2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 23–34. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Necessary conditions of existence for an elliptic equation with source term and measure data

involving p-Laplacian ^{∗}

Marie-Françoise Bidaut-Véron

Abstract

We study the nonnegative solutions to equation

−Δ_{p}u = u^{q} + λν,

in a bounded domain Ω of ℝ^{N}, where 1 < p < N, q > p − 1, ν is a nonnegative Radon measure on Ω, and λ > 0 is a parameter. We give necessary conditions on ν for existence, with λ small enough, in terms of capacity. We also give a priori estimates of the solutions.

1 Introduction

Let Ω be a bounded regular domain in ℝ^{N}. We denote by M(Ω) the set of Radon measures on Ω, M^{+}(Ω) the set of nonnegative ones, and by M_{b}(Ω), M_{b}^{+}(Ω) the subsets of bounded ones. We consider the quasilinear elliptic problem with a source term:

−Δ_{p}u = − div(|∇u|^{p−2}∇u) = |u|^{q−1}u + μ, in Ω,

u = 0, on ∂Ω, (1.1)

with 1 < p < N, q > p − 1, and μ ∈ M_{b}^{+}(Ω). We look for conditions on the measure μ 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

μ = λν, λ ≥ 0,

where ν ∈ M_{b}^{+}(Ω) is fixed and λ is a parameter. Recall a result of [3] in case p = 2 , N ≥ 3, which gives a necessary and sufficient condition for existence:

∗Mathematics Subject Classifications: 35J70, 35B45, 35D05.

Key words: Degenerate quasilinear equations, measure data, capacities, a priori estimates.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

Theorem 1.1 ([3]) The following problem:

\begin{matrix}−Δu = u^{q} + λν, in Ω, \\ u = 0, on ∂Ω, \end{matrix}  (1.2)

where ν ∈ M_{b}^{+}(Ω), ν ≠ 0, has a nonnegative solution (in the integral sense) if and only if

λ ∫_{Ω} φdν ≤ \frac{q − 1}{q^{q′}} ∫_{Ω} φ^{1−q′}(−Δφ)^{q′}dx,  (1.3)

for any φ ∈ W_{0}^{1,∞}(Ω) ∩ W^{2,∞}(Ω) such that −Δφ ≥ 0, with compact support in Ω.

Thus if q is subcritical, that means q < N/(N − 2), problem (1.2) always admits a solution for λ small enough. In case q ≥ N/(N − 2), in order to obtain existence, the measure μ = λν has to be small enough, and also not to charge some small sets, in particular the point sets (this was first observed in [15]). More precisely, if the measure is compactly supported, from [3], condition (1.3) implies that

∫_{K} dν ≤ C cap_{2,q′}(K, ℝ^{N}), for every compact set K ⊂ Ω,  (1.4)

where for any domain Ω and any m ∈ ℕ^{∗} and r > 1, cap_{m,r} is the capacity associated to the Sobolev space W_{0}^{m,r}(Ω), defined by

cap_{m,r}(K, Ω) = inf {∥ψ∥_{W_{0}^{m,r}(Ω)}^{r} : ψ ∈ D(Ω), 0 ≤ ψ ≤ 1, ψ = 1 on K } .

In fact it was proved in [2] that (1.4) is also sufficient:

Theorem 1.2 ([2]) Assume that ν has a compact support in Ω. Then problem (1.2) has a solution for any λ ≥ 0 small enough if and only if there exists C > 0 such that (1.4) holds.

Condition (1.4) implies that μ 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 ν ∈ L^{s}(Ω) with 1 ≤ s < N/2q′ (resp. s = 1) such that problem (1.2) admits no solution, for any λ > 0.

Concerning problem (1.1) with p ≠ 2, the question is much harder, because the full duality argument used in [3] 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 2 we recall the usual notions of entropy solutions, which suppose that the measure is bounded; this leads to assume that u^{q} ∈ L^{1}(Ω). We denote by

\overline{P} = \frac{N(p − 1)}{N − p}

the critical exponent linked to the p-Laplacian, and we set

q^{∗} = q/(q − p + 1),

(hence q^{∗} = q′ if p = 2). In Section 3 we prove our main result:

Theorem 1.3 Let ν ∈ M_{b}^{+}(Ω) and λ ≥ 0. Assume that problem

\begin{matrix}−Δ_{p}u = u^{q} + λν, in Ω, \\ u = 0, on ∂Ω, \end{matrix}  (1.5)

has a nonnegative entropy solution (hence u^{q} ∈ L^{1}(Ω)). Then for any R > pq^{∗}, there exists C = C(N, p, q, R, Ω) > 0 such that

λ ∫_{Ω} φdν + ∫_{Ω} u^{q}φ dx ≤ C( ∫_{Ω} φ^{1−R}|∇φ|^{R}dx)^{pq^{∗}/R},  (1.6)

for any φ ∈ W_{0}^{1,p}(Ω) ∩ W^{1,s}(Ω) (s > N) such that 0 ≤ φ ≤ 1 in Ω. And for any α < 0, there exists C = C(α, N, p, q, R, Ω) > 0 such that

∫_{Ω}(u + 1)^{α−1}|∇u|^{p}φ dx ≤ C(1 + ∫_{Ω} u^{q}φ dx)( ∫_{Ω} φ^{1−R}|∇φ|^{R}dx)^{p/R}. (1.7)

This Theorem gives a priori estimate not only of the size of the measure, but also of the integral ∫_{Ω} u^{q}φ dx, independently on u. In the case p = 2, this was first remarked by [12] when μ = 0 ; it was the starting point for proving L^{∞} universal estimates. It was also used in [7] and [8] for obtaining a priori estimates with a general measure μ. As a consequence we deduce the following:

Theorem 1.4 If problem (1.5) has a solution, then, for any R > pq^{∗}, there exists C = C(N, p, q, R, Ω) > 0 such that

λ ∫_{K} dν ≤ C (cap_{1,R}(K, Ω))^{pq^{∗}/R}, for every compact set K ⊂ Ω. (1.8)

and if ν has a compact support in Ω, there exists C = C(N, p, q, R, μ) > 0 such that

λ ∫_{K} dν ≤ C (cap_{1,R}(K, ℝ^{N}))^{pq^{∗}/R}, for every compact set K ⊂ Ω. (1.9)

In particular, if q > \overline{P}, then ν does not charge the point sets. Moreover for any 1 ≤ s < N/pq^{∗}, there exists a function ν ∈ L^{s}(Ω) such that for any λ > 0, problem (1.5) admits no solution.

In Section 4, we mention some partially or fully open problems linked to this study. We refer to [5] for more complete results for problem (1.1) with possible signed measure μ, and for the problem with an absorption term

\begin{matrix}−Δ_{p}u + |u|^{q−1}u = μ, in Ω, \\ u = 0, on ∂Ω. \end{matrix}  (1.10)

2 Entropy solutions

First recall some well-known results concerning the problem

\begin{matrix}−Δ_{p}u = μ, in Ω, \\ u = 0, on ∂Ω, \end{matrix}  (2.1)

with μ ∈ M_{b}(Ω). We set

P_{0} = \frac{2N}{N + 1}, P_{1} = 2 − \frac{1}{N} ,

so that 1 < P_{0} < P_{1}, and P > P_{0} ⟺ \overline{P} > 1. When p > P_{1}, problem (2.1) admits at least a solution u in the sense of distributions, such that u ∈ W_{0}^{1,r}(Ω) for any 1 ≤ r < \overline{P} . In the general case, one can define a notion of entropy or renormalized solutions in four equivalent ways, see [11], which allow to give a sense to the gradient in any case: they are solutions such that ∇T_{k}(u) ∈ L_{loc}^{1}(Ω) for any k > 0, where

T_{k}(s) = \begin{cases}s, & if |s| ≤ k, \\ k sign(s), & if |s| > k, \end{cases}  (2.2)

and the gradient of u, denoted by y = ∇u is defined by

∇(T_{k}(u)) = y × 1_{{|u|≤k}} a.e. in Ω.  (2.3)

For any p > 1 there exists at least an entropy solution of (2.1), and it is unique if μ ∈ L^{1}(Ω). 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

u^{p−1} ∈ L^{s}(Ω), for any 1 ≤ s < N/(N − p),
|∇u|^{p−1} ∈ L^{r}(Ω), for any 1 ≤ r < N/(N − 1).

Thus the gradient is well defined in L^{1}(Ω) if and only if p > P_{1} and u itself is in L^{1}(Ω) if and only if p > P_{0}.

Recall that any measure μ ∈ M_{b}(Ω) can be decomposed as

μ = μ_{0} + μ_{s}^{+} − μ_{s},

where μ_{0} ∈ M_{0,b}(Ω), set of bounded measures such that

μ_{0}(B) = 0 for any Borel set B ⊂ Ω such that cap_{1,p}(B, Ω) = 0; (2.4)

and μ_{s}^{+}, μ_{s}^{−} are nonnegative and concentrated on a set E with cap_{1,p}(E, Ω) = 0. If μ ∈ M_{b}^{+}(Ω), then μ_{0} is nonnegative, and μ = μ_{0} + μ_{s}^{+}.

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 Ω, and

T_{k}(u) ∈ W_{0}^{1,p}(Ω) for every k > 0,  (2.5)

and the gradient defined by (2.3) satisfies

|∇u|^{p−1} ∈ L^{r}(Ω), for any 1 ≤ r < N/(N − 1),  (2.6)

and u satisfies

∫_{Ω} |∇u|^{p−2}∇u.∇(h(u)φ)dx = ∫_{Ω} h(u)φdμ_{0}

+ h(+∞) ∫_{Ω} φdμ_{s}^{+} − h(−∞) ∫_{Ω} φdμ_{s}^{−},

for any h ∈ W^{1,∞}(ℝ) and h′ has a compact support, and any φ ∈ W^{1,s}(Ω) for some s > N, such that h(u)φ ∈ W_{0}^{1,p}(Ω).

In the same way, for given μ = μ_{0} + μ_{s}^{+} ∈ M_{b}^{+}(Ω), a nonnegative entropy solution u of problem (1.1) will be a measurable function u such that u^{q} ∈ L^{1}(Ω) and u is an entropy solution of problem

\begin{matrix}−Δ_{p}u = μ − u^{q} in Ω, \\ u = 0 on ∂Ω. \end{matrix}

In particular

∫_{Ω} |∇u|^{p−2}∇u.∇(h(u)φ)dx + ∫_{Ω} u^{q}h(u)φ dx = ∫_{Ω} h(u)φdμ_{0} + h(+∞) ∫_{Ω} φdμ_{s}^{+},

for any h and φ as above.

3 Proofs and comments

Proof of Theorem 1.3 Let μ = λν = μ_{0} + μ_{s}^{+}, where μ_{0} ∈ M_{0,b}(Ω) and μ_{s}^{+} is singular, and let α ∈ (1 − p, 0) be a parameter. For any k > 0, we set u_{k} = T_{k}(u), and, for any ε ∈ (0, k),

h_{α,k,ε}(r) = (T_{k}(r^{+}) + ε)^{α} = \begin{cases}ε^{α}, & if r ≤ 0, \\ (r + ε)^{α}, & if 0 ≤ r ≤ k, \\ (k + ε)^{α}, & if r ≥ k. \end{cases}

28 Existence of solutions to an elliptic equation We choose in (2) the test functions h = h_{α,k,ε}, and φ ∈ W_{0}^{1,p}(Ω) ∩ W^{1,s}(Ω), with s > N and φ ≥ 0 in Ω, and obtain

∫_{Ω}(u_{k} + ε)^{α}φdμ_{0} + (k + ε)^{α} ∫_{Ω} φdμ_{s}^{+} + ∫_{Ω}(u_{k} + ε)^{α}u^{q}φ dx
+|α| ∫_{Ω} ∫_{Ω}(u_{k} + ε)^{α−1}|∇u_{k}|^{p}φ dx
= ∫_{Ω}(u_{k} + ε)^{α}|∇u|^{p−2}∇u.∇φ dx
≤ ∫_{Ω}(u_{k} + ε)^{α}|∇u_{k}|^{p−1}|∇φ|dx + ∫_{{u≥k}}(u_{k} + ε)^{α}|∇u|^{p−1}|∇φ|dx
≤\frac{|α|}{2} ∫_{Ω}(u_{k} + ε)^{α−1}|∇u_{k}|^{p} φ dx + C ∫_{Ω}(u_{k} + ε)^{α+p−1}φ^{1−p}|∇φ|^{p} dx
+ (k + ε)^{α} ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx,

where C = C(α) > 0.

Now from Hölder inequality, setting θ = q/(p − 1 + α) > 1, ∫_{Ω}(u_{k} + ε)^{α+p−1}φ^{1−p}|∇φ|^{p} dx

≤ ( ∫_{Ω}(u_{k} + ε)^{q}φ dx)^{1/θ}( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/θ′}.

In particular for any k > 1,

\frac{|α|}{2} ∫_{Ω} ∫_{Ω}(u_{k} + ε)^{α−1}|∇u_{k}|^{p}φ dx

≤ C( ∫_{Ω}(u_{k}+ε)^{q}φ dx)^{1/θ}( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/θ′}+∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx. (3.1) Letting ε tend to 0, we get

\frac{|α|}{2} ∫_{Ω} u_{k}^{α−1}|∇u_{k}|^{p}φ dx ≤C( ∫_{Ω} u_{k}^{q}φ dx)^{1/θ}( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/θ′}
+ ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx.  (3.2)

Choosing now h(u) = 1 in (2), with the same φ, we find

∫_{Ω} φdμ_{0} + ∫_{Ω} φdμ_{s}^{+} + ∫_{Ω} u^{q}φ dx = ∫_{Ω} |∇u|^{p−2}∇u.∇φ dx
≤ ∫_{Ω} u_{k}^{(α−1)/p′}|∇u|^{p−1}u_{k}^{(1−α)/p′}|∇φ|dx + ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx
≤( ∫_{Ω} u_{k}^{α−1}|∇u_{k}|^{p} φ dx)^{1/p′}( ∫_{Ω} u_{k}^{(1−α)(p−1)}φ^{1−p}|∇φ|^{p}dx)^{1/p}
+ ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx.  (3.3)

Since q > p − 1, we can fix α ∈ (1 − p, 0) such that τ = q/(1 − α)(p − 1) > 1. From (3.2) and (3.3), we derive

∫Ωφdμ + ∫_{Ω} u^{q}φ dx
≤( ∫_{Ω} u_{k}^{α−1}|∇u_{k}|^{p} φ dx)^{1/p′}( ∫_{Ω} u_{k}^{q}φ dx)^{1/τp}( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{1/τ′p}
+ ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx
≤(C( ∫_{Ω} u_{k}^{q}φ dx)^{1/θ}( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/θ′} + ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx)^{1/p′}

× ( ∫_{Ω} u_{k}^{q}φ dx)^{1/τp} (∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{1/τ′p} + ∫_{{u≥k}} |∇u|^{p−1}|∇φ|dx.

Now we can let k tend to ∞, since u^{q} + |∇u|^{p−1} ∈ L^{1}(Ω). It follows that

∫_{Ω} φdμ + ∫_{Ω} u^{q}φ dx ≤ C( ∫_{Ω} u^{q}φ dx)^{1/p′θ+1/τp}  (3.4)

× ( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/p′θ′}( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{1/τ′p},

with a new C = C(α, N, p, q). Since 1/θ′p′ + 1/τ′p = 1/q^{∗} = 1 − (1/θp′ + 1/τp), we find in particular

( ∫Ωu^{q}φ dx)^{1−(p−1)/q}
=( ∫_{Ω} u^{q}φ dx)^{(1/p′θ′+1/τ′p)}
≤C( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/p′θ′}( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{1/τ′p}.

Consequently

∫Ωu^{q}φ dx
≤C( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{τ′p/(τ′p+p′θ′)}( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{p′θ′/(τ′p+p′θ′)}.

Notice that τ < q/(p − 1) < θ, then from Hölder inequality,

∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx ≤ ( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{θ′/τ′}( ∫_{Ω} φ dx)^{1−θ′/τ′}
≤ C( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{θ′/τ′},

with a new constant C = C(N, p, q, α, Ω), since 0 ≤ φ ≤ 1. Therefore

∫_{Ω} u^{q}φ dx ≤ C( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{q^{∗}/τ′},  (3.5)

with a new constant C > 0. Moreover, from (3.4) and (3.5),

∫_{Ω} φdμ ≤ C( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{(q^{∗}−1+1/p′+1/p)/τ′},

then

∫_{Ω} φdμ + ∫_{Ω} u^{q}φ dx ≤ C( ∫_{Ω} φ^{1−τ′p}|∇φ|^{τ′p}dx)^{q^{∗}/τ′}.

We can choose |α| sufficiently small, such that

pq^{∗} < pτ′ = q/(q − p + 1 − |α|(p − 1)) ≤ R;

thus we deduce (1.6) from Hölder inequality. Also, for any α < 0, with |α| small enough, from (3.1), taking ε = 1 and letting k tend to ∞, we obtain

\frac{|α|}{2} ∫_{Ω} ∫_{Ω}(u + 1)^{α−1}|∇u|^{p}φ dx
≤C( ∫_{Ω}(u + 1)^{q}φ, dx)^{1/θ}( ∫_{Ω} φ^{1−pθ′}|∇φ|^{pθ′}dx)^{1/θ′}
≤C(1 + ∫_{Ω} u^{q}φ dx)( ∫_{Ω} φ^{1−R}|∇φ|^{R}dx)^{p/R}.

Then (1.7) follows for any α < 0. □

When p = 2, Theorem 1.1 naturally gives a stronger result, since any set with 1, R - capacity zero for some R > 2q′ has also a 2, q′- capacity zero, see [1]. The capacity involved in Theorem 1.3 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.

Proof of Theorem 1.4 Let ψ_{n} ∈ D(Ω) such that 0 ≤ ψ_{n} ≤ 1 and ψ_{n} ≥ χ_{K} and ∥ψ_{n}∥_{W^{1,R}(Ω)}^{R} tends to cap_{1,R}(K, Ω) as n tends to ∞. Choosing φ = ψ_{n}^{R} in (1.6), we deduce that

λ ∫_{K} dν ≤ C( ∫_{Ω} |∇ψ_{n}|^{R}dx)^{pq^{∗}/R} ≤ C ∥ψ_{n}∥_{W^{1,R}(Ω)}^{R} ,

with new constants C = C(N, p, q, R, Ω), and (1.8) follows. If ν has a compact support X in Ω, then (1.9) holds after localization on a neighborhood of X. Assume moreover that q > \overline{P}, then we can choose R such that pq^{∗} < R < N. Thus any point set {a} of Ω has a 1, R - capacity zero, hence ν({a}) = 0. Moreover taking K = \overline{B(x_{0}, r)} with r > 0 small enough, we derive

λ ∫_{B(x_{0},r)} dν ≤ Cr^{N−R},  (3.6)

with C = C(N, p, q, R, x_{0}, Ω). For any 1 ≤ s < N/pq^{∗}, we can construct a function ν ∈ L^{s}(Ω) with a singularity in |x − x_{0}|^{−k} with pq^{∗} < k < N/s, and with compact support in Ω, such that for any λ > 0, λν does not satisfy (3.6) for pq^{∗} < R < k. Then there exists no solution of problem (1.5). □

4 Open problems

Problem 1: Can we obtain sufficient conditions of existence? In the subcritical case q < \overline{P}, at least when p > P_{0}, the existence of solutions of problem (1.1), with possibly signed measure μ, is shown in [13]. 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 ≠ 2 a consequence of Theorem 1.1:

Theorem 4.1 ([3]) Assume that N ≥ 3, and ν ∈ L^{s}(Ω), for some s ≥ 1. If q > N/(N − 2) and s ≥ N/2q′, or q = N/(N − 2) and s > N/2q′, then problem (1.2) has a solution for λ small enough.

Problem 2: Can we solve problems (2.1) and (1.5) if μ is not bounded? Let us begin by the case without reaction term. For any x ∈ Ω, denote by ρ(x) the distance from x to ∂Ω. When p = 2, problem (2.1) is well posed in fact for any measure μ, possibly unbounded, such that ∫_{Ω} ρd|μ| < ∞ : it admits a unique integral solution

u(x) = G(μ) = ∫_{Ω} G(x, y)dμ(y),  (4.1)

where G is the Green kernel. And u is also the weak solution of the problem in the sense that u ∈ L^{1}(Ω) and

∫_{Ω} u(−Δξ)dx = ∫_{Ω} ξdμ,  (4.2)

for any ξ ∈ C^{1}(\overline{Ω}) vanishing on ∂Ω with ∇ξ is Lipschitz continuous, see [7]. The case where μ is a function f, such that ∫_{Ω} ρfdx < ∞, was first considered by Brézis, see [17]. The problem is open when p ≠ 2 : up to now we have no existence results concerning equation (2.1) when μ 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 [14] and specified in [9]:

Theorem 4.2 ([14]) Let ν ∈ M^{+}(Ω), ν ≠ 0 such that ∫_{Ω} ρdν < ∞. Then problem (1.2) has a solution such that G(u^{q}) < ∞, a.e. in Ω, for any λ ≥ 0 small enough, if and only if there exists C > 0 such that

G(G^{q}(ν)) ≤ CG(ν), a.e. in Ω.  (4.3)

Notice that condition G(u^{q}) < ∞ a.e. in Ω, is satisfied as soon as ∫_{Ω} ρfu^{q}dx < ∞, and the solutions are taken in the integral sense. More recently new existence results and a priori estimates were given in [8], covering the case of measures μ such that ∫_{Ω} ρ^{γ}dμ < ∞ for some 0 ≤ γ ≤ 1. Condition (4.3) allows to obtain a supersolution, and then a solution by using an iterative scheme. It is available for much more general linear operators, see [14] and [16]. It seems to be difficult to extend to nonlinear ones, since it is based on a representation formula. However Kalton and Verbitski [14] also gave necessary and sufficient in terms of capacity with weights, extending the result of [2] to general measures:

Theorem 4.3 ([14]) Let ν ≠ 0 be a nonnegative Radon measure on Ω. Then problem (1.2) has a solution for any λ ≥ 0 small enough if and only if there exists C > 0 such that

∫_{K} dν ≤ C cap_{2,q′,ρ}(K), for every compact set K ⊂ Ω,

where

cap_{2,q′,ρ}(K) = inf { ∫_{Ω} w^{q′}ρ^{1−q′}dx : w ≥ 0, Gw ≥ ρχ_{K} a.e. in Ω}.

One can ask if results of this type can be obtained for the p-Laplacian, using capacities of order 1 with suitable weights.

References

[1] D. R. Adams and L. I. Hedberg, Functions spaces and potential theory, Grundlehren der Mathematischen wissenchaften, Springer Verlag, Berlin, 314 (1996).

[2] D. Adams et M. Pierre, Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier, 41 (1991), 117-135.

[3] P. Baras and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré, Anal. non Lin. 2 (1985), 185-212.

[4] M-F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arc. Rat. Mech. Anal. 107 (1989), 293-324.

[5] M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation with absorption or source term and measure data, Adv. in Nonlinear Studies (to appear).

[6] M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.

[7] M-F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Revista Matematica Iberoamericana, 16 (2000), 477-513.

[8] M-F. Bidaut-Véron and C. Yarur, Semilinear elliptic equations and systems with measure data: existence and a priori estimates, Advances in Diff. Equ., 7 (2002), 257-296.

[9] H. Brézis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. 8 (1998), 223-262.

[10] H. Brézis and P.L. Lions, A note on isolated singularities for linear elliptic equation, Math. Anal. Appl. Adv. Math. Suppl. Stud. 7A (1981), 263-266.

[11] Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Sup. Pisa, 28 (1999), 741-808.

[12] De Figuereido, P.L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63.

[13] N. Grenon, Existence results for some semilinear elliptic equations with measure data, Ann. Inst. H. Poincaré, 19 (2002), 1-11.

[14] N. Kalton and I. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc., 351, 9 (1999), 3441-3497.

[15] P. L. Lions, Isolated singularities in semilinear problems, J. Diff. Eq., 38 (1980), 441-450.

[16] I. Verbitski, Superlinear equations, potential theory and weighted norm inequalities, Nonlinear Analysis, Funct. Spaces Appl., Vol. 6. (1999), 223-269.

[17] L. Véron, Singularities of solutions of second order quasilinear equations, Pitman Research Notes in Math., Longman Sci. & Tech. 353 (1996).

Marie-Françoise Bidaut-Véron Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, Parc de Grandmont, 37200 Tours, France e-mail: veronmf@univ-tours.fr