2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 47–52. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A remark on some nonlinear elliptic problems ^{∗}

Lucio Boccardo

Abstract

We shall prove an existence result of W_{0}^{1,p}(Ω) solutions for the boundary value problem

− div a(x, u, ∇u) = F in Ω

u = 0 on ∂Ω (0.1)

with right hand side in W^{−1,p′}(Ω). The features of the equation are that no restrictions on the growth of the function a(x, s, ξ) with respect to s are assumed and that a(x, s, ξ) with respect to ξ is monotone, but not strictly monotone. We overcome the difficulty of the uncontrolled growth of a thanks to a suitable definition of solution (similar to the one introduced in [1] for the study of the Dirichlet problem in L^{1}) and the difficulty of the not strict monotonicity thanks to a technique (the L^{1}-version of Minty’s Lemma) similar to the one used in [5].

1 Introduction and assumptions

We deal with boundary value problems with differential operators A defined as

A(v) = − div (a(x, v, ∇v))

where Ω is a bounded domain of ℝ^{N}, N ≥ 2, a : Ω × ℝ × ℝ^{N} → ℝ^{N} is a Carathéodory function (that is, measurable with respect to x in Ω for every (s, ξ) in ℝ×ℝ^{N}, and continuous with respect to (s, ξ) in ℝ×ℝ^{N} for almost every x in Ω). We assume that there exist a real positive constant α, a continuous function β(s) and a nonnegative function k in L^{p′}(Ω), where 1 < p, such that for almost every x in Ω, for every s in ℝ, for every ξ and η in ℝ^{N}

a(x, s, ξ) · ξ ≥ α|ξ|^{p} ,  (1.1)
[a(x, s, ξ) − a(x, s, η)] · [ξ − η] ≥ 0 ,  (1.2)
|a(x, s, ξ)| ≤ (k(x) + [β(s)|ξ|]^{p−1}) .  (1.3)

∗Mathematics Subject Classifications: 35J60, 35J65, 35J70.

Key words: Dirichlet problem in L^{1}”, uncontrolled growth.

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

Published October 21, 2002.

Thus, A is not well defined on the whole W_{0}^{1,p}(Ω), but only in W_{0}^{1,p}(Ω)∩L^{∞}(Ω). Note that if we assume

[a(x, s, ξ) − a(x, s, η)] · [ξ − η] > 0, ξ ≠ η,  (1.4)
|a(x, s, ξ)| ≤ (k(x) + |s|^{p−1} + |ξ|^{p−1}),  (1.5)

instead of (1.2), (1.3), A turns out to be pseudomonotone, coercive and is hence surjective on W_{0}^{1,p}(Ω) (see [6, 7, 8, 10]). A model operator for our setting is

− ∑_{i} \frac{∂}{∂x_{i}} ((1 + |v|^{γ_{i}}χ_{E_{i}})\frac{∂v}{∂x_{i}} )

where γ_{i} ≥ 0 and χ_{E_{i}} is the characteristic function of the measurable subset E_{i} ⊂ Ω. Concerning the right hand side of (0.1), we assume that

F ∈ W^{−1,p′}(Ω).  (1.6)

The aim of this note is to prove existence of solutions for (0.1) under the weaker assumption (1.2), without using the almost everywhere convergence of the gradients of the approximate equations, since this is impossible to prove in our setting. The main tools of our proof are a version of Minty’s Lemma, similar to that one used in [5] to study nonlinear boundary value problems in L^{1}, and a definition of solution, similar to the one introduced in [1] for the Dirichlet problem in L^{1}. Other existence results of finite energy solutions, similar to the one of Theorem 2.2, can be found [3] (see also [9, 13, 2, 4] for existence results and [12, 14] for uniqueness results).

2 Existence

We recall that, for k > 0 and s in ℝ, the truncating function is defined as

T_{k}(s) = min {k, max{−k, s}} .

The composition of functions in W_{0}^{1,p}(Ω) with T_{k} will play an important role in our approach to the existence of solutions of (0.1). More precisely, we will use the following definition of solution, which is similar to the one introduced in [1] for the Dirichlet problem in L^{1}.

Definition 2.1 A function u ∈ W_{0}^{1,p}(Ω) is a T -solution of (0.1) if

\begin{matrix}u ∈ W_{0}^{1,p}(Ω), ∀φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) : \\ ∫_{Ω} a(x, u, ∇u)∇T_{k}[u − φ] = ⟨F, T_{k}[u − φ]⟩ \end{matrix}  (2.1)

Theorem 2.2 Under the assumptions (1.1), (1.2), (1.3), (1.6) there exists a T -solution u of (2.1).

Remark 2.3 Note that, even if a(x, s, ξ) is unbounded with respect to s, the integral in (2.1) is well defined, since ∇T_{k}[u − φ] is not zero on the subset {x ∈ Ω : |u(x) − φ(x)| ≤ k}, that is in subsets where u is bounded.

Remark 2.4 If in (2.1) we take φ = 0, we have

∫_{Ω} a(x, u, ∇u)∇T_{k}(u) = ⟨F, T_{k}(u)⟩,

so that Lebesgue and Fatou Theorems imply that a(x, u, ∇u)∇u ∈ L^{1}(Ω), but we are not able to prove that a(x, u, ∇u) ∈ L^{1}(Ω), which would imply that u is a solution in D′(Ω), that is

u ∈ W_{0}^{1,p}(Ω) : ∫_{Ω} a(x, u, ∇u)∇ϕ = ⟨F, ϕ⟩, ∀ϕ ∈ D(Ω).

Proof of Theorem 2.2. Consider the approximate problems

u_{n} ∈ W_{0}^{1,p}(Ω) : − div(a(x, T_{n}(u_{n}), ∇u_{n}) = F.  (2.2)

The solutions u_{n} exist thanks to the Leray-Lions existence theorem (see [10]). Moreover, the use of u_{n} as test function in (2.2) and the assumption (1.1) imply that the sequence {u_{n}} is bounded in W_{0}^{1,p}(Ω). Thus, there exists a function u ∈ W_{0}^{1,p}(Ω) and a subsequence {u_{n_{j}}} such that u_{n_{j}} converges weakly to u in W_{0}^{1,p}(Ω) and almost everywhere. Now we use an idea of G.J. Minty ([11]), in the framework of [5]: thanks to the monotonicity of a(x, s, ξ) with respect to ξ, if φ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω) and n > k + ∥φ∥_{L^{∞}(Ω)}, we have that

∫_{Ω} a(x, u_{n_{j}}, ∇φ)∇T_{k}[u_{n_{j}} − φ] ≤ ⟨F, T_{k}[u_{n_{j}} − φ]⟩.

The weak convergence of the sequence {u_{n_{j}}} in W_{0}^{1,p}(Ω) and the remark that ∇T_{k}[u_{n_{j}} − φ] is not zero on the subset {x ∈ Ω : |u_{n_{j}}(x) − φ(x)| ≤ k} (subset of {x ∈ Ω : |u_{n_{j}}(x)| ≤ k + ∥φ∥_{L^{∞}(Ω)}}) allow to pass to the limit in the previous inequality, so that

∫_{Ω} a(x, u, ∇φ)∇T_{k}[u − φ] ≤ ⟨F, T_{k}[u − φ]⟩.

Let h and k be positive real numbers, let t belong to (−1, 1), and let ψ be a function in W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω). Choose φ = T_{h}(u) + t T_{k}[u − ψ] in the previous inequality. Setting G_{k}(s) = s − T_{k}(s), we obtain

I = ∫_{Ω} a(x, u, ∇T_{h}(u) + t∇T_{k}[u − ψ])∇T_{k}(G_{h}(u) − t T_{k}[u − ψ])
≤ ⟨F, T_{k}(G_{h}(u) − t, T_{k}[u − ψ])⟩ = J .

We then have

I = ∫_{{|G_{h}(u)−t T_{k}[u−ψ]|≤k}} a(x, u, ∇T_{h}(u) + t ∇T_{k}[u − ψ])∇G_{h}(u)

−t ∫_{{|G_{h}(u)−t T_{k}[u−ψ]|≤k}} a(x, u, ∇T_{h}(u) + t ∇T_{k}[u − ψ])∇T_{k}[u − ψ]

= H + L .

Choosing h ≥ k + ∥ψ∥_{L^{∞}(Ω)}, we have |T_{k}[u − ψ]| = k on the set {|u| ≥ h}; on the same set, we have ∇T_{h}(u) = 0. Since ∇G_{h}(u) is different from zero only on {|u| ≥ h}, we obtain

H = ∫_{{|G_{h}(u)−t T_{k}[u−ψ]|≤k}} a(x, u, 0)∇G_{h}(u) = 0 ,

being a(x, s, 0) = 0 as a consequence of (1.1). Since ∇T_{k}[u − ψ] is different from zero only on the set {x ∈ Ω : |u(x) − ψ(x)| ≤ k}, and on this set |u| ≤ k + ∥ψ∥_{L^{∞}(Ω)} ≤ h, then

{|G_{h}(u) − t T_{k}[u − ψ]| ≤ k} ∩ {|u − ψ| ≤ k}
= {| − t T_{k}[u − ψ]| ≤ k} ∩ {|u − ψ| ≤ k}
= {|u − ψ| ≤ k} ,

where the last passage is due to the fact that |t| < 1. Hence,

L = −t ∫_{Ω} a(x, u, ∇u + t T_{k}[u − ψ])∇T_{k}[u − ψ] ,

and so, for h ≥ k + ∥ψ∥_{L^{∞}(Ω)}, we have

I = −t ∫_{Ω} a(x, u, ∇u + t T_{k}[u − ψ])∇T_{k}[u − ψ] .

On the other hand, we have, since |t| < 1, T_{k}(s) is odd and u ∈ W_{0}^{1,p}(Ω),

lim_{h→+∞}⟨F, T_{k}(G_{h}(u) − t T_{k}[u − ψ])⟩ = −t⟨F, T_{k}[u − ψ]⟩

We thus have proved that

−t ∫_{Ω} a(x, u, ∇u + t T_{k}[u − ψ])∇T_{k}[u − ψ] ≤ −t⟨F, T_{k}[u − ψ]⟩ ,

for every ψ ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω), and for every k > 0. Choosing t > 0, dividing by t, and then letting t tend to zero, we obtain

∫_{Ω} a(x, u, ∇u)∇T_{k}[u − ψ] ≥ ⟨F, T_{k}[u − ψ]⟩ ,

while the reverse inequality is obtained choosing t < 0, dividing by −t, and then letting t tend to zero. This completes the proof of Theorem 2.2. □

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Lucio Boccardo Dipartimento di Matematica, Università di Roma I, Piazza A. Moro 2, 00185 Roma, Italia

e-mail: boccardo@mat.uniroma1.it