Journal of Convex Analysis Volume 3 (1996), No. 2, 361–365

Summability of the Solutions of Nonlinear Elliptic Equations With Right Hand Side Measures

Lucio Boccardo

Dipartimento di Matematica, Università di Roma I, P.le A. Moro 2, I-00185 Roma, Italia. e-mail: boccardo@gpxrme.sci.uniroma1.it

Thierry Gallouet

UMR 128, ENS-Lyon, F-69364 Lyon Cedex 7, France. e-mail: thierry.gallouet@umpa.ens-lyon.fr

Received July 19, 1994 Revised manuscript received March 20, 1995

1. Introduction

A classical theorem by Guido Stampacchia states that, if A(x) = (a_{ij}(x)) is an elliptic bounded measurable matrix, there exists a solution (obtained by duality) u ∈ W_{0}^{1,q}(Ω), for every q < \frac{N}{N−1}, of the Dirichlet problem

\{ \begin{matrix}−div (A(x) ∇u) = μ & in Ω, \\ u = 0 & on ∂Ω, \end{matrix}  (1.1)

where μ is a bounded measure, and Ω is a bounded open subset of IR^{N}. Of course, the Stampacchia method is restricted to a linear setting. For example, consider the Dirichlet problem for the so-called p-laplacian (2 − \frac{1}{N} < p < N)

\{ \begin{matrix}−div (|∇u|^{p−2} ∇u) = μ & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}  (1.2)

Then the duality method cannot be used. Moreover the solutions of (1.2) are not, in general, critical points of the convex functional

∫_{Ω} |∇v|^{p} dx − ∫_{Ω} v dμ.

For instance, the minimum can be −∞. We proved by approximation the existence of solutions u ∈ W_{0}^{1,q}(Ω), q < \frac{N(p−1)}{N−1} of nonlinear Dirichlet problems when the right hand side is a bounded measure.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

362 L. Boccardo, T. Gallouet / Summability of the solutions nonlinear elliptic equations

We begin stating the hypotheses that will hold throughout the paper. Let Ω be a bounded, open subset of IR^{N}, N ≥ 2. Let p be a real number such that 1 < p ≤ N. Let a : Ω × IR^{N} → IR^{N} be a Carathéodory function such that the following holds:

a(x, ξ) · ξ ≥ ρ_{1} |ξ|^{p} ,  (1.3)

for almost every x ∈ Ω, for every ξ ∈ IR^{N}, where ρ_{1} is a positive constant;

|a(x, ξ)| ≤ λ(x) + ρ_{2} |ξ|^{p−1} ,  (1.4)

for almost every x ∈ Ω, for every ξ ∈ IR^{N}, where ρ_{2} is a positive constant, and λ belongs to L^{p′}(Ω);

(a(x, ξ) − a(x, η)) · (ξ − η) > 0 ,  (1.5)

for almost every x ∈ Ω, for every ξ and η in IR^{N}, with ξ ≠ η. Let us define the differential operator

A(u) = −div (a(x, ∇u)) .

Thanks to (1.3), (1.4) and (1.5), A is a monotone and coercive differential operator acting between W_{0}^{1,p}(Ω) and W^{−1,p′}(Ω); hence, it is surjective. We look for weak solutions, i.e., for functions u such that

\{ \begin{matrix}u ∈ W_{0}^{1,1}(Ω), \\ ∫_{Ω} a(x, ∇u) · ∇v dx = ∫_{Ω} v μ, ∀v ∈ C_{0}^{∞}(Ω) , \end{matrix}  (1.6)

We recall the following results (see [1] or [2]).

Theorem 1.1. There exists a solution u of (1.6), and u ∈ W_{0}^{1,q}(Ω), for every q < \frac{N(p−1)}{N−1} = q_{0}.

Remark 1.2. If p > N, then μ ∈ W^{−1,p′}(Ω), and the existence theorem is a consequence of the classical theorems on the surjectivity of monotone operators.

Remark 1.3. We observe that q_{0} > 1 if and only if p > 2 − \frac{1}{N}.

Remark 1.4. If μ is the Dirac mass, and Ω is a ball, we can check that the result of Theorem 1.1 is optimal.

If μ has a density f with respect to the Lebesgue measure, and f ∈ L^{1}(Ω), we recall the following results.

Theorem 1.5. If f ∈ L^{m}(Ω), 1 < m < \frac{Np}{Np−p+N} = (p^{∗})′, then u ∈ W_{0}^{1,(p−1)m^{∗}}(Ω).

Remark 1.6. If m = (p^{∗})′, then L^{m}(Ω) ⊂ W^{−1,p′}(Ω), and W_{0}^{1,(p−1)m^{∗}}(Ω) = W_{0}^{1,p}(Ω).

Remark 1.7. If m tends to 1, we cannot deduce Theorem 1.1 from Theorem 1.5 (see Remark 1.4, and the counterexample in Section 2 of [3]). In order to obtain a solution in W_{0}^{1,q_{0}}(Ω), a sufficient condition is given by the following theorem.

Theorem 1.8. If ∫_{Ω} |f| log(1 + |f|) dx < ∞, then there exists a solution u ∈ W_{0}^{1,q_{0}}(Ω). That is there exists a function u such that

\{ \begin{matrix}u ∈ W_{0}^{1,1}(Ω), \\ ∫_{Ω} a(x, ∇u) · ∇v dx = ∫_{Ω} f v dx, ∀v ∈ C_{0}^{∞}(Ω) , \end{matrix}  (1.7)

We point out that if m ∈ (1, (p^{∗})′) then, roughly speaking, the function “m → summability of (u, |∇u|)” is increasing. On the other hand, if m > (p^{∗})′, only the function “m → summability of u” is increasing . Other results concerning existence, uniqueness, regularity and generalizations can be found in the papers quoted in the references.

2. Regularity results

In this section we present an improvement of the summability of the solution u of the Dirichlet problem (1.6).

Theorem 2.1. The solution u of (1.6) given by Theorem 1.1 is such that, for every β > \frac{1}{p−1},

\frac{u}{[log(2 + |u|)]^{β}} ∈ L^{q_{0}^{∗}}(Ω) ,  (2.1)

\frac{|∇u|}{[log(2 + |u|)]^{β}} ∈ L^{q_{0}}(Ω) .  (2.2)

Proof. Let {f_{k}} be a sequence of smooth functions converging to μ in the sense of the measures. Consider the approximate problems

\{ \begin{matrix}−div (a(x, ∇u_{k})) = f_{k} & in Ω, \\ u_{k} = 0 & on ∂Ω, \end{matrix}  (2.3)

that have a unique solution u_{k} ∈ W_{0}^{1,p}(Ω). Define, for β > \frac{1}{p−1} and for s ∈ IR,

ψ(s) = \frac{s}{log(2 + |s|)^{β}} ,

and, for n ∈ IN,

φ_{n}(s) = \{ \begin{matrix}0 & if 0 ≤ s ≤ n, \\ s − n & if n < s < n + 1, \\ 1 & if s > n + 1, \\ −φ_{n}(−s) & if s < 0. \end{matrix}

The use of φ_{n}(u_{k}) as test function in (2.3), and the boundedness of {f_{k}} in L^{1}(Ω), imply that

∫_{B_{n}} |∇u_{k}|^{p} dx ≤ c_{1} ,  (2.4)

where

B_{n} = {x ∈ Ω : n ≤ |u_{k}| < n + 1}

(observe that B_{n} is the set in Ω where the derivative of φ_{n}(u_{k}) is different from zero). Then, if α > 1,

∫_{Ω} \frac{|∇u_{k}|^{p}}{(2 + |u_{k}|) [log(2 + |u_{k}|)]^{α}} dx = ∑_{n=0}^{∞} ∫_{B_{n}} \frac{|∇u_{k}|^{p}}{(2 + |u_{k}|) [log(2 + |u_{k}|)]^{α}} dx

≤ c_{1} ∑_{n=0}^{∞} \frac{1}{(2 + n) [log(2 + n)]^{α}} ≤ c_{2} .

Sobolev and Hölder inequalities yield, setting g(u_{k}) = (2 + |u_{k}|) [log(2 + |u_{k}|)]^{α} for the sake of simplicity,

c_{3} (∫_{Ω} |ψ(u_{k})|^{q_{0}^{∗}} dx)^{\frac{q_{0}}{q^{∗}_{0}}} ≤ ∫_{Ω} |∇ψ(u_{k})|^{q_{0}} dx = ∫_{Ω} |∇u_{k}|^{q_{0}} |ψ′(u_{k})|^{q_{0}} dx

= ∫_{Ω} \frac{|∇u_{k}|^{q_{0}}}{g(u_{k})^{\frac{q_{0}}{p}}} |ψ′(u_{k})|^{q_{0}} g(u_{k})^{\frac{q_{0}}{p}} dx

≤ (∫_{Ω} \frac{|∇u_{k}|^{p}}{g(u_{k})} dx)^{\frac{q_{0}}{p}} (∫_{Ω} |ψ′(u_{k})|^{\frac{pq_{0}}{p−q_{0}}} g(u_{k})^{\frac{q_{0}}{p−q_{0}}} dx)^{1− \frac{q_{0}}{p}}

≤ c2^{\frac{q_{0}}{p}} (∫_{Ω} |ψ′(u_{k})|^{\frac{pq_{0}}{p−q_{0}}} (2 + |u_{k}|)^{q_{0}^{∗}} [log(2 + |u_{k}|)]^{\frac{αq_{0}}{p−q_{0}}} dx)^{1− \frac{q_{0}}{p}} .

Now we point out that

\begin{aligned}|ψ′(s)|^{\frac{pq_{0}}{p−q_{0}}} (2 + |s|)^{q_{0}^{∗}} [log(2 + |s|)]^{\frac{αq_{0}}{p−q_{0}}} & ≤ [log(2 + |s|)]^{αq_{0}^{∗}−βpq_{0}^{∗}} (2 + |s|)^{q_{0}^{∗}} \\ & ≤ c_{4} (1 + |ψ(s)|^{q_{0}^{∗}}) , \end{aligned}

if αq_{0}^{∗} − βpq_{0}^{∗} ≤ −βq_{0}^{∗}; the previous inequality is verified if α ≤ β(p − 1), and this explains the bound on β given in the statement. Thus

∫_{Ω} |ψ(u_{k})|^{q_{0}^{∗}} dx ≤ c_{5} , ∫_{Ω} |∇ψ(u_{k})|^{q_{0}} dx ≤ c_{6} .

Then the methods of [1], [2] allow us to pass to the limit in the approximate equations, and we can prove that there exists a solution u of (1.6) such that both (2.1) and (2.2) hold true.

Acknowledgment. The authors would like to thank the organizers (G. Buttazzo and G. Bouchitté) of the International Conference held in Luminy on June 1993, where some previous results were presented together with the summability Theorem 2.1.

References

[1] L. Boccardo, T. Gallouet: Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989) 149–169,

[2] L. Boccardo, T. Gallouet: Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations 17 (1992) 641–655,

[3] L. Orsina: Solvability of linear and semilinear eigenvalue problems with L^{1} data, Rend. Sem. Mat. Univ. Padova 90 (1993) 207–238

[4] P. Bènilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J.L. Vazquez: An L^{1} theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, to appear.

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