbf{I}}_{\frac{p(q_2+p-1)}{q_2},\frac{q_1{q}_2}{q_1{q}_2-(p-1{)}^2}}}(E)\text{ for all Borel sets }E.$$

$${\mathrm{Cap}}_{{\mathbf{I}}_{\alpha ,\beta }}(K)=0\forall K\text{ compact},$$

$$\frac{p(q_1{q}_2+(p-1)\max \{q_1,q_2\})}{q_1{q}_2-(p-1{)}^2}\ge N.$$

$$\left.\begin{array}[]{ll}-\Delta_{p}u\geq v^{q_{1}}&\text{in }\;\mathbb{R}^{N}\\ -\Delta_{p}v\geq u^{q_{2}}&\text{in }\;\mathbb{R}^{N},\\ \end{array}\right.$$

$$\left.\begin{array}[]{ll}-\Delta_{p}u=v^{q_{1}}+\mu&\text{in }\;\Omega\\ -\Delta_{p}v=u^{q_{2}}+\eta&\text{in }\;\Omega\\ \phantom{-\Delta_{p}}u=v=0&\text{on }\;\partial\Omega,\end{array}\right.$$

$$\eta (E)+{\int }_E{\left({\mathbf{W}}_{1,p}^{\frac{d(x)}{4}}[\mu ](x)\right)}^{q_2}dx\le c_{{}_{10}}{\mathrm{Cap}}_{{\mathbf{G}}_{\frac{p(q_1+p-1)}{q_1},\frac{q_1{q}_2}{q_1{q}_2-(p-1{)}^2}}}(E)\text{ for all Borel sets }E\subset K.$$

$$\eta (K)+{\int }_K{\left({\mathbf{W}}_{1,p}^{2R}[\mu ]\right)}^{q_2}dx\le c_{{}_{11}}{\mathrm{Cap}}_{{\mathbf{G}}_{\frac{p(q_1+p-1)}{q_1},\frac{q_1{q}_2}{q_1{q}_2-(p-1{)}^2}}}(K),$$

$$v\le c_{{}_{13}}{\mathbf{W}}_{1,p}^R[\omega ],u\le c_{{}_{14}}{\mathbf{W}}_{1,p}^R[{\left({\mathbf{W}}_{1,p}^R[\omega ]\right)}^{q_1}]+c_{{}_{12}}{\mathbf{W}}_{1,p}^R[\mu ]$$

$${\mathrm{Cap}}_{{\mathbf{G}}_{\alpha ,\beta }}(\{x_0\})>0$$

$$\left.\begin{array}[]{ll}-\Delta_{p}u=v^{q_{1}}+\mu&\text{in }\;\Omega\\ -\Delta_{p}v=u^{q_{2}}+a\delta_{x_{0}}&\text{in }\;\Omega\\ \phantom{-\Delta_{p}}u=v=0&\text{on }\;\partial\Omega,\end{array}\right.$$

$$\begin{array}[]{lll}(i)&N<{\displaystyle\frac{pq_{2}(q_{1}+p-1)}{q_{1}q_{2}-(p-1)^{2}}},\\[8.53581pt] (ii)&a\leq c_{{}_{15}},\\[5.69054pt] (iii)&{\displaystyle\int_{K}}\left(\mathbf{W}^{2R}_{1,p}[\mu]\right)^{q_{2}}dx\leq c_{{}_{16}}.\end{array}$$

$$v\le c_{{}_{21}}{W}_{1,p}^R[\omega ],u\le c_{{}_{22}}{\mathbf{W}}_{1,p}^R[{\left(W_{1,p}^R[\omega ]\right)}^{q_1}]+c_{{}_{20}}{\mathbf{W}}_{1,p}^R[\mu ]$$

$$W_{1,p}^R[\omega ]={\mathbf{W}}_{1,p}^R\left[{\left({\mathbf{W}}_{1,p}^R[\mu ]\right)}^{q_2}\right]+a^{\frac{1}{p-1}}{\left({\left|x-x_0\right|}^{-\frac{N-p}{p-1}}-R^{-\frac{N-p}{p-1}}\right)}_{+}.$$

$$F_k[u]=S_k(\lambda (D^{2}u)),$$

$$S_k(\lambda )=\sum _{1\le i_1<...<i_k\le N}{\lambda }_{i_1}...{\lambda }_{i_k}.$$

$$F_k[u]={\left[D^{2}u\right]}_k,$$

$$S_{k-1}(\kappa )\ge c_0>0on\partial \Omega ,$$

Full text

2010 Mathematics Subject Classification. 35J70, 35J60, 45G15, 31C15.

Key words: $p$-Laplacian, $k$-Hessian, Bessel and Riesz capacities, measures, maximal functions.

1. 1 Introduction and Main results
2. 2 Estimates on potentials
3. 3 Quasilinear Dirichlet problems
4. 4 p-superharmonic functions and quasilinear equations in $\mathbb{R}^{N}$
5. 5 Hessian equations
6. 6 Further results

1 Introduction and Main results

Let $\Omega\subset\mathbb{R}^{N}$ be either a bounded domain or the whole $\mathbb{R}^{N}$, $p>1$ and $k\in\{1,2,...,N\}$. We denote by

[TABLE]

the p-Laplace operator and by

[TABLE]

the k-Hessian operator where $\lambda_{1},...,\lambda_{N}$ are the eigenvalues of the Hessian matrix $D^{2}u$. In the work [20], Phuc and Verbitsky obtained necessary and sufficient conditions for existence of nonnegative solutions to the following equations

[TABLE]

and

[TABLE]

Their conditions involve the continuity of the measures with respect to Bessel or Riesz capacities and Wolff potentials estimates. For example, if $\Omega$ is bounded and $\mu$ has compact support in $\Omega$, they proved that it is equivalent to solve (1.3), or to have

[TABLE]

for some constant $c_{1}>0$ where $\text{Cap}_{\mathbf{G}_{p},\frac{q}{q+1-p}}$ is a Bessel capacity, or to have

[TABLE]

for some constant $c_{2}>0$, where $R=2\text{ diam}(\Omega)$ and ${\bf W}^{R}_{1,p}[\mu_{B}]$ denotes the $R$-truncated Wolff potential of the measure $\mu_{B}=\chi_{{}_{B}}\mu$. Concerning the k-Hessian operator in a bounded $(k-1)$-convex domain $\Omega$, they proved that if $\mu$ has compact support, the problem (1.6) with $q>k$ admits a nonnegative solution if and only if

[TABLE]

for some $c_{3}$. In turn this condition is equivalent to

[TABLE]

for some $c_{4}>0$. The results concerning the linear case $p=2$ and $k=1$, can be found in [2, 3, 27].

The natural counterpart of equation (1.3) and (1.6) for systems:

[TABLE]

and

[TABLE]

where $q_{1},q_{2}>p-1,s_{1},s_{2}>k$ and $\mu,\eta$ are Radon measures. If $\Omega=\mathbb{R}^{N}$, we consider the same equations, except that the boundary conditions are replaced by $\inf_{\mathbb{R}^{N}}u=\inf_{\mathbb{R}^{N}}v=0$ and our statements involve the Riesz potentials and their associated capacities $\operatorname{Cap}_{I_{\alpha,\beta}}$. Our main results are the following.

Theorem A Let $1<p<N$, $q_{1},q_{2}>0$ and $q_{2}q_{1}>(p-1)^{2}$. Let $\mu,\eta$ be nonnegative Radon measures in $\mathbb{R}^{N}$. If the following system

[TABLE]

admits a nonnegative p-superharmonic solution $(u,v)$ then there exists a positive constant $c_{5}$ depending on $N,p,q_{1},q_{2}$ such that

[TABLE]

Conversely, if $\mu$ and $\eta$ are bounded, there exists $c_{6}>0$ depending on $N,p,q_{1},q_{2}$ such that if $0<q_{1}<\frac{N(p-1)}{N-p}$ and (1.20) holds with $c_{5}$ replaced by $c_{6}$, then (1.19) admits a nonnegative p-superharmonic solution $(u,v)$ satisfying

[TABLE]

in $\mathbb{R}^{N}$ for some $c_{7},c_{8},c_{9}>0$ where $d\omega=\left(\mathbf{W}_{1,p}[\mu]\right)^{q_{2}}dx+d\eta$ .

We notice that the left-hand side in (1.20) is not symmetric in $\eta$ and $\mu$ and the capacity in the right-hade side is not symmetric in $q_{1}$ and $q_{2}$. Hence the following symmetrized inequality holds

[TABLE]

It is known that

[TABLE]

if $\alpha\beta\geq N$, the first part of above implies the following Liouville theorem, obtained by another method in [9, Th 5.3 -(i)].

Corollary B Assume that

[TABLE]

Any nonnegative p-superharmonic solution $(u,v)$ of inequalities

[TABLE]

is trivial, i.e. $u=v=0$.

Classical Liouville results for one equation or inequality, are proved in [4], [5], [11], [22].

When $\Omega$ is bounded domain, we have a similar result in which we denote by $d$ the distance function to the boundary $x\mapsto d(x)=\mbox{\rm dist}\,(x,\partial\Omega)$.

Theorem C Let $1<p<N$, $q_{1},q_{2}>0$ and $q_{2}q_{1}>(p-1)^{2}$. Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain and $\mu,\eta$ nonnegative Radon measures in $\Omega$. If the following problem

[TABLE]

admits a nonnegative renormalized solution $(u,v)$, then then for any compact set $K\subset\Omega$, there exists a positive constant $c_{{}_{10}}$ depending on $N,p,q_{1},q_{2}$ and $\text{dist}(K,\partial\Omega)$ such that

[TABLE]

Conversely, let $\mu$ and $\eta$ be bounded with the property that there exists $c_{{}_{11}}>0$ depending on $N,p,q_{1},q_{2}$ and $R=2diam\,(\Omega)$ such that if $0<q_{1}<\frac{N(p-1)}{N-p}$ and

[TABLE]

for all compact set $K\subset\Omega$, then (1.24) admits a nonnegative renormalized solution $(u,v)$ satisfying

[TABLE]

in $\Omega$, where $d\omega=\left(\mathbf{W}_{1,p}^{R}[\mu]\right)^{q_{2}}dx+d\eta$.

It is known that

[TABLE]

if and only if $\alpha\beta>N$. Thus, as an application in a partially subcritical case we have,

Corollary D Let the assumptions on $p$, $q_{1}$, $q_{2}$, $\Omega$ and $R$ of Theorem C be satisfied, $x_{0}\in\Omega$, $a>0$ and $\mu$ be a nonnegative Radon measures in $\Omega$. If the following problem

[TABLE]

admits a nonnegative renormalized solution $(u,v)$, then there exist positive constants $c_{{}_{15}}=c_{{}_{15}}(N,p,q_{1},q_{2},d(x_{0}))$ and, for any compact subset $K$ of $\Omega$, c_{{}_{16}}=c_{{}_{16}}(N,p,q_{1},q_{2},\mbox{\rm dist}\,(K,\partial\Omega){\color[rgb]{1,0,0})}, such that

[TABLE]

Conversely, assuming that $\mu$ is bounded, there exist positive constants $c_{{}_{17}}=c_{{}_{17}}(N,p,q_{1},q_{2},d(x_{0}))$, $c_{{}_{18}}=c_{{}_{18}}(N,p,q_{1},q_{2})$ such that if $0<q_{1}<\frac{N(p-1)}{N-p}$ and $(\ref{sub1})$ holds with $c_{{}_{15}}$ and $c_{{}_{16}}$ replaced respectively by $c_{{}_{17}}$ and $c_{{}_{18}}$, then there exists a nonnegative renormalized solution $(u,v)$ of $(\ref{2hvMT1a1})$ satisfying

[TABLE]

in $\Omega$, where

[TABLE]

Concerning the $k$-Hessian operator we recall some notions introduced by Trudinger and Wang [23, 24, 25], and we follow their notations. For $k=1,...,N$ and $u\in C^{2}(\Omega)$ the k-Hessian operator $F_{k}$ is defined by

[TABLE]

where $\lambda(D^{2}u)=\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{N})$ denotes the eigenvalues of the Hessian matrix of second partial derivatives $D^{2}u$ and $S_{k}$ is the k-th elementary symmetric polynomial that is

[TABLE]

Since ${D^{2}}u$ is symmetric, it is clear that

[TABLE]

where we denote by $[A]_{k}$ the sum of the k-th principal minors of a matrix $A=(a_{ij})$. In order that there exists a smooth k-admissible function which vanishes on $\partial\Omega$, the boundary $\partial\Omega$ must satisfy a uniformly (k-1)-convex condition, that is

[TABLE]

for some positive constant $c_{0}$, where $\kappa=(\kappa_{1},\kappa_{2},...,\kappa_{n-1})$ denote the principal curvatures of $\partial\Omega$ with respect to its inner normal. We also denote by $\Phi^{k}(\Omega)$ the class of upper-semicontinuous functions $\Omega\to[Õ-\infty,\infty)$ which are $k$-convex, or subharmonic in the Perron sense (see Definition 5.1). In this paper we prove the following theorem (in which expression $\mathbb{E}[q]$ is the largest integer less or equal to $q$)

Theorem E Let $2k<N,s_{1},s_{2}>0$, $s_{1}s_{2}>k^{2}$. Let $\Omega$ be a bounded uniformly (k-1)-convex domain in $\mathbb{R}^{N}$ with diameter $R$. Let $\mu=\mu_{1}+f$ and $\eta=\eta_{1}+g$ be nonnegative Radon measures where $\mu_{1},\eta_{1}$ has compact support in $\Omega$ and $f,g\in L^{l}(\Omega)$ for some $l>\frac{N}{2k}$. If the following problem

[TABLE]

admits a nonnegative solutions $(u,v)$, continuous near $\partial\Omega$, with $-u$ and $-v$ elements of $\Phi^{k}(\Omega)$, then for any compact set $K\subset\Omega$, there exists a positive constant $c_{{}_{23}}$ depending on $N,k,s_{1},s_{2}$ and $dist(K,\partial\Omega)$ such that there holds

[TABLE]

Conversely,, if $\mu$ and $\eta$ are bounded, there exist a positive constant $c_{{}_{24}}$ depending on $N,k,s_{1},s_{2}$ and $diam\,(\Omega)$ such that, if $k\leq s_{1}<\frac{Nk}{N-2k}$ and

[TABLE]

for all Borel set $K\subset\Omega$, then (1.31) admits a nonnegative solution $(u,v)$, continuous near $\partial\Omega$, with $-u,-v\in\Phi^{k}(\Omega)$ satisfying

[TABLE]

in $\Omega$ for some constants $c_{j}$ ($j=27,28,29$) depending on $N,k,s_{1},s_{2}$, and $diam\,(\Omega)$.

If $\Omega$ is replaced by the whole space we prove,

Theorem F Let $2k<N,s_{1},s_{2}>0$, $s_{1}s_{2}>k^{2}$. Let $\mu,\eta$ be a nonnegative Radon measures in $\mathbb{R}^{N}$. If the following problem

[TABLE]

admits a nonnegative solutions $(u,v)$ with $-u$ and $-v$ belonging to $\Phi^{k}(\mathbb{R}^{N})$, then there exists a positive constant $c_{{}_{30}}$ depending on $N,k,s_{1},s_{2}$ such that there holds

[TABLE]

Conversely,, if $\mu$ and $\eta$ are bounded, there exists positive constant $c_{{}_{31}}$ depending on $N,k,s_{1},s_{2}$ such that, if $0<s_{1}<\frac{Nk}{N-2k}$ and (1.36) holds with $c_{{}_{31}}$ instead of $c_{{}_{30}}$, then (1.35) admits a nonnegative solution $(u,v)$ with $-u$ and $-v$ in $\Phi^{k}(\mathbb{R}^{N})$ satisfying

[TABLE]

in $\mathbb{R}^{N}$, where the $c_{j}$ ($j=32,33,34$) depend on $N,k,s_{1},s_{2}$.

As in the p-Laplace case, we have a Liouville property for Hessian systems.

Corollary G Assume that

[TABLE]

Any nonnegative solution (u,v) of inequalities

[TABLE]

with $-u$ and $-v$ in $\Phi^{k}(\mathbb{R}^{N})$ is trivial.

2 Estimates on potentials

Throughout this article $c_{{}_{j}}$, j=1,2,…, denote structural positive constants and $c_{{}_{N}}$ is the volume of the unit ball in $\mathbb{R}^{N}$. The following inequality will be used several times in the sequel.

Lemma 2.1

Let $\kappa,\gamma,\theta\in\mathbb{R}$, such that $\kappa,\gamma>0$. Let $h:(0,\infty)\to(0,\infty)$ be nondecreasing. Then,

[TABLE]

for some $c_{{}_{35}}>0$ depending on $\kappa$, $\gamma$, $\theta$.

Proof. Case 1: $\gamma\leq 1$. Since there holds

[TABLE]

we deduce

[TABLE]

where $c_{{}_{\gamma,\theta}}=2^{\frac{\gamma}{4}}\max\{1,2^{-\frac{\gamma\theta}{4}}\}$ and $2^{\frac{j_{0}}{4}}t<R\leq 2^{\frac{j_{0}+1}{4}}t$ if $R<\infty$ and $j_{0}=\infty$ if $R=\infty$. By Fubini’s theorem,

[TABLE]

which is (2.1).

Case 2: $\gamma>1$. Since

[TABLE]

we obtain

[TABLE]

by Fubini’s theorem, which completes the proof.          $\square$

We recall that if $\alpha>0$, $1<\beta<\frac{N}{\alpha}$ and $\mu$ belongs to the set of positive Radon measures in $\mathbb{R}^{N}$ that we denote $\mathfrak{M}^{+}(\mathbb{R}^{N})$, the Wolff potential of $\mu$ is defined by

[TABLE]

and if $R>0$, the $R$-truncated Wolff potential of $\mu$ is

[TABLE]

If $\mu$ is a Radon measure on a Borel set $G$, it’s Wolff potential (or truncated Wolff potential) is the potential of its extension by [math] in $G^{c}$. We start with the following composition estimate on Wolff potentials.

Lemma 2.2

Let $1<\beta<N/\alpha$. Then for any $q>0$ and $\mu\in\mathfrak{M}^{+}(\mathbb{R}^{N})$ we have

[TABLE]

in $\mathbb{R}^{N}$ for some $c_{{}_{36}}>0$ depending on $\alpha,\beta,N,q$. Moreover, if $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$, there holds

[TABLE]

in $\mathbb{R}^{N}$, where $c_{{}_{37}}>0$ depends on $\alpha,\beta,N,q$.

Proof. For any $x\in\mathbb{R}^{N}$, using the fact if $y\in B_{t}(x)$ then $B_{t}(x)\subset B_{2t}(y)$, we have

[TABLE]

where $c_{{}_{38}}=c_{{}_{38}}(\alpha,\beta,N,q)>0$, which proves (2.4).

In order to prove (2.4) we recall the following estimate on Wolff potentials [7]

[TABLE]

where $L^{\frac{(p-1)N}{N-\alpha\beta},\infty}$ denotes the weak-$L^{\frac{(p-1)N}{N-\alpha\beta}}$ space. In particular, since $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$,

[TABLE]

Applying this inequality to $\omega=\chi_{B_{2r}(x)}\mu$ yields

[TABLE]

We claim that

[TABLE]

Since $B_{r}(y)\subset B_{2r}(x)$ for any $y\in B_{t}(x),r\geq t$, we have

[TABLE]

Hence,

[TABLE]

Using Lemma 2.1, we infer

[TABLE]

which completes the proof.           $\square$

The following is a version of Lemma 2.2 for truncated Wolff potentials,

Lemma 2.3

Let $1<\beta<N/\alpha$ and $q>0$. If $\delta\in(0,1)$ there holds for any $\mu\in\mathfrak{M}^{+}(\mathbb{R}^{N})$

[TABLE]

in $\Omega$. Moreover, if $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$, there holds for any $\mu\in\mathfrak{M}^{+}(\mathbb{R}^{N})$,

[TABLE]

in $\mathbb{R}^{N}$.

Proof. For any $x\in\Omega$,

[TABLE]

Since $\delta d(y)\geq\frac{7\delta}{8}d(x)$ for all $y\in B_{\frac{t}{8}}(x)$, provided $0<t<\delta d(x)$,

[TABLE]

Hence

[TABLE]

which implies (2.10).

Because of (2.8), it is sufficient to prove that there holds

[TABLE]

in order to obtain (2.11). Since $B_{\rho}(y)\subset B_{2\rho}(x)$ for any $y\in B_{r}(x),\rho\geq r$, we have

[TABLE]

Therefore

[TABLE]

We infer (2.12) by Lemma 2.1, which completes the proof.           $\square$

The next two propositions link Wolff potentials of a measure with Riesz capaciticies (in the case of whole space) and truncated Wolff potentials with Bessel capaciticies (in the bounded domain case). Their proof can be found in [20, 21] (and [8] with a different method).

Proposition 2.4

Let $1<\beta<N/\alpha$, $q>\beta-1$, $\nu\in\mathfrak{M}^{+}(\mathbb{R}^{N})$. Then, the following statements are equivalent:

(a) The inequality

[TABLE]

holds for any compact set $K\subset\mathbb{R}^{N}$, for some $c_{{}_{48}}>0$.

(b) The inequality

[TABLE]

holds for any ball $B_{t}(x)\subset\mathbb{R}^{N}$, for some $c_{{}_{49}}>0$.

(c) The inequality

[TABLE]

holds for some $c_{{}_{50}}>0$.

Proposition 2.5

Let $1<\beta<N/\alpha$, $q>\beta-1$, $R>0$ and $\nu\in\mathfrak{M}_{b}^{+}(B_{R}(x_{0}))$ for some $x_{0}\in\mathbb{R}^{N}$. Then, the following statements are equivalent:

(a) The inequality

[TABLE]

holds for any compact set $K\subset\mathbb{R}^{N}$, for some $c_{{}_{51}}=c_{{}_{51}}(R)>0$.

(b) The inequality

[TABLE]

holds for any ball $B_{t}(x)\subset\mathbb{R}^{N}$, for some $c_{{}_{52}}=c_{{}_{52}}(R)>0$.

(c) The inequality

[TABLE]

holds for some $c_{{}_{53}}=c_{{}_{53}}(R)>0$.

In the following statement we obtain capacitary estimates on combination of measures.

Proposition 2.6

Let $\eta,\mu$ be in $\mathfrak{M}^{+}(\mathbb{R}^{N})$. Assume that $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$ and $qs>(\beta-1)^{2}$.

(i) If there holds

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$, then

[TABLE]

where $\omega=\left(\mathbf{W}_{\alpha,\beta}[\mu]\right)^{s}+\eta$.

(ii) If there holds

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$, then

[TABLE]

where $\omega=\chi_{{}_{B_{R}(x_{0})}}\left(\mathbf{W}^{2R}_{\alpha,\beta}[\mu]\right)^{s}+\chi_{{}_{B_{R}(x_{0})}}\eta$.

Proof. Statement (i): We assume that (2.19) holds. Put $\omega=\left(\mathbf{W}_{\alpha,\beta}[\mu]\right)^{s}+\eta$ and apply (2.19) to $K=\overline{B}_{2\rho}(x)$. Since by homogeneity

[TABLE]

we deduce from (2.19)

[TABLE]

which is equivalent to

[TABLE]

We apply Proposition 2.4 to $\nu=\omega$ with $(\alpha,\beta,q)=\left(\frac{\alpha\beta(q+\beta-1)}{q+(\beta-1)^{2}},\frac{(\beta-1)^{2}}{q}+1,s\right)$, (2.19) implies

[TABLE]

By Lemma 2.2, (2.20) is equivalent to

[TABLE]

Therefore, it is enough to show that (2.23) and (2.24) imply (2.25). In fact, since for $t>0$

[TABLE]

we apply (2.24) and obtain

[TABLE]

So, it is enough to show that

[TABLE]

Since $B_{r}(y)\subset B_{2r}(x)$ for any $y\in B_{t}(x),r\geq t$, we have

[TABLE]

It follows from Lemma 2.1 and (2.23) that

[TABLE]

which is (2.26).

Statement (ii): We assume that (2.21) holds. Put $d\omega=\chi_{{}_{\Omega}}\left(\mathbf{W}_{\alpha,\beta}[\mu]\right)^{s}+\chi_{{}_{\Omega}}\eta$, then

[TABLE]

As in the proof of statement (i), the above inequality is equivalent to

[TABLE]

Applying Proposition 2.5 with $\nu=\omega$ and $(\alpha,\beta,q)=\left(\frac{\alpha\beta(q+\beta-1)}{q+(\beta-1)^{2}},\frac{(\beta-1)^{2}}{q}+1,s\right)$,

[TABLE]

By Lemma 2.3, (2.22) is equivalent to

[TABLE]

Therefore, it is sufficient to prove that (2.27) and (2.28) imply (2.29). Actually, since

[TABLE]

for all $0<t<4R$, thus applying (2.28), we obtain

[TABLE]

So, it is sufficient to show that for any $x\in B_{R}(x_{0})$

[TABLE]

Since $B_{r}(y)\subset B_{2r}(x)$ for any $y\in B_{t}(x)$ with $r\geq t$, we have

[TABLE]

Combining this with Lemma 2.1 and (2.27) yields

[TABLE]

Therefore, (2.29) follows since $\mathbf{W}^{16R}_{\alpha,\beta}[\omega]\leq c_{{}_{67}}\mathbf{W}^{4R}_{\alpha,\beta}[\omega]$ in $B_{R}(x_{0})$.           $\square$

Proposition 2.7

Let $\eta,\mu$ be in $\mathfrak{M}^{+}(\mathbb{R}^{N})$. Assume that $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$ and $qs>(\beta-1)^{2}$. Let $(u_{m},v_{m})$ be nonnegative measurable funtions in $\mathbb{R}^{N}$ verifying, for all $m\geq 0$,

[TABLE]

for some $c^{*}>0$ and $(u_{0},v_{0})=0$. Then, there exists a constant $M^{*}>0$ depending only on $N,\alpha,\beta,q,s,c^{*}$ such that if the measure $d\omega=\left(\mathbf{W}_{\alpha,\beta}[\mu]\right)^{s}dx+d\eta$ satisfies

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$, then

[TABLE]

for some constants $c_{{}_{68}},c_{{}_{69}},c_{{}_{70}}$ depending only on $N,\alpha,\beta,q,s$ and $c^{*}$.

Proof. By Proposition 2.6, (2.31) implies

[TABLE]

We set

[TABLE]

and choose $M^{*}>0$ such that

[TABLE]

We claim that

[TABLE]

Clearly, by definition of $c_{{}_{68}},c_{{}_{69}}$ and $c_{{}_{70}}$, we have (2.34) for $m=0,1$. Next we assume that (2.34) holds for all integer $m\leq l$ for some $l\in\mathbb{N}_{+}^{*}$, then

[TABLE]

and

[TABLE]

Thus, (2.34) holds true for $m=l+1$. Hence, (2.34) is valid for all $l\geq 0.$           $\square$

The next result is an adaptation of Proposition 2.7 to truncated Wolff potentials.

Proposition 2.8

Let $\eta,\mu$ be in $\mathfrak{M}_{b}^{+}(B_{R}(x_{0}))$. Assume that $0<q<\frac{N(\beta-1)}{N-\alpha\beta}$ and $qs>(\beta-1)^{2}$. Let $(u_{m},v_{m})$ be nonnegative measurable funtions in $\mathbb{R}^{N}$ such that for all $m\geq 0$

[TABLE]

and $(u_{0},v_{0})=0$. If we set $d\omega=\left(\mathbf{W}_{\alpha,\beta}^{2R}[\mu]\right)^{s}dx+d\eta$, there exists a constant $M_{*}>0$ depending only on $N,\alpha,\beta,q,s,R$ and $c_{*}$ such that if

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$, then

[TABLE]

in $B_{R}(x_{0})$ for some constants $c_{{}_{72}},c_{{}_{73}},c_{{}_{74}}$ depending only on $N,\alpha,\beta,q,s,R$ and $c_{*}$.

Proof. The proof is similar to the one of Proposition 2.7 and we omit the details.           $\square$

Proposition 2.9

Let $1<\beta<N/\alpha$ and $q,s>0$ such that $qs>(\beta-1)^{2}$.

(i) Assume that $\eta$ and $\mu$ belong to $\mathfrak{M}_{b}^{+}(\mathbb{R}^{N})$ and $(u,v)$ are nonnegative measurable functions satisfying

[TABLE]

for some $c_{{}_{75}}>0$. Then there exists a constant $c_{{}_{76}}>0$ depending only on $N,\alpha,\beta,q,s$ and $c_{{}_{75}}$ such that

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$.

(ii) Assume that $\eta$ and $\mu$ belong to $\mathfrak{M}_{b}^{+}(\Omega)$ and $(u,v)$ are nonnegative functions satisfying

[TABLE]

for some $c_{{}_{77}}>0$. Then for any $\Omega^{\prime}\subset\subset\Omega$, there exists a constant $c_{{}_{78}}>0$ depending only on $n,\alpha,\beta,q,s,c_{{}_{77}}$ and $dist(\Omega^{\prime},\partial\Omega)$ such that

[TABLE]

for any compact set $K\subset\Omega^{\prime}$.

Proof. (i): Set $\omega=u^{s}+\eta$, then

[TABLE]

By (2.4) in Lemma 2.2, we get

[TABLE]

which implies

[TABLE]

Applying Proposition 2.4 to $\mu=\omega$ with $(\alpha,\beta,q)=\left(\frac{\alpha\beta(q+\beta-1)}{q+(\beta-1)^{2}},\frac{(\beta-1)^{2}}{q}+1,s\right)$, we get (2.38).

(ii) We define $\omega$ as above and we have

[TABLE]

which leads to

[TABLE]

by inequality (2.10) in Lemma 2.3. Let $M_{\omega}$ denote the centered Hardy-Littlewood maximal function which is defined for any $f\in L_{loc}^{1}(\mathbb{R}^{N},d\omega)$ by

[TABLE]

Let $K\subset\Omega$ be compact. Set $r_{K}=dist(K,\partial\Omega)$ and $\Omega_{K}=\{x\in\Omega:d(x,K)<r_{K}/2\}$. Then, for any Borel set $E\subset K$,

[TABLE]

Since $M_{\omega}$ is a bounded linear map on $L^{p}(\mathbb{R}^{N},d\omega)$ for any $p>1$ and

[TABLE]

we obtain

[TABLE]

where $\omega_{E}=\chi_{{}_{E}}\omega$. Note that if $x\in\Omega$ and $d(x)\leq r_{K}/8$, then $B_{t}(x)\subset\Omega\backslash\Omega_{K}$ for all $t\in(0,\frac{\delta d(x)}{2})$; indeed, for all $y\in B_{t}(x)$

[TABLE]

thus

[TABLE]

which implies $y\notin\Omega_{K}$. We deduce that

[TABLE]

and

[TABLE]

Hence we obtain

[TABLE]

Applying Proposition 2.5 with $\mu=\chi_{{}_{K\cap B_{2^{-6}\delta r_{{}_{K}}}\!\!\!\!\!\!(x)}}\omega$ we get (2.40), which completes the proof.                    $\square$

3 Quasilinear Dirichlet problems

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$. If $\mu\in\mathfrak{M}_{b}(\Omega)$, we denote by $\mu^{+}$ and $\mu^{-}$ respectively its positive and negative parts in the Jordan decomposition. We denote by $\mathfrak{M}_{0}(\Omega)$ the space of measures in $\Omega$ which are absolutely continuous with respect to the $c^{\Omega}_{1,p}$-capacity defined on a compact set $K\subset\Omega$ by

[TABLE]

We also denote $\mathfrak{M}_{s}(\Omega)$ the space of measures in $\Omega$ with support on a set of zero $c^{\Omega}_{1,p}$-capacity. Classically, any $\mu\in\mathfrak{M}_{b}(\Omega)$ can be written in a unique way under the form $\mu=\mu_{0}+\mu_{s}$ where $\mu_{0}\in\mathfrak{M}_{0}(\Omega)\cap\mathfrak{M}_{b}(\Omega)$ and $\mu_{s}\in\mathfrak{M}_{s}(\Omega)\cap\mathfrak{M}_{b}(\Omega)$. It is well known that any $\mu_{0}\in\mathfrak{M}_{0}(\Omega)\cap\mathfrak{M}_{b}(\Omega)$ can be written under the form $\mu_{0}=f-div~{}g$ where $f\in L^{1}(\Omega)$ and $g\in L^{p^{\prime}}(\Omega,\mathbb{R}^{N})$.

For $k>0$ and $s\in\mathbb{R}$ we set $T_{k}(s)=\max\{\min\{s,k\},-k\}$. If $u$ is a measurable function defined in $\Omega$, finite a.e. and such that $T_{k}(u)\in W^{1,p}_{loc}(\Omega)$ for any $k>0$, there exists a measurable function $v:\Omega\to\mathbb{R}^{N}$ such that $\nabla T_{k}(u)=\chi_{{}_{\{|u|\leq k\}}}v$ a.e. in $\Omega$ and for all $k>0$. We define the gradient a.e. $\nabla u$ of $u$ by $v=\nabla u$. We recall the definition of a renormalized solution given in [12].

Definition 3.1

Let $\mu=\mu_{0}+\mu_{s}\in\mathfrak{M}_{b}(\Omega)$. A measurable function $u$ defined in $\Omega$ and finite a.e. is called a renormalized solution of

[TABLE]

if $T_{k}(u)\in W^{1,p}_{0}(\Omega)$ for any $k>0$, $|{\nabla u}|^{p-1}\in L^{r}(\Omega)$ for any $0<r<\frac{N}{N-1}$, and $u$ has the property that for any $k>0$ there exist $\lambda_{k}^{+}$ and $\lambda_{k}^{-}$ belonging to $\mathfrak{M}_{b}^{+}\cap\mathfrak{M}_{0}(\Omega)$, respectively concentrated on the sets $u=k$ and $u=-k$, with the property that $\mu_{k}^{+}\rightharpoonup\mu_{s}^{+}$, $\mu_{k}^{-}\rightharpoonup\lambda_{s}^{-}$ in the narrow topology of measures and such that

[TABLE]

for every $\varphi\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$.

Remark 3.2

We recall that if $u$ is a renormalized solution to problem (3.1), then $\frac{|\nabla u|^{p}}{(|u|+1)^{r}}\in L^{1}(\Omega)$ for all $r>1$. Furthermore, $u\geq 0$ $a.e.$ in $\Omega$ if $\mu\in\mathfrak{M}_{b}^{+}(\Omega)$.

The following general stability result has been proved in [12, Th 4.1].

Theorem 3.3

Let $\mu=\mu_{0}+\mu_{s}^{+}-\mu_{s}^{-},$ with $\mu_{0}=F-\operatorname{div}g\in\mathfrak{M}_{0}(\Omega)$ and $\mu_{s}^{+}$, $\mu^{-}_{s}$ belonging to $\mathfrak{M}_{s}^{+}(\Omega).$ Let $\mu_{n}=F_{n}-\operatorname{div}g_{n}+\rho_{n}-\eta_{n}$ with $F_{n}\in L^{1}(\Omega)$, $g_{n}\in(L^{p^{\prime}}(\Omega))^{N}$ and $\rho_{n}$, $\eta_{n}$ belonging to $\mathfrak{M}_{b}^{+}(\Omega)$. Assume that $\{F_{n}\}$ converges to $F$ weakly in $L^{1}(\Omega)$, $\{g_{n}\}$ converges to $g$ strongly in $(L^{p^{\prime}}(\Omega))^{N}$ and $(\operatorname{div}g_{n})$ is bounded in $\mathfrak{M}_{b}(\Omega)$; assume also that $\{\rho_{n}\}$ converges to $\mu_{s}^{+}$ and $\{\eta_{n}\}$ to $\mu_{s}^{-}$ in the narrow topology. If $\{u_{n}\}$ is a sequence of renormalized solutions of (3.1) with data $\mu_{n}$, then, up to a subsequence, it converges a.e. in $\Omega$ to a renormalized solution $u$ of problem (3.1). Furthermore, $T_{k}(u_{n})$ converges to $T_{k}(u)$ in $W_{0}^{1,p}(\Omega)$ for any $k>0$.

We also recall the following estimate [20, Th 2.1].

Proposition 3.4

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$. Then there exists a constant $C>0$, depending on $p$ and $N$ such that if $\mu\in\mathfrak{M}^{+}_{b}(\Omega)$ and $u$ is a nonnegative renormalized solution of problem (3.1) with data $\mu$, there holds

[TABLE]

Proof of Theorem C. The condition is necessary. Assume that (1.24) admits a nonnegative renormalized solutions $(u,v)$. By Proposition 3.4 there holds

[TABLE]

Hence, we infer (1.25) from Proposition 2.9-(ii).

Sufficient conditions. Let $\{(u_{m},v_{m})\}_{m\in\mathbb{N}}$ be a sequence of nonnegative renormalized solutions of the following problems for $m\in\mathbb{N}$,

[TABLE]

with initial condition $(u_{0},v_{0})=0$. The sequences $\{u_{m}\}$ and $\{v_{m}\}$ can be constructed in such a way that they are nondecreasing (see e.g. [21]). By Proposition 3.4 we have

[TABLE]

where $R=2\,diam\,(\Omega)$. Thus, by Proposition 2.8 there exists a constant $M_{*}>0$ depending only on $N,p,q_{1},q_{2},R$ such that if

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$ with $d\omega=\left(\mathbf{W}_{1,p}^{R}[\mu]\right)^{q_{2}}dx+d\eta$, then

[TABLE]

in $\Omega$, and

[TABLE]

This implies that $\{u_{m}\},\{v_{m}\}_{m\in\mathbb{N}}$ are well defined and nondecreasing. Thus $\{(u_{m},v_{m})\}$ converges a.e in $\Omega$ to some functions $(u,v)$ which satisfies (1.27) in $\Omega$. Furthermore, we deduce from (3.6) and the monotone convergence theorem that $u_{m}^{q_{1}}\to u^{q_{1}}$ and $v_{m}^{q_{2}}\to u^{q_{2}}$ in $L^{1}(\Omega)$. Finally we infer that $u$ is a renormalized solution of (1.24) by Theorem 3.3.          $\square$

4 p-superharmonic functions and quasilinear equations in $\mathbb{R}^{N}$

We recall some definitions and properties of $p$-superharmonic functions (see e.g. [13], [14], [15] for general properties and [28] for a simple presentation).

Definition 4.1

A function $u$ is said to be $p$-harmonic in $\mathbb{R}^{N}$ if $u\in W^{1,p}_{loc}(\mathbb{R}^{N})$ and $-\Delta_{p}u=0$ in $\mathcal{D}^{\prime}(\mathbb{R}^{N})$; it is always $C^{1}$. A function $u$ is called a $p$-supersolution in $\mathbb{R}^{N}$ if $u\in W^{1,p}_{loc}(\mathbb{R}^{N})$ and $-\Delta_{p}u\geq 0$ in $\mathcal{D}^{\prime}(\mathbb{R}^{N})$.

Definition 4.2

A lower semicontinuous (l.s.c) function $u:\mathbb{R}^{N}\to(-\infty,\infty]$ is called $p$-super-harmonic if $u$ is not identically infinite and if, for all open $D\subset\subset\mathbb{R}^{N}$ and all $v\in C(\overline{D})$, $p$-harmonic in $D$, $v\leq u$ on $\partial D$ implies $v\leq u$ in $D$.

Let $u$ be a $p$-superharmonic in $\mathbb{R}^{N}$. It is well known that $u\wedge k:=\min\{u,k\}\in W^{1,p}_{loc}(\mathbb{R}^{N})$ is a p-supersolution for all $k>0$ and $u<\infty$ a.e in $\mathbb{R}^{N}$, thus, $u$ has a gradient (see the previous section). We also have $|\nabla u|^{p-1}\in L^{q}_{loc}(\mathbb{R}^{N})$, $\frac{|\nabla u|^{p}}{(|u|+1)^{r}}\in L^{1}_{loc}(\mathbb{R}^{N})$ and $u\in L^{s}_{loc}(\mathbb{R}^{N})$ for $1\leq q<\frac{N}{N-1}$ and $r>1$, $1\leq s<\frac{N(p-1)}{N-p}$ (see [13, Theorem 7.46]). Thus for any $0\leq\varphi\in C^{1}_{c}(\Omega)$, by the dominated convergence theorem,

[TABLE]

Hence, by the Riesz Representation Theorem, there is a nonnegative Radon measure denoted by $\mu[u]$, called the Riesz measure, such that $-\Delta_{p}u=\mu[u]$ in $\mathcal{D}^{\prime}(\mathbb{R}^{N})$.

The following weak convergence result for Riesz measures proved in [26] will be used to obtain the existence of $p$-superharmonic solutions to quasilinear equations.

Proposition 4.3

Suppose that $\{u_{n}\}$ is a sequence of nonnegative $p$-superharmonic functions in $\mathbb{R}^{N}$ that converges a.e to a $p$-superharmonic function $u$. Then the sequence of measures $\{\mu[u_{n}]\}$ converges to $\mu[u]$ in the weak sense of measures.

The proof of the next result can be found in [20].

Proposition 4.4

Let $\mu$ be a measure in $\mathfrak{M}^{+}(\mathbb{R}^{N})$. Suppose that ${\bf W}_{1,p}[\mu]<\infty$ a.e. Then there exists a nonnegative $p$-superharmonic function $u$ in $\mathbb{R}^{N}$ such that $-\Delta_{p}u=\mu$ in $\mathcal{D}^{\prime}(\mathbb{R}^{N})$, $\inf_{\mathbb{R}^{N}}u=0$ and

[TABLE]

for almost all $x$ in $\mathbb{R}^{N}$, where the constant $c_{{}_{86}}$ is the one of Proposition 3.4. Furthermore any $p$-superharmonic function $u$ in $\mathbb{R}^{N}$, such that $\inf_{\mathbb{R}^{N}}u=0$ satisfies (4.1) with $\mu=-\Delta_{p}u$.

Proof of Theorem A. The condition is necessary. Assume that (1.24) admits a nonnegative $p$-superharmonic functions $(u,v)$. By Proposition 4.4 there holds

[TABLE]

Hence, we obtain (1.20) from Proposition 2.9-(i).

*The condition is sufficient. * Let $\{(u_{m},v_{m})\}_{m\in\mathbb{N}}$ be a sequence of nonnegative $p$-superharmonic solutions of the following problems for $m\in\mathbb{N}$,

[TABLE]

with $(u_{0},v_{0})=(0,0)$. As in the proof of Theorem C we can assume that $\{u_{m}\}$ and $\{v_{m}\}$ are nondecreasing. By Proposition 4.4 we have

[TABLE]

Thus, by Proposition 2.7 there exists a constant $c>0$ depending only on $N,p,q_{1},q_{2}$ such that, if

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$ with $d\omega=\left(\mathbf{W}_{1,p}[\mu]\right)^{q_{2}}dx+d\eta$, then there holds in $\Omega$,

[TABLE]

and

[TABLE]

This implies that $\{u_{m}\},\{v_{m}\}$ are well defined and nondecreasing. Thus $\{(u_{m},v_{m})\}$ converges a.e in $\mathbb{R}^{N}$ to some functions $(u,v)$ which satisfies (1.27) in $\mathbb{R}^{N}$. Furthermore, we infer from (3.6) and the monotone convergence theorem that $u_{m}^{q_{1}}\to u^{q_{1}},v_{m}^{q_{2}}\to u^{q_{2}}$ in $L^{1}_{loc}(\mathbb{R}^{N})$. By Proposition 4.3 we deduce that $(u,v)$ are nonnegative $p$-superharmonic solutions of (1.19).——–           $\square$

5 Hessian equations

In this section $\Omega\subset\mathbb{R}^{N}$ is either a bounded domain with a $C^{2}$ boundary or the whole $\mathbb{R}^{N}$. For $k=1,...,N$ and $u\in C^{2}(\Omega)$ the k-hessian operator $F_{k}$ is defined by

[TABLE]

where $\lambda(D^{2}u)=\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{N})$ denotes the eigenvalues of the Hessian matrix of second partial derivative $D^{2}u$ and $S_{k}$ is the k-th elementary symmetric polynomial that is

[TABLE]

We can see that

[TABLE]

where for a matrix $A=(a_{ij})$, $[A]_{k}$ denotes the sum of the k-th principal minors. We assume that $\partial\Omega$ is uniformly (k-1)-convex, that is

[TABLE]

for some positive constant $c_{0}$, where $\kappa=(\kappa_{1},\kappa_{2},...,\kappa_{n-1})$ denote the principal curvatures of $\partial\Omega$ with respect to its inner normal.

Definition 5.1

An upper-semicontinuous function $u:\Omega\to[-\infty,\infty)$ is k-convex (k-subharmonic) if, for every open set $\Omega^{\prime}\subset\overline{\Omega}^{\prime}\subset\Omega$ and for every function $v\in C^{2}(\Omega^{\prime})\cap C(\overline{\Omega^{\prime}})$ satisfying $F_{k}[v]\leq 0$ in $\Omega^{\prime}$, the following implication is true

[TABLE]

We denote by $\Phi^{k}(\Omega)$ the class of all $k$-subharmonic functions in $\Omega$ which are not identically equal to $-\infty$.

The following weak convergence result for $k$-Hessian operators proved in [24] is fundamental in our study.

Proposition 5.2

Let $\Omega$ be either a bounded uniformly (k-1)-convex in $\mathbb{R}^{N}$ or the whole $\mathbb{R}^{N}$. For each $u\in\Phi^{k}(\Omega)$, there exists a nonnegative Radon measure $\mu_{k}[u]$ in $\Omega$ such that

1* $\mu_{k}[u]=F_{k}[u]$ for $u\in C^{2}(\Omega)$.*

2* If $\{u_{n}\}$ is a sequence of k-convex functions which converges a.e to $u$, then $\mu_{k}[u_{n}]\rightharpoonup\mu_{k}[u]$ in the weak sense of measures.*

As in the case of quasilinear equations with measure data, precise estimates of solutions of k-Hessian equations with measures data are expressed in terms of Wolff potentials. The next results are proved in [24, 17, 20].

Theorem 5.3

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded $C^{2}$, uniformly (k-1)-convex domain. Let $\mu$ be a nonnegative Radon measure in $\Omega$ which can be decomposed under the form

[TABLE]

where $\mu_{1}$ is a measure with compact support in $\Omega$ and $f\in L^{q}(\Omega)$ for some $q>\frac{N}{2k}$ if $k\leq\frac{N}{2}$, or $p=1$ if $k>\frac{N}{2}$. Then there exists a nonnegative function $u$ in $\Omega$, continuous near $\partial\Omega$, such that $-u\in\Phi^{k}(\Omega)$ and u is a solution of the problem

[TABLE]

Furthermore, any nonnegative function $u$ such that $-u\in\Phi^{k}(\Omega)$ which is continuous near $\partial\Omega$ and is a solution of above equation, satisfies

[TABLE]

where $c_{{}_{88}}$ is a positive constant independent of $x,u$ and $\Omega$.

Theorem 5.4

Let $\mu$ be a measure in $\mathfrak{M}^{+}(\mathbb{R}^{N})$ and $2k<N$. Suppose that ${\bf W}_{\frac{2k}{k+1},k+1}[\mu]<\infty$ a.e. Then there exists $u$, $-u\in\Phi^{k}(\mathbb{R}^{N})$ such that $\inf_{\mathbb{R}^{N}}u=0$ and $F_{k}[-u]=\mu~{}~{}\text{ in }~{}~{}\mathbb{R}^{N}$ and

[TABLE]

for all $x$ in $\mathbb{R}^{N}$. Furthermore, if $u$ is a nonnegative function such that $\inf_{\mathbb{R}^{N}}u=0$ and $-u\in\Phi^{k}(\mathbb{R}^{N})$, then (5.2) holds with $\mu=F_{k}[-u]$.

Proof of Theorem E. The condition is necessary. Assume that (1.31) admits a nonnegative solution $(u,v)$, continuous near $\partial\Omega$, such that $-u,-v\in\Phi^{k}(\Omega)$ and $u^{s_{2}},v^{s_{1}}\in L^{1}(\Omega)$. Then by Theorem 5.3 we have

[TABLE]

Using the part 2 of Proposition 2.9, we conclude that (1.32) holds.

The condition is sufficient. We define a sequence of nonnegative functions $u_{m},v_{m}$, continuous near $\partial\Omega$ and such that $-u_{m},-v_{m}\in\Phi^{k}(\Omega)$, by the following iterative scheme for $m\geq 0$,

[TABLE]

Clearly, we can assume that $\{u_{m}\}$ is nondecreasing as in [21]. By Theorem 5.3 we have

[TABLE]

where $R=2\,diam\,(\Omega)$.

Then, by Proposition 2.8, there exists a constant $M_{*}>0$ depending only on $N,p,q_{1},q_{2},R$ such that if

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$ with $d\omega=\left({\bf W}^{R}_{\frac{2k}{k+1},k+1}\mu]\right)^{s_{2}}dx+d\eta$, then there holds,

[TABLE]

in $\Omega$, for all $m\in\mathbb{N}$, for some positive constants $c_{{}_{72}},c_{{}_{73}}$ and $c_{{}_{74}}$ depending only on $N,k,s_{1},s_{2},R$. Note that we can write

[TABLE]

and

[TABLE]

where $\Omega_{\delta}=\{x\in\Omega:d(x,\partial\Omega)>\delta\}$ and $\delta>0$ is small enough and since $u_{m}$ is continuous near $\partial\Omega$, then $v_{m}^{s_{1}}+\mu,u_{m}^{s_{2}}+\eta$ satisfy the assumptions of the data in Theorem 5.3. Therefore the sequence $\{u_{m}\}$ is well defined and nondecreasing. Thus, $\{u_{m}\}$ converges a.e in $\Omega$ to some function $u$ which satisfies (1.34) in $\Omega$. Furthermore, by the monotone convergence theorem there holds $v_{m}^{s_{1}}\to v,u_{m}^{s_{2}}\to u$ in $L^{1}(\Omega)$. Finally, by Proposition 5.2, we infer that (1.31) admits a nonnegative solutions $u,v$, continuous near $\partial\Omega$, with $-u,-v\in\Phi^{k}(\Omega)$ satisfying (1.34).           $\square$

Proof of Theorem F The condition is necessary. Assume that (1.31) admits nonnegative solution $(u,v)$, such that $-u,-v\in\Phi^{k}(\mathbb{R}^{N})$ and $u^{s_{2}},v^{s_{1}}\in L^{1}_{loc}(\mathbb{R}^{N})$. Then by Theorem 5.3 we have

[TABLE]

Using Proposition 2.9-(ii), we conclude that (1.32) holds.

The condition is sufficient. We defined a sequence of nonnegative functions $u_{m},v_{m}$, continuous near $\partial\Omega$ and such that $-u_{m},-v_{m}\in\Phi^{k}(\Omega)$, by the following iterative scheme for $m\geq 0$,

[TABLE]

As in the previous proofs $\{u_{m}\}$ is nondecreasing. By Theorem 5.3 we have

[TABLE]

Then, by Proposition 2.7, there exists a constant $M^{*}>0$ depending only on $N,p,q_{1},q_{2},R$ such that if

[TABLE]

for any compact set $K\subset\mathbb{R}^{N}$ with $d\omega=\left({\bf W}_{\frac{2k}{k+1},k+1}\mu]\right)^{s_{2}}dx+d\eta$, then

[TABLE]

in $\Omega$, for all $m\in\mathbb{N}$, where $c_{{}_{68}}$, $c_{{}_{69}}$ and $c_{{}_{70}}$ depend on $N,k,s_{1},s_{2},R$. Therefore the sequence $\{u_{m}\}$ is well defined and nondecreasing. Thus, $\{u_{m}\}$ converges a.e in $\Omega$ to some function $u$ for which (1.37) is satisfied in $\mathbb{R}^{N}$. Furthermore, by the monotone convergence theorem we have $v_{m}^{s_{1}}\to v,u_{m}^{s_{2}}\to u$ in $L^{1}_{loc}(\mathbb{R}^{N})$. Finally, by Proposition 5.2, we obtain that (1.31) admits a nonnegative solutions $u,v$ with $-u,-v\in\Phi^{k}(\mathbb{R}^{N})$ satisfying (1.37).           $\square$

6 Further results

The method exposed in the previous sections, can be applied to types of problems. We give below an example for a semilinear system in $\mathbb{R}^{N}_{+}=\{x=(x^{\prime},x_{{}_{N}}),x^{\prime}\in\mathbb{R}^{N-1},x_{{}_{N}}>0\}$.

[TABLE]

where we have identified $\partial\mathbb{R}^{N}_{+}$ and $\mathbb{R}^{N-1}$. We denote by $\mathbf{P}$ (resp. $\mathbf{G}$) the Poisson kernel in $\mathbb{R}^{N}_{+}$ (resp the Green kernel in $\mathbb{R}^{N}$). The Poisson potential and the Green potential, $\mathbf{P}[.]$ and $\mathbf{G}[.]$, associated to $-\Delta$ are defined respectively by

[TABLE]

see [18]. We set $\rho(x)=x_{{}_{N}}$ and define the capacity $\operatorname{Cap}_{\alpha,s}^{\rho}$ by

[TABLE]

for all Borel set $K\subset\mathbb{R}^{N}$, where $\mathbf{I}_{\alpha}$ is the Riesz kernel of order $\alpha$ in $\mathbb{R}^{N}$.

Theorem 6.1

Let $1\leq q_{1}<\frac{N}{N-1}$, $q_{1}q_{2}>1$. If there exists a constant $\tilde{c}>0$ such that if

[TABLE]

for all Borel sets $K\subset\mathbb{R}_{+}^{N}$ and $G\subset\mathbb{R}^{N-1}$, then the problem (6.1) admits a solution.

All solutions in above theorem are understood in the usual very weak sense: $u\in L^{1}_{loc}(\mathbb{R}^{N}_{+}\cap B)$, $u^{q_{2}},v^{q_{1}}\in L^{1}_{\rho}(\mathbb{R}^{N}_{+}\cap B)$ for any ball $B$ and

[TABLE]

for any $\xi\in C^{2}(\overline{\mathbb{R}^{N}_{+}})\cap C_{c}(\mathbb{R}^{N})$ with $\xi=0$ on $\partial\mathbb{R}^{N}_{+}$. It is well-known that such a solution $u$ satisfies

[TABLE]

To prove Theorem 6.1 we need the following basic estimate,

Lemma 6.2

Assume that $0<q_{1}<\frac{N}{N-1}$. Then for any $\omega\in\mathfrak{M}^{+}_{b}(\mathbb{R}^{N})$,

[TABLE]

where $c_{{}_{89}}>0$ depends on $q_{1},q_{2}$ and $N$.

Proof. The proof of Lemma 6.2 is similar to the one of Lemma 2.2 and details are omitted. Note that if $\omega\in\mathfrak{M}_{b}(\overline{\mathbb{R}^{N}_{+}})$ it is extended by [math] in $\mathbb{R}^{N}_{-}$.           $\square$

Remark 6.3

The condition $0<q_{1}<\frac{N}{N-1}$ is a necessary and sufficient condition in order $\left(\mathbf{I}_{1}[\omega]\right)^{q_{1}}$ be locally integrable in $\mathbb{R}^{N}$ for any $\omega\in\mathfrak{M}^{+}_{b}(\mathbb{R}^{N})$.

Theorem 6.4

Let $q_{1}\geq 1$, $q_{1}q_{2}>1$ and $\omega\in\mathfrak{M}_{b}(\overline{\mathbb{R}^{N}_{+}})$. If

[TABLE]

for some $c_{{}_{90}}>0$, then

[TABLE]

Proof. Step 1. For any compact $K\subset\left\{x\in\mathbb{R}^{N}_{+}:\mathbf{I}_{\frac{q_{1}+2}{q_{1}}}[f\rho\,\chi_{{}_{\mathbb{R}^{N}_{+}}}](x)>\lambda\right\}$, we have

[TABLE]

by assumption and the definition of the capacity. Hence,

[TABLE]

This implies an estimate in Lorentz space,

[TABLE]

Step 2. Since, for any $g\in C_{c}(\mathbb{R}^{N}_{+})$,

[TABLE]

we infer, using duality between $L^{p,1}$ and $L^{p^{\prime},\infty}$, Holder’s inequality therein and (6.5), that

[TABLE]

Therefore,

[TABLE]

Step 3. Taking $g=\chi_{{}_{B_{t}(x)}}$ and since for $q_{1}\geq 1$

[TABLE]

we deduce that for almost all $x\in\mathbb{R}^{N}_{+}$,

[TABLE]

from (6.6), which implies

[TABLE]

since $\int_{B_{r}(x)}\chi_{{}_{\mathbb{R}^{N}_{+}}}\rho\,dy\asymp r^{N}\max\{x_{{}_{N}},r\}$ for any $x\in\mathbb{R}^{N}_{+},r>0$ where the symbol $\asymp$ is defined by

[TABLE]

It implies also

[TABLE]

from which follows

[TABLE]

Therefore, if the following inequality holds

[TABLE]

it will imply (6.4).

Step 4. We claim that (6.9) holds. Since $B_{r}(y)\in B_{2r}(x)$, $y\in B_{t}(x),r\geq t$,

[TABLE]

By integration by part,

[TABLE]

We have

[TABLE]

[TABLE]

by (6.7) and

[TABLE]

Thus,

[TABLE]

and we obtain (6.9).           $\square$

Lemma 6.5

Let $\alpha>0$, $s>1$ such that $\alpha+\frac{2}{s^{\prime}}<N-1$ where $s^{\prime}=\frac{s}{s-1}$. For all $\eta\in\mathfrak{M}^{+}(\mathbb{R}^{N-1})$, there holds

[TABLE]

where $I_{\beta}$ is the Riesz potential of order $\beta$ in $\mathbb{R}^{N-1}$. As a consequence, we have

[TABLE]

Proof. We have

[TABLE]

By using Lemma 2.1 we obtain

[TABLE]

On the other hand, by [20, Proposition 5.1], there holds

[TABLE]

Combining (6.12), (6.13) and (6.14) we obtain (6.10). Moreover, we deduce (6.11) from (6.10) and [1, Theorem 2.5.1], which ends the proof.           $\square$

Proof of Theorem 6.1 The following estimates are cclassical

[TABLE]

Thus,

[TABLE]

where $\omega(x)=\rho(\mathbf{P}[\sigma_{1}])^{q_{2}}+\sigma_{2}$ in $\mathbb{R}^{N}$. Therefore, we infer that if

[TABLE]

for some $c_{{}_{103}}>0$ small enough, then (6.1) admits a positive solution $(u,v)$. On the other hand, we deduce (6.18) from Lemma 6.2 and Theorem 6.4. The proof is complete.           $\square$

Remark 6.6

The system

[TABLE]

where $d(.)\mu,d(.)\lambda$ belong to $\mathfrak{M}_{b}^{+}(\Omega)$, $\sigma_{1},\sigma_{2}$ to $\mathfrak{M}^{+}(\partial\Omega)$ and the $\epsilon_{j}$ are positive numbers, is analyzed in [10, Th 4.6]. Therein it is proved that if

[TABLE]

which is equivalent to a capacitary estimate, and

[TABLE]

and if the $\epsilon_{j}$ are small enough, then (6.19) admits a positive solution. Now condition (6.21) is a subcriticality assumption (for at least one of the two exponents $q_{j}$) since there is no condition on the boundary measures.