Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 10, 1-16; http://www.math.u-szeged.hu/ejqtde/

BOUNDED WEAK SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS

ABDERRAHMANE EL HACHIMI, JAOUAD IGBIDA

Abstract. In this work, we are concerned with a class of elliptic problems with both absorption terms and critical growth in the gradient. We suppose that the data belong to L^{m}(Ω) with m > n/2 and we prove the existence of bounded weak solutions via L^{∞}- estimates. A priori estimates and Stampacchia’s L^{∞}-regularity are our main ingredient.

1. Introduction

In this work, we intend to study the Dirichlet problem for some nonlinear elliptic equations whose model example is:

(P) \{ \begin{matrix}−Δu + a(x)u|u|^{r−1} = β(u)|∇u|^{2} + f(x) in Ω, \\ u|_{∂Ω} = 0. \end{matrix}

where Ω is a bounded open set in ℝ^{N}, N > 2, Δ denotes the Laplace operator and β is a continuous nonincreasing real function , with β ∈ L^{1} (ℝ). The real function a (x) is nonnegative and bounded in L^{∞} (Ω). Under suitable conditions on the data, we shall study existence and regularity of solutions for problem (P).

These kind of problems have been treated in a large literature starting from the classical references [18] and [19]. Later, many works have been devoted to elliptic problems with lower order terms having quadratic growth with respect to the gradients (see e.g. [8], [9], [13], [15], [16], [17], [22] and the references therein).

The general problem (P), though being physically natural, does not seem to have been studied in the literature. So for special situations, in the case where a = 0, β is constant and f = 0, this equation may be considered as the stationary part of equation

u_{t} − Δu = ϵ|∇u|^{2} ,

EJQTDE, 2009 No. 10, p. 1 which appears in the physical theory of growth and roughening of surfaces. It is well known as the Kardar-Parisi-Zhang equation (see [16]). It presents also the viscosity approximation as ϵ → +∞ of Hamilton-Jacobi equations from stochastic control theory (see [21]).

Key words and phrases. Nonlinear elliptic equations, critical growth, absorption terms, existence, a priori estimates, weak solutions.

For the simpler case where a = 0, β is a constant (we can assume β = 1 without loss of generality) and f ∈ L^{\frac{N}{2}} ; that is when (P) of the form

(1.1)  −Δu = |∇u|^{2} + f (x) in Ω,
u = 0 on ∂Ω,

the problem has been studied in [17], where the change of variable v = e^{u} − 1 leads to the following problem

(1.2)  −Δv = f (x) (v + 1) in Ω,
v = 0 on ∂Ω.

Then, provided that f ∈ L^{\frac{N}{2}} , it is proved there that (1.1) admits a unique solution in W_{0}^{1,2} (Ω).

In the case where a = 0, f ∈ L^{q} with q > \frac{N}{2} , and β is a continuous nonnegative function satisfying supplementary conditions according to each situation, for instance β (s) = \frac{1}{\sqrt{(1+s^{2})^{3}}} , or β (s) = \frac{e^{|s|}}{(1+s^{2})}, a pri-

ori estimates have been proved in [1] and [6] to obtain existence and regularity results, while uniqueness have been shown in [2].

In this paper, to prove existence of bounded weak solutions for (P), we assume that f ∈ L^{m}, m > \frac{N}{2} , β is a continuous real function nonincreasing with β ∈ L^{1} (ℝ) and a is a nonnegative bounded real function. We shall obtain a solution by an approximating process. Using a priori estimates and Stampacchia’s L^{∞}-regularity results we shall show that the approximated solutions converges to a solution of problem (P).

2. Preliminaries and main results

In this section, we present some notations and assumptions. We also recall some concepts and results which will be used in our further considerations. We will refer the reader to the corresponding references.

EJQTDE, 2009 No. 10, p. 2

Throughout this paper Ω will denotes a bounded open set in ℝ^{N} with N > 2. We denote by c a positive constant which may only depend on the parameters of our problem, its value my vary from line to line.

For 1 ≤ q ≤ N we denote q^{∗} = \frac{N q}{N−q}. Moreover, we denote N′ = \frac{N}{N−1} and its Sobolev conjugate by N_{0} = \frac{N}{N−2}.

For k > 0 we define the truncature at level ±k as

T_{k}(s) = min{k, max{s, −k}}.

We also consider G_{k}(s) = s−T_{k} (s) = (|s| − k)^{+} sign (s) . We introduce T_{0}^{1,2} (Ω) as the set of all measurable functions u : Ω → ℝ^{N} such that T_{k} (u) ∈ W_{0}^{1,2} (Ω) for all k > 0. We point out that T_{0}^{1,2} (Ω) ∩ L^{∞} (Ω) = W_{0}^{1,2} (Ω) ∩ L^{∞} (Ω).

For a measurable function u belonging to T_{0}^{1,2} (Ω) , a gradient can be defined as a measurable function which is also denoted by ∇u and satisfies ∇T_{k} (u) = ∇u χ_{[|u|<k]} for all k > 0 (see e.g. [3]).

We are going to investigate the solution of the following nonlinear elliptic problem

(P) \{ \begin{matrix}−Δu + a(x)u|u|^{r−1} = β(u)|∇u|^{2} + f(x) in Ω, \\ u|_{∂Ω} = 0, \end{matrix}

where Ω denotes a bounded open set in ℝ^{N} with N > 2. u denote a real function depending on x in ℝ^{N}.

We denote by γ the real function

(2.1)  γ (s) = ∫_{0}^{s} β (σ) dσ.

We assume that r > 1, and that

(2.2)  f ∈ L^{m} (Ω) with m > \frac{N}{2} .

Both functions β and a have to satisfy certain structural assumptions which are described by:

(A) There exists a_{0} such that a ≥ a_{0} > 0 a.e in Ω and a ∈ L^{∞} (Ω). (B) The real function β is continuous nonincreasing with β ∈ L^{1} (ℝ) .

Without loss of generality we assume β (0) = 0 EJQTDE, 2009 No. 10, p. 3

By a weak solution of problem (P) we mean a function u, such that both functions β(u) |∇u|^{2} and a(x)u|u|^{r−1} are integrable, and the following equality holds

(2.3)  ∫_{Ω} ∇u∇ϕ + ∫_{Ω} a(x)u|u|^{r−1}ϕ = ∫_{Ω} β(u)|∇u|^{2}ϕ + ∫_{Ω} f ϕ,

for any test function ϕ in H_{0}^{1} (Ω) ∩ L^{∞} (Ω).

Theorem 2.1. Let f in L^{m} (Ω) , m > \frac{N}{2} and r > 1. Then, under the assumptions (A) and (B) the problem (P) has at least one solution which belongs to W^{1,2} (Ω) ∩ L^{∞} (Ω).

□

3. Fundamental estimates

3.1. Estimates on general problem. In this sections we prove some basic estimate for regular elliptic problem. The main tools for proving theorem 2.1 are a priori estimates together with compactness arguments applied to a sequences of bounded approximating solutions. We shall study the nonlinear elliptic equation

(3.1)  −Δu = B(x, u, ∇u) + f(x) in Ω,

under the assumption

(3.2)  u|_{∂Ω} = 0.

Where B(x, s, ξ) = b (s, ξ) − a (x, s) |s|^{r−1}; a (., .) and b (., .) are two functions satisfying the following hypothesis:

(a_{1}) a (x, s) : Ω × R → R is measurable in x ∈ R^{N} for any fixed

s ∈ R and continuous in s for a.e. x. (a_{2}) There exists a constant c > 0 such that for all s and almost

every x

a (x, s) ≥ a (x) s + c.

(a_{3}) For any α > 0 the function

a_{α} (x) = sup_{|s|≤α}{a (x, s) |s|^{r−1}}

is integrable over Ω. (b_{1}) b (s, ξ) : R × R^{N} → R is measurable in s ∈ R for any fixed

ξ ∈ R^{N} and continuous in ξ for a.e. s. EJQTDE, 2009 No. 10, p. 4 (b_{2}) The real function b satisfies

b (s, ξ) ≤ β (s) |ξ|^{2} , for all s and ξ .

Let us note that if u is a weak solution of (3.1), then it satisfy the following equality

∫_{Ω} ∇u∇ψ = ∫_{Ω} B(x, u, ∇u)ψ + ∫_{Ω} f ψ,

for all ψ in H_{0}^{1} (Ω) ∩ L^{∞} (Ω).

We will now prove the following basic results. If u is a weak solution of (3.1) we denote u_{k} = T_{k}(u). Then we have the following estimates:

Lemma 3.1. Let u be a weak solution of problem (3.1). Then u satisfies

||u||_{L^{r}(Ω)} ≤ c.

Proof. We consider for m > 1 the following function

\begin{matrix}ψ_{m} (s) = (m − 1) ∫_{0}^{s} \frac{1}{(1+t)^{m}} & if s ≥ 0, \\ ψ_{m} (s) = −ψ_{m} (−s) & if s ≤ 0. \end{matrix}

Taking ψ_{m} (u) as test function in (3.1), where m is such that 0 < m − 1 < r −1, we obtain

∫_{Ω} ∇u ∇ψ_{m}(u) = ∫_{Ω} B(x, u, ∇u)ψ_{m}(u) + ∫_{Ω} fψ_{m} (u) .

∫_{Ω} ∇u ∇ψ_{m}(u)+∫_{Ω} a(x, u)|u|^{r−1}ψ_{m}(u) ≤ ∫_{Ω} b(u, ∇u)ψ_{m}(u)+∫_{Ω} fψ_{m} (u) .

∫_{Ω} ∇u ∇ψ_{m}(u)+∫_{Ω} a(x)u|u|^{r−1}ψ_{m}(u) ≤ ∫_{Ω} β (u) |∇u|^{2}ψ_{m}(u)+∫_{Ω} fψ_{m} (u) .

Then we have

∫_{Ω} |∇u|^{2} ψ′_{m}(u)+∫_{Ω} a(x)u|u|^{r−1}ψ_{m}(u) ≤ ∫_{Ω} β (u) |∇u|^{2}ψ_{m}(u)+∫_{Ω} fψ_{m} (u) .

(m − 1) ∫_{Ω} |∇u|^{2} \frac{1}{(1 + |u|)^{m}} + ∫_{Ω} a(x)u|u|^{r−1}ψ_{m}(u) ≤ ∫_{Ω} fψ_{m} (u) .

(3.3) (m−1) ∫_{Ω} |∇u|^{2} \frac{1}{(1 + |u|)^{m}} +∫_{Ω} a_{0} u |u|^{r−1}ψ_{m}(u) ≤ ∫_{Ω} fψ_{m} (u) .

Since s|s|^{r−1}ψ(s) is nonnegative, then using the fact that

s|s|^{r−1}ψ_{m}(s) ≥ |s|^{r}ψ_{m}(1), (ψ_{m}(1) = 1 − 2^{−m+1}), for s > 1. EJQTDE, 2009 No. 10, p. 5 We get for all u

∫_{Ω} |u|^{r} ≤ ∫_{Ω} u|u|^{r−1} \frac{ψ_{m}(u)}{ψ_{m}(1)} + |Ω|.

Then we obtain

∫_{Ω} |u|^{r} ≤ c.
□

Lemma 3.2. Let u be a weak solution of problem (3.1). Then, for N > 2 and r > 1, one has

∇u ∈ L^{q} (Ω) for any q, 1 ≤ q < N′ = \frac{N}{N − 1}

and

u ∈ L^{q^{∗}} (Ω) where q^{∗} = \frac{qN}{N − q} .

Proof. Let q ∈ [1, N′[, where N′ = \frac{N}{N−1}. We note that N′ ∈]1, N[.

We chose m such that 0 < m < m_{0} = (N′ − q)\frac{N−1}{N−q} and we use Hölder’s inequality to obtain (3.4)

∫_{Ω} |∇u|^{q}dx ≤ (∫_{Ω} |∇u|^{2} \frac{1}{(1 + |u|)^{m}} dx)^{\frac{q}{2}} (∫_{Ω} (1 + |u|)^{m \frac{q}{2−q}} dx)^{\frac{2−q}{2}} .

Moreover m < m_{0} is equivalent to m\frac{q}{2−q} < q^{∗}, thus we get for any ϵ > 0, that

(3.5)  (1 + |u|)^{m \frac{q}{2−q}} ≤ ϵ|u|^{q^{∗}} + c (ϵ) .

From (3.3) and (3.4), we obtain

(3.6)  ∫_{Ω} |∇u|^{q}dx ≤ c_{1} (∫_{Ω} |u|^{q^{∗}}dx)^{\frac{2−q}{2}} + c_{2}.

Since q < N , then from Sobolev’s inequality, we have

||u − ũ||_{L^{q^{∗}}(Ω)} ≤ ||∇u ||_{L^{q}(Ω)},

where

ũ := mes(Ω)^{−1} ∫_{Ω} u(x) dx.

This implies that

(3.7)  (∫_{Ω} |u|^{q^{∗}}dx)^{1/q^{∗}} ≤ (∫_{Ω} |∇u|^{q}dx)^{1/q} + |Ω|^{\frac{1}{q^{∗}}} ||ũ||_{r}.

EJQTDE, 2009 No. 10, p. 6 From lemma 3.1, we have for r > 1 that

||ũ||_{r}^{r} ≤ \frac{1}{|Ω|} ∫_{Ω} |u|^{r}dx ≤ c.

Using (3.6) and (3.7), we deduce that

||u ||_{L^{q^{∗}}(Ω)} ≤ \frac{1}{2} (∫_{Ω} |u|^{q^{∗}}dx)^{\frac{2−q}{2q}} + c.

Now, since N > 2, \frac{2−q}{2q} < \frac{1}{q^{∗}} = \frac{N−q}{qN}, we obtain

||u ||_{L^{q^{∗}}(Ω)} ≤ c.

□

Lemma 3.3. For a weak solution of (3.1).The following estimates

(3.8)  ∫_{[|u|>k]} |∇u|^{2}dx ≤ c ∫_{Ω} |f G_{k} (u) |,

and

(3.9)  ∫_{Ω} |∇u_{k}|^{2}dx ≤ c ∫_{Ω} |f u_{k}|,

hold for all k > 0. Proof. We define the following functions

φ_{k,h} (s) = G_{k} (T_{h} (s)) ,

ψ_{k,h} (s) = e^{γ(T_{k}(s))}φ_{k,h} (s) .

Taking ψ_{k,h} (u) as a test function in (3.1), we obtain

∫_{Ω} β(u_{k})ψ_{k,h}(u)∇u∇u_{k} + ∫_{Ω} e^{γ(u_{k})}∇u∇u_{k}φ_{k,h}(u)

= ∫_{Ω} B(x, u, ∇u)ψ_{k,h}(u) + ∫_{Ω} f ψ_{k,h}(u).

∫_{Ω} β(u_{k})ψ_{k,h}(u)∇u∇u_{k}+∫_{Ω} e^{γ(u_{k})}∇u∇u_{k}φ_{k,h}(u)+∫_{Ω} a(x, u)|u|^{r−1}ψ_{k,h}(u)

≤ ∫_{Ω} b(u, ∇u)ψ_{k,h}(u) + ∫_{Ω} f ψ_{k,h}(u). EJQTDE, 2009 No. 10, p. 7 (3.10) ∫_{Ω} β(u_{k})ψ_{k,h}(u)∇u∇u_{k}+∫_{Ω} e^{γ(u_{k})}∇u∇u_{k}φ_{k,h}(u)+∫_{Ω} a(x)u|u|^{r−1}ψ_{k,h}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}ψ_{k,h}(u) + ∫_{Ω} f ψ_{k,h}(u).

We note that

∫_{Ω} β(u_{k})ψ_{k,h}(u)∇u∇u_{k} = ∫_{[u<k]} β(u)e^{γ(u)}φ_{k,h}(u)|∇u|^{2}.

Applying monotone convergence theorem, we have

lim_{h→+∞} ∫_{Ω} β(u_{k})φ_{k,h}(u)∇u∇u_{k} = ∫_{Ω} β(u)e^{γ(u)}G_{k}(u)|∇u|^{2},

lim_{h→+∞} ∫_{Ω} β(u)|∇u|^{2}ψ_{k,h}(u) = ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}G_{k}(u).

Letting h tend to infinity in (3.10) and applying Lebesgue’s dominated convergence theorem, we obtain

∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}G_{k}(u)+∫_{Ω} e^{γ(u)}∇u∇G_{k}(u)+∫_{Ω} a(x)u|u|^{r−1}e^{γ(u)}G_{k}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}G_{k}(u) + ∫_{Ω} f e^{γ(u)}G_{k}(u).

Then, we obtain

∫_{Ω} e^{γ(u)}∇u∇G_{k}(u) ≤ ∫_{Ω} f G_{k}(u).

Therefore, we have

∫_{[|u|>k]} |∇G_{k}(u)|^{2}dx ≤ c ∫_{Ω} |f G_{k} (u) |,

which implies that (3.8) is satisfied.

To prove the second assertion, let us take

ϕ_{k,h} = e^{γ(u_{k})}u_{h}

as a test function in (3.1), we obtain

∫_{Ω} β(u_{k})ϕ_{k,h}(u)∇u∇u_{k} + ∫_{Ω} e^{γ(u_{k})}∇u∇u_{k}

= ∫_{Ω} B(x, u, ∇u)ϕ_{k,h}(u) + ∫_{Ω} f ϕ_{k,h}(u).

∫_{Ω} β(u_{k})ϕ_{k,h}(u)∇u∇u_{k} + ∫_{Ω} e^{γ(u_{k})}∇u∇u_{k} + ∫_{Ω} a(x, u)|u|^{r−1}ϕ_{k,h}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}ϕ_{k,h}(u) + ∫_{Ω} f ϕ_{k,h}(u).

(3.11) ∫_{Ω} β(u_{k})ϕ_{k,h}(u)∇u∇u_{k} + ∫_{Ω} e^{γ(u_{k})}∇u∇u_{k} + ∫_{Ω} a(x)u|u|^{r−1}ϕ_{k,h}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}ϕ_{k,h}(u) + ∫_{Ω} f ϕ_{k,h}(u).

The monotone Convergence Theorem yields

lim_{h→+∞} ∫_{Ω} β(u_{k})ϕ_{k,h}(u)∇u∇u_{k} = ∫_{Ω} β(u)e^{γ(u)}u_{k}|∇u|^{2},

lim_{h→+∞} ∫_{Ω} β(u)|∇u|^{2}ϕ_{k,h}(u) = ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}u_{k}.

Now, applying Lebesgue’s dominated convergence theorem in (3.11), we obtain

∫_{Ω} β(u)e^{γ(u)}|∇u|^{2}T_{k}(u) + ∫_{Ω} e^{γ(u)}∇u∇u_{k} + ∫_{Ω} a(x)u|u|^{r−1}e^{γ(u)}u_{k}

≤ ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}u_{k} + ∫_{Ω} f e^{γ(u)}u_{k}.

After simplifications we have

∫_{Ω} e^{γ(u)}∇u∇u_{k} ≤ ∫_{Ω} f e^{γ(u)}u_{k}.

Therefore, we get

∫_{[|u|<k]} e^{γ(u)}|∇u|^{2}dx ≤ c ∫_{Ω} |f u_{k}|.

Finally, by Fatou’s lemma, we deduce that

∫_{Ω} |∇u_{k})|^{2}dx ≤ c ∫_{Ω} |f u_{k}|.
□

Lemma 3.4. There exists a constant c such that the solution of problem (3.1) satisfies

∫_{[|u|≥k]} |b(u, ∇u)| ≤ c.

Proof. Let us consider

φ_{k}(s) = γ (G_{k}(s) + k sign(s)) − γ (k sign(s)) ,

ψ_{k,h}(s) = φ_{k}(T_{h}(s)).

Taking e^{γ(u_{h})}ψ_{k,h}(u) as test function in (3.1), we obtain ∫_{Ω} β(u_{h})e^{γ(u_{h})}ψ_{k,h}(u)∇u∇u_{k} + ∫_{Ω} e^{γ(u_{h})}∇u∇ψ_{k,h}(u)

≤ ∫_{Ω} B(x, u, ∇u)e^{γ(u_{h})}ψ_{k,h}(u) + ∫_{Ω} f e^{γ(u_{h})}ψ_{k,h}(u).

∫_{Ω} β(u_{h})e^{γ(u_{h})}ψ_{k,h}(u)∇u∇u_{k}+∫_{Ω} e^{γ(u_{h})}∇u∇ψ_{k,h}(u)+∫_{Ω} a(x, u)|u|^{r−1}e^{γ(u_{h})}ψ_{k,h}(u) ≤ ∫_{Ω} β(u)|∇u|^{2}e^{γ(u_{h})}ψ_{k,h}(u) + ∫_{Ω} f e^{γ(u_{h})}ψ_{k,h}(u).

(3.12) ∫_{Ω} β(u_{h})e^{γ(u_{h})}ψ_{k,h}(u)∇u∇u_{k}+∫_{Ω} e^{γ(u_{h})}∇u∇ψ_{k,h}(u)+∫_{Ω} a(x)u|u|^{r−1}e^{γ(u_{h})}ψ_{k,h}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}e^{γ(u_{h})}ψ_{k,h}(u) + ∫_{Ω} f e^{γ(u_{h})}ψ_{k,h}(u).

We note that

∫_{Ω} β(u_{h})e^{γ(u_{h})}ψ_{k,h}(u)∇u∇u_{k} = ∫_{[u<h]} β(u)e^{γ(u)}φ_{k,h}(u)|∇u|^{2}. From Monotone Convergence Theorem, we have

lim_{h→+∞} ∫_{Ω} β(u_{h})e^{γ(u_{h})}∇u∇u_{k}ψ_{k,h}(u) = ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}φ_{k}(u).

Letting h tend to infinity in (3.12) and applying Lebesgue’s dominated convergence theorem, we obtain

∫_{Ω} β(u)e^{γ(u)}|∇u|^{2}φ_{k}(u)+∫_{Ω} e^{γ(u)}∇u∇φ_{k}(u)+∫_{Ω} a(x)u|u|^{r−1}e^{γ(u)}φ_{k}(u)

≤ ∫_{Ω} β(u)|∇u|^{2}e^{γ(u)}φ_{k}(u) + ∫_{Ω} f e^{γ(u)}φ_{k}(u).

Then, we have

∫_{Ω} e^{γ(u)}∇u∇φ_{k}(u) ≤ ∫_{Ω} f e^{γ(u)}φ_{k}(u) + c, EJQTDE, 2009 No. 10, p. 10 which yields

∫_{[|u|<h]} φ′_{k}(u)|∇u|^{2} ≤ c ∫_{Ω} f φ_{k}(u) for all k > 0.

Therefore from Fatou’s lemma, we obtain

∫_{Ω} φ′_{k}(u)|∇u|^{2} ≤ c ∫_{Ω} f φ_{k}(u) for all k > 0,

and then we have

∫_{Ω} |β(u)|χ_{[|u|≥k]}|∇u|^{2} ≤ c.

Finally, this implies that

(3.13)  ∫_{[|u|≥k]} |b(u, ∇u)| ≤ c.
□

3.2. Estimates on the approximating solutions. This section is devoted to study the limiting process of the approximating problem. We consider the following sequence of problems which we denote by (P_{n}) :

(3.14)  −Δu_{n} = B_{n}(x, u_{n}, ∇u_{n}) + f_{n} (x) in Ω,

under the assumption

u_{n}|_{∂Ω} = 0.

Where B_{n}(x, u_{n}, ∇u_{n}) = b_{n}(u_{n}, ∇u_{n}) − a_{n}(x, u_{n})|u_{n}|^{r−1}, a_{n} and b_{n} are two sequences of functions defined by

a_{n}(x, s) = a(x)T_{n}(s), f_{n} = T_{n}(f) and b_{n}(s, ξ) = T_{n}(β(s))|ξ|^{2}.

From standard result by Leray and Lions (see e.g. [19]) there exist weak solutions, for problem(3.14), which we denote by u_{n} ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω) satisfying for all v ∈ H_{0}^{1}(Ω) ∩ L^{∞}(Ω)

∫_{Ω} ∇u_{n}∇v = ∫_{Ω} B_{n}(x, u_{n}, ∇u_{n})v + ∫_{Ω} f_{n} v.

It yields that,

(3.15) ∫_{Ω} ∇u_{n}∇v + ∫_{Ω} a_{n}(x, u_{n})|u_{n}|^{r−1}v = ∫_{Ω} b_{n}(u_{n}, ∇u_{n})v + ∫_{Ω} f_{n} v.

EJQTDE, 2009 No. 10, p. 11

Hence the previous result of the precedent section can be applied. Using lemma (3.3) we deduce that there exist a constant c such that

∫_{Ω} |∇G_{k}(u_{n})|^{2}dx ≤ c ∫_{Ω} |f G_{k} (u) |.

Applying Hölder’s inequality, we obtain

∫_{Ω} |∇G_{k}(u_{n})|^{2}dx ≤ c ||f ||_{L^{m}(Ω)} (∫_{Ω} |G_{k} (u_{n}) |^{m′})^{\frac{1}{m′}} .

Using Sobolev’s imbedding theorem, we obtain for N = \frac{2N}{N−2} that

(∫_{Ω} |e^{G_{k}(u_{n})}|^{N})^{\frac{2}{N}} ≤ c ||f ||_{L^{m}(Ω)} (∫_{Ω} |G_{k} (u_{n}) |^{m′})^{\frac{1}{m′}} .

Denoting A_{K} = {|u| ≥ k}, we get

(∫_{Ω} |G_{k}(u_{n})|^{N})^{\frac{2}{N}} ≤ c |A_{K}|^{\frac{1}{m′} − \frac{1}{N}} (∫_{Ω} |G_{k} (u_{n}) |^{N})^{\frac{1}{N}} .

Thus

(∫_{Ω} |G_{k}(u_{n})|^{N})^{\frac{1}{N}} ≤ c |A_{K}|^{\frac{1}{m′} − \frac{1}{N}} .

In this stage by using Stampacchia’s L^{∞}-regularity procedure (see [24]) we obtain that u_{n} is bounded uniformly in L^{∞}(Ω). That is

||u_{n}||_{L^{∞}(Ω)} ≤ c,

where c > 0 is a constant that only depends on the parameters of the problem.

Using lemma 3.3 we obtain

(3.16)  ∫_{Ω} |∇u_{n}|^{2}dx ≤ c,

that is u_{n} is bounded in H_{0}^{1}(Ω).

Afterwards we consider λ > max{|β(s)|; |s| ≤ k}. Then

∫_{Ω} |b_{n}(u_{n}, ∇u_{n})| = ∫_{Ω∩[|u_{n}|<k]} |b_{n}(u_{n}, ∇u_{n})| + ∫_{Ω∩[|u_{n}|≥k]} |b_{n}(u_{n}, ∇u_{n})|
≤ λ ∫_{Ω} |∇T_{k}(u_{n})|^{2} + ∫_{[|u_{n}|≥k]} |b_{n}(u_{n}, ∇u_{n})|.

EJQTDE, 2009 No. 10, p. 12

From (3.16) and lemma (3.4) we obtain

(3.17)  ∫_{Ω} |b_{n}(u_{n}, ∇u_{n})|dx ≤ c,

that is b_{n}(u_{n}, ∇u_{n}) equi-integrable.

Lemma 3.5. Let u_{n} a sequence of functions satisfying (3.15 ). Then

(3.18)  lim_{n,m→+∞} ∫_{Ω} |∇u_{n} − ∇u_{m}|dx = 0.

Proof. We consider A_{ϵ}^{m,n} = {|u_{n} − u_{m}| ≤ ϵ} ∩ Ω.

We apply the weak formulation (3.15) successively to u_{n} and u_{m} and substitute v by the function defined by ξ = inf(u_{n} − u_{m}, ϵ) if u_{n} ≥ u_{m} and ξ = − inf(u_{m} − u_{n}, ϵ) if u_{n} ≤ u_{m}.

After substraction, we obtain

∫_{A_{ϵ}^{m,n}} |∇u_{n} − ∇u_{m}|^{2}dx ≤ ϵ(∫_{Ω} |f_{n} + B_{n}(x, u_{n}, ∇u_{n})|dx

+ ∫_{Ω} |f_{m} + B_{m}(x, u_{m}, ∇u_{m})|dx).

The equi-integrability of f_{n} and B_{n}(x, u_{n}, ∇u_{n}) gives

(3.19)  ∫_{A_{ϵ}^{m,n}} |∇u_{n} − ∇u_{m}|^{2}dx ≤ ϵ c.

Let us now observe that by Hölder’s inequality, we have (3.20)

∫_{Ω} |∇u_{n}−∇u_{m}|dx ≤ c (∫_{A_{ϵ}^{m,n}} |∇u_{n} − ∇u_{m}|^{2}dx)^{\frac{1}{2}} +∫_{A{m,n \atop ϵ}} |∇u_{n}−∇u_{m}|dx;

where A_{ϵ}^{m,n} = {|u_{n} − u_{m}| ≥ ϵ} ∩ Ω.

Since lim_{n,m→+∞} ∫_{A}_{ϵ}^{m,n} |∇u_{n} − ∇u_{m}|d = 0 (the measure of A_{ϵ}^{m,n} tends

to 0 for n, m tending to +∞), then from (3.19) and (3.20), we have

lim_{n,m→+∞} ∫_{Ω} |∇u_{n} − ∇u_{m}|dx = 0.

□

EJQTDE, 2009 No. 10, p. 13

4. Existence and regularity results

Let u_{n} be the solution of the approximating problem. Then for all ψ ∈ H_{0}^{1} (Ω) ∩ L^{∞} (Ω) , we have

∫_{Ω} ∇u_{n}∇ψ = ∫_{Ω} B_{n}(x, u_{n}, ∇u_{n})ψ + ∫_{Ω} f_{n} ψ.

∫_{Ω} ∇u_{n}∇ψ + ∫_{Ω} a_{n}(x, u_{n})|u_{n}|^{r−1}ψ = ∫_{Ω} b_{n}(u_{n}, ∇u_{n})ψ + ∫_{Ω} f_{n} ψ.

From the construction of f_{n} we have

f_{n} → f in L^{1}(Ω) for n tending to + ∞.

From lemma (3.2) the solution u_{n} is bounded independently on n in W^{1,q} (Ω) , for any q, 1 ≤ q < q_{0}. Then, up to a subsequence ,that we denote again by u_{n}, there exist u ∈ W^{1,q} (Ω) , for any q, 1 ≤ q < q_{0}, such that u_{n} converge to u weakly in W^{1,q} (Ω) , for any q, 1 ≤ q < q_{0}. From Rellich-Kondrachov’s theorem we have the almost every where convergence in Ω. That is

u_{n} → u weakly in W^{1,q} (Ω) for any q, 1 ≤ q < q_{0}.
(4.1)  u_{n} → u almost every where in Ω .
a_{n}(x, u_{n}) → a(x, u) almost every where in Ω.

Taking into account the equi-integrability of u_{n} in L^{r} (Ω) , it follows that of a_{n}(x, u_{n})|u_{n}|^{r−1} in L^{1} (Ω) . Hence, we have

(4.2)  a_{n}(x, u_{n})|u_{n}|^{r−1} → a(x, u)|u|^{r−1} in L^{1} (Ω) .

From lemma 3.5 we have up to a subsequence u_{n}, that

(4.3)  ∇u_{n} → ∇u almost every where in Ω .

Since ∇u_{n} is bounded in L^{q} (Ω) for any q, 1 ≤ q < N′, we have

∇u_{n} → ∇u in L^{q} (Ω) for any q, 1 ≤ q < N′,

and then we conclude that

Δu_{n} → Δu in L^{1} (Ω) .

Now, we have from (4.1) and (4.3) that

β_{n}(u_{n})|∇u_{n}|^{2} → β(u)|∇u|^{2} almost every where in Ω.
b_{n}(u_{n}, ∇u_{n}) → b(u, ∇u) almost every where in Ω.
B (x, u_{n}, ∇u_{n}) → B (x, u, ∇u) almost every where in Ω.

EJQTDE, 2009 No. 10, p. 14 From (3.17) we obtain that

b_{n}(u_{n}, ∇u_{n}) → β(u)|∇u|^{2} in L^{1} (Ω) ,

and from (4.2), that

B (x, u_{n}, ∇u_{n}) → B (x, u, ∇u) in L^{1} (Ω) .

Which conclude to the desired convergence result.

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(Received November 17, 2008)

Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles, Faculté des sciences, b. P. 20, El Jadida, Maroc

E-mail address: aelhachi@yahoo.fr

Jaouad Igbida UFR Mathématiques Appliquées et Industrielles, Faculté des sciences, B. P. 20, El Jadida, Maroc

E-mail address: jigbida@yahoo.fr