QUASILINEAR ELLIPTIC EQUATIONS WITH NATURAL GROWTH 1 ...

Differential and Integral Equations

Volume xx, Number xxx, , Pages xx–xx

QUASILINEAR ELLIPTIC EQUATIONS WITH NATURAL GROWTH Boumediene Abdellaoui∗ Département de Mathématiques, Université Aboubekr Belka¨ıd, Tlemcen Tlemcen 13000, Algeria Lucio Boccardo Dipartimento di Matematica, Universit` a di Roma I Piazza A. Moro 2, 00185 Roma, Italy Ireneo Peral∗ and Ana Primo∗ Departamento de Matemáticas, U. Aut´ onoma de Madrid 28049 Madrid, Spain (Submitted by: James Serrin) Abstract. In this paper we deal with the problem −div (a(x, u)∇u) + g(x, u, ∇u) = λh(x)u + f in Ω, u = 0 on ∂Ω. The main goal of the work is to get hypotheses on a, g and h such that the previous problem has a solution for all λ > 0 and f ∈ L1 (Ω). In particular, we focus our attention in the model equation with a(x, u) = 1 (1 + |u|m ), g(x, u, ∇u) = m |u|m−2 u|∇u|2 and h(x) = . 2 |x|2

1. Introduction This work is devoted to some results concerning to the following quasilinear elliptic equations with natural growth: −div (a(x, u)∇u) + g(x, u, ∇u) = λh(x)u + f in Ω, (1.1) u = 0 on ∂Ω. Assume that 0 ∈ Ω, bounded open set in RN with N ≥ 3,

(1.2)

λ ≥ 0, f ∈ L1 (Ω),

(1.3)

Accepted for publication: June 2007. AMS Subject Classifications: 35D05, 35D10, 35J20, 35J25, 35J70, 46E30, 46E35. ∗ Partially supported by projects MTM2004-02223, MEC, Spain and CCG06UAM/ESP-0340, CAM-UAM.. 1

2

Boumediene Abdellaoui, Lucio Boccardo, Ireneo Peral, and Ana Primo

and a(x, s) : Ω×R → R, g(x, s, ξ) : Ω×R×RN → R are measurable functions with respect to x and continuous with respect to (s, ξ). We assume the existence of α, σ, μ ∈ R+ and β, θ continuous, increasing (possibly unbounded) functions of a real variable such that 0 < α ≤ a(x, s) ≤ β(s),

(1.4)

|g(x, s, ξ)| ≤ θ(s)|ξ| ,

(1.5)

2

g(x, s, ξ)s ≥ 0,

∀ s ∈ R,

(1.6)

and for |s| ≥ σ we have

(1.7) |g(x, s, ξ)| ≥ μ|ξ|2 . We focus our attention on the model equation with a(x, u) = (1 + |u|m ), m−2 u|∇u|2 and h(x) = 1 . We have the associated g(x, u, ∇u) = m 2 |u| |x|2 functional of energy u2 λ 1 m 2 (1 + |u| )|∇u| dx − dx − f (x)u(x) dx, m > 1, J(u) = 2 Ω 2 Ω |x|2 Ω whose Euler-Lagrange equation is m−2 u|∇u|2 = λ u + f in Ω, −div ((1 + |u|m )∇u) + m 2 |u| |x|2 u = 0 on ∂Ω.

(1.8)

For λ ≡ 0, equations of the form (1.1) have been widely studied in the literature. We refer to [3], [7], [8] and the references therein. In the case where m > 0 and λ = 0, in [6], the existence of finite-energy solutions with only the weak assumption f ∈ L1 (Ω) is proved. This is thanks to the presence of the lower-order term with quadratic dependence with respect to the gradient and the condition (1.6). Notice that this result is somewhat surprising because it is not true in the linear case (m = 0). If λ > 0, m = 0 and f ∈ Lp (Ω), in [10] it is proved that the classical result ∗∗ u ∈ Lm (Ω) holds only if λ < [N (p − 1)(N − 2p)]/[p2 ]. In the case where a(x, u) ≡ 1 and g(x, u, ∇u) = |∇u|q , q > 1, and under the hypothesis (2.1) on h, it is proved in [1] (see also [2]) that problem (1.1) has a positive solution for all λ > 0 and for all f ∈ L1 (Ω) such that f ≥ 0. This means that the term |∇u|q produces a strong regularizing effect and then breaks down any resonance effect of the linear zero-order term. The main goal of this work is to extend the previous results to a general operator under the conditions (1.4)–(1.7). Since we will not assume any sign condition on f , we will then prove the existence of a nontrivial solution for all λ > 0 and f ∈ L1 (Ω). We point out that the gradient-dependent term g

Quasilinear elliptic equations with natural growth

3

improves the regularity of the solutions, whereas the zero-order term makes smaller the summability of the solutions. The paper is organized as follows. We start by analyzing the meaning of the hypothesis (2.1) on h and giving some preliminary results that we will use later. Section 3 is devoted to proving results of existence. We begin by proving an approximating existence result by taking a regularized version of the main problem. The result follows using closely the arguments of [15] and [9]. The second part of the section is devoted to proving the main existence result by using a priori estimates and then passing to the limit in the approximating problem. In the last section we study the problem of regularity of the solution depending on the regularity of f . 2. Preliminary results We say that h is an admissible weight in the sense that the following hypothesis holds: 1 2 dx 2 |∇φ| Ω h 0, h ∈ L1 (Ω) and C(h) = inf > 0. (2.1) 1,2 φ∈W0 (Ω)\{0} Ω h|φ| dx Remark 2.1. It is obvious that if h satisfies (2.1), then h ∈ W −1,2 (Ω) ∩ L1 (Ω). Moreover, (2.1) implies that h |u| dx < ∞ for all u ∈ W01,2 (Ω). a) Ω

b) Defining H : W01,2 (Ω) → R by

H, u ≡

hu dx,

(2.2)

Ω

we have that H is a linear continuous form on W01,2 (Ω). Thus we get the existence of F = (f1 , f2 , . . . , fN ) ∈ (L2 (Ω))N such that h = −div(F ), and then hu dx = F , ∇u dx. H, u ≡ Ω

Ω

As a consequence, the duality product is equivalent to the first integral and H = F (L2 (Ω))N . Proposition 2.2. Assume that h satisfies (2.1) and consider H defined by (2.2).

4


Define hn (x) = min{h(x), n} and the corresponding linear continuous form R Hn : W01,2 (Ω) → u

→ Hn (u) =

hn u dx. Ω

The following statements hold. i) Hn → H strongly in W −1,2 (Ω). ii) Assume that un u weakly in W01,2 (Ω), un ≥ 0 and un → u almost everywhere; then hn un → hu strongly in L1 (Ω). Proof. i) As in Remark 2.1 we also have that Hn , u = hn u dx; Ω

that is, the duality is realized by the integral, and moreover 1 2 dx 2 |∇φ| Ω inf > 0. C(hn ) = φ∈W01,2 (Ω)\{0} Ω hn |φ| dx Notice that

Hn W −1,2 =

u

≤

u

sup

≤1 1,2 W0 (Ω)

|Hn , u | ≤

sup

≤1 1,2 W0 (Ω)

Ω

u

sup

≤1 1,2 W0 (Ω)

|

hn u| Ω

h|u| ≤ H W −1,2 .

Then {Hn }n∈N , up to a subsequence, converges weakly in W −1,2 (Ω). As for all u ∈ W01,2 (Ω), the Lebesgue theorem proves that hn |u| → h|u|

strongly in L1 (Ω), then {Hn }n∈N H.

Therefore,

H W −1,2 ≤ lim inf Hn W −1,2 ≤ lim sup Hn W −1,2 ≤ H W −1,2 , n→∞

n→∞

and hence,

Hn → H, strongly in W −1,2 (Ω). ii) According to the strong convergence proved in i), we have that if un u weakly in W01,2 (Ω), then hn un dx → hu dx = H, u as n → ∞. Hn , un = Ω

Ω


5

If we assume that, moreover, un ≥ 0 and un → u almost everywhere, then by using a well-known result in [5] we obtain that hn un → hu strongly in L1 (Ω). (See also [14], Theorem 1.9, page 21.) 3. Existence of solution In this subsection we prove the existence of a solution to problem (1.1). The existence is obtained for admissible weights h; i.e., h satisfies (2.1), for all λ > 0 and f ∈ L1 (Ω). The existence of solution will be proved by approximation. Our techniques will follow those of [4] and [1]. Define gn (x, s, ξ) =

g(x, s, ξ) , 1 + n1 |g(x, s, ξ)|

where

Tk (s) =

and an (x, s) = a(x, Tn (s)),

s, sk |s| ,

if |s| ≤ k, if |s| > k.

First, we consider the problem (1.1) with f, h ∈ Ls (Ω), s >

N 2.

Theorem 3.1. Assume (1.2), (1.4), (1.5), (1.6), (1.7), λ ≥ 0 and f, h ∈ Ls (Ω), with s > N2 . Then there exists a solution u ∈ W01,2 (Ω) ∩ L∞ (Ω) to problem (1.1) in the distributional sense, ⎧ ⎪ u ∈ W01,2 (Ω) ∩ L∞ (Ω), g(x, u, ∇u) ∈ L1 (Ω), ⎪ ⎪ ⎪ ⎨ a(x, u)∇u∇ϕ dx + g(x, u, ∇u) ϕ dx (3.1) Ω Ω ⎪ ⎪ ⎪ 1,2 ⎪ f ϕ dx, ∀ϕ ∈ W0 (Ω). = λ Ω h(x)uϕ dx + ⎩ Ω

Proof. Thanks to the Leray-Lions existence theorem ([12]), for every fixed k > 0 we find a sequence of solutions {um,k } to the problems −div (am (x, um,k )∇um,k ) + gm (x, um,k , ∇um,k ) (3.2) um ∈ W01,2 (Ω) ∩ L∞ (Ω), = λh(x)Tk (um ) + f, for all m ∈ N. Using the fact that f, h ∈ Ls (Ω) for s > N2 and using the Stampacchia regularity theorem [15] and the assumption (1.6), we get the existence of a positive constant C such that

um,k ∞ ≤ C(k, g, f, h, Ω).

(3.3)

6


Taking now um,k as test function in (3.2), we obtain |∇um,k |2 dx + gm (x, um,k , ∇um,k )um,k dx α Ω Ω h(x)um,k dx + f um,k dx ≤ λk Ω Ω 2 |∇um,k | dx + Cε λk + f um,k dx. ≤ λkε Ω

Ω

Hence we get the existence of a positive constant C(k, g, f, Ω) such that α |∇um,k |2 dx ≤ C(k, g, f, Ω); Ω

therefore, um,k uk weakly in W01,2 (Ω). It is clear that

uk ∞ ≤ C(k, g, f, h, Ω), where C(k, g, f, h, Ω) is defined in (3.3). To get the strong convergence of {um,k } in W01,2 (Ω) we follow closely the arguments used in [9]. 2 Let φ(s) = seT s , where T is a positive constant that we will choose later. Using φ(um,k − uk ) as a test function in (3.2) we get am (x, um,k )∇um,k φ (um,k − uk )∇(um,k − uk ) dx Ω gm (x, um,k , ∇um,k )φ(um,k − uk ) dx (3.4) + Ω = (λh(x)Tk (um,k ) + f )φ(um,k − uk ) dx. Ω

Direct computation shows that am (x, um,k )∇um,k φ (um,k − uk )∇(um,k − uk ) dx Ω φ (um,k − uk )|∇(um,k − uk )|2 dx + o(1). ≥α Ω

Using the fact that um,k and uk are bounded and that um,k uk weakly in W01,2 (Ω) we obtain gm (x, um,k , ∇um,k )φ(um,k − uk ) dx Ω |∇um,k |2 θ(um,k )|φ(um,k − uk )|dx (3.5) ≤ Ω


7

≤

|∇(um,k − uk )|2 θ(C)|φ(um,k − uk )|dx, Ω

where we have used the fact that um,k ∞ ≤ C. Since (λh(x)Tk (um,k ) + f )φ(um,k − uk ) dx = o(1) as

m → ∞,

Ω

we conclude that {αφ (um,k − uk ) − θ(C)|φ(um,k − uk )|}|∇(um,k − uk )|2 dx ≤ o(1). Ω

Choosing T large enough such that αT s2 + α − θ(C)s ≥ we obtain that 1 2

α , 2

|∇(um,k − uk )|2 dx ≤ o(1). Ω

Hence, the strong convergence holds. Then using the fact that {un } is bounded in L∞ (Ω) and the above strong convergence we can pass to the limit in problem (3.2) to conclude that uk is a solution to −div (a(x, uk )∇uk ) + g(x, uk , ∇uk ) = λh(x)Tk (uk ) + f, (3.6) uk ∈ W01,2 (Ω) ∩ L∞ (Ω), with f, h ∈ Ls (Ω), s > N2 . Taking uk as test function in (3.6) it follows that |∇uk |2 dx + |g(x, uk , ∇uk )uk | ≤ λ h(x)u2k dx + f uk dx. α Ω

Ω

Ω

Ω

Applying Sobolev’s and Young’s inequalities we obtain that for all ε, ε¯ > 0 there exist Cε , Cε¯, β > 0 such that β |∇uk |2 dx ≤ λ Cε¯ h N + Cε f N = C(ε, ε¯, f, h, λ) uniformly in k Ω

L

2

L

2

and, since f, h ∈ Ls (Ω) with s > N2 , we can prove that the sequence {uk } is uniformly bounded in L∞ (Ω). Moreover, uk u in W01,2 (Ω). As a consequence, we also have that u ∈ W01,2 (Ω) ∩ L∞ (Ω). To get the existence result we have to prove that uk → u in W01,2 (Ω), but this follows immediately using previous computations.

8


Assume now that hypotheses (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7) hold, and let h(x) be an admissible weight in the sense of hypothesis (2.1). Thanks to Theorem 3.1 we can consider the approximated Dirichlet problems, un ∈ W01,2 (Ω) ∩ L∞ (Ω) :

(3.7)

− div (a(x, un )∇un ) + g(x, un , ∇un ) = λhn (x)Tk (un ) + fn , where {fn } is a sequence of smooth functions converging uniformly to f in L1 (Ω) (for example we can take fn = Tn (f )) and hn (x) = min{h(x), n}. The use in (3.7) of the test function Tk (un ) yields for any k > 0 (see [6]), 2 a(x, un )|∇Tk (un )| dx + g(x, un , ∇un )Tk (un ) Ω Ω hn (x)un dx + k fn dx. ≤ λk Ω

Ω

Since g(x, un , ∇un )Tk (un ) dx g(x, un , ∇un )un dx + k = Ω

{|un (x)|≤k}

{|un (x)|≥k}

g(x, un , ∇un )

un dx, |un |

from (1.6) and (1.7), by setting k = σ, we have that for all ∀ε > 0 2 |∇Tσ (un )| + μσ |∇un |2 α Ω {x∈Ω:σ≤|un (x)|} ≤ σλεC(h) |∇un |2 dx + C (ε, k, f ). Ω

Choosing ε small enough, we conclude that {un } is bounded in W01,2 (Ω), and then there exists u ∈ W01,2 (Ω) such that un u weakly in W01,2 (Ω) and un → u strongly in Lp (Ω), ∀p < 2∗ = N2N −2 . + − − Consider un = max{un , 0} and un = max{−un , 0}, then un = u+ n − un . We prove the following lemma. Lemma 3.2. It follows that lim |g(x, un , ∇un )| dx k→∞ {x∈Ω: |un (x)|≥k+1} = lim |∇un |2 dx = 0. k→∞ {x∈Ω: |un (x)|≥k+1}

(3.8)


9

Proof. Using T1 (Gk (un )) as a test function in (3.7) and by the condition (1.6) we obtain that for all k > 0, |g(x, un , ∇un )| dx (3.9) {x∈Ω: k+1≤|un (x)|} ≤λ hn (x)|un | dx + |f | dx {x∈Ω: k≤|un (x)|}

{x∈Ω: k≤|un (x)|}

so that the choice k = σ − 1 gives the following estimate: |∇un |2 dx μ {x∈Ω:k+1≤|un (x)|} ≤λ hn (x)|un | dx + {x∈Ω: k≤|un (x)|}

{x∈Ω: k≤|un (x)|}

(3.10) |f | dx.

The measure of the set {x ∈ Ω : |un (x)| ≥ k} tends to zero as k tends to infinity uniformly in n, since un → u strongly in Lp (Ω), ∀p < 2∗ . Proposition 3.3. Fixing k > 0, it follows that Tk (un ) → Tk (u) strongly in W01,2 (Ω). Proof. We denote wn = Tk (un ) and w = Tk (u). It is immediate that

wn − w W 1,2 (Ω)

(3.11)

0

≤ (wn+ − w+ )+ W 1,2 (Ω) + (wn− − w− )+ W 1,2 (Ω) + (wn − w)− W 1,2 (Ω) . 0

0

0

We estimate each term of the second member. On the one hand, for every l > 0,

(wn+ − w+ )+ W 1,2 (Ω) = Tl (wn+ − w+ )+ W 1,2 (Ω) + Gl (wn+ − w+ )+ W 1,2 (Ω) . 0 0 0 (3.12) The first term on the right hand can be estimated in the following way. We take Tl (wn+ − w+ )+ as test function in (3.7): + + + 2 α |∇Tl (wn − w ) | dx + gn (x, un , ∇un )Tl (wn+ − w+ )+ dx Ω Ω + + + hn (x)un Tl (wn − w ) dx + fn Tl (wn+ − w+ )+ dx. ≤λ Ω

Ω

Thanks to condition (1.6), it immediately follows that for every l > 0,

Tl (wn+ − w+ )+ W 1,2 (Ω) goes to zero as l and n tend to infinity. 0

10


The second term on the right hand can be estimated in the following way. For every l > 0, |∇(wn+ − w+ )|2 dx

Gl (wn+ − w+ )+ W 1,2 (Ω) = + 0 + {x∈Ω:l≤wn (x)−w (x)} + 2 ≤2 |∇wn | dx + 2 |∇w+ |2 dx. + {x∈Ω:l≤wn (x)}

+ {x∈Ω:l≤wn (x)}

From (3.10),

Gl (wn+

2λ ≤ μ

− w ) W 1,2 (Ω) hn (x)wn+ dx + 0 {x∈Ω: l−1≤wn (x)} 2 + |f | dx + 2 |∇w+ |2 dx. + μ {x∈Ω: l−1≤wn+ (x)} {x∈Ω:l≤wn (x)} + +

Since the measure of the set {x ∈ Ω : wn+ (x) ≥ l} tends to zero as l tends to infinity uniformly in n, then passing to the limit as l and n tend to infinity 1,2 + + + + and since u+ n u weakly in W0 (Ω), it follows that Gl (wn −w ) W01,2 (Ω) goes to zero. Therefore, it follows that lim (wn+ − w+ )+ W 1,2 (Ω) = 0.

n→∞

0

(3.13)

Following the same arguments, we have that lim (wn− − w− )+ W 1,2 (Ω) = 0.

n→∞

0

(3.14)

We continue by estimating the third term on the right hand in (3.11). 2 Consider the function φ(s) = seT s , used above, with T > 0 to be chosen later. Using φ((wn − w)− ) = φ((w − wn )+ ) as test function in (3.7) it follows that a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx (3.15) Ω g(x, un , ∇un )φ((w − wn )+ ) dx ≤ (λh(x)un + fn )φ((w − wn )+ ) dx. + Ω

Ω

We analyze term by term in the previous equation (3.15). On the left hand, it follows that a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx Ω a(x, un )∇un φ ((w − wn )+ )∇((w − wn )) dx = {x∈Ω:wn ≤w≤k}


= +

{x∈Ω:wn ≤w≤k}

{x∈Ω:wn ≤w≤k}

11

a(x, un )|∇((w − wn ))|2 φ ((w − wn )+ ) dx a(x, un )∇wφ ((w − wn )+ )∇((w − wn )) dx.

Since un u in W01,2 (Ω), then arguing by duality we conclude that a(x, un )∇un φ ((w − wn )+ )∇((w − wn )+ ) dx (3.16) Ω a(x, un )|∇((wn − w)− )|2 φ ((w − wn )+ ) dx + o(1). = Ω

Now we estimate the second term on the left hand. Taking φ((w − wn )+ ) as a test function in (3.7), g(x, un , ∇un )φ((w − wn )+ ) dx Ω g(x, un , ∇un )φ((w − wn )+ ) dx = {x∈Ω:wn ≤w≤k} ≤ θ(un )|∇wn |2 φ((w − wn )+ ) dx {x∈Ω:wn ≤w≤k} = θ(un )|∇(wn − w)|2 φ((w − wn )+ )dx {x∈Ω:wn ≤w≤k} − θ(un )|∇w|2 φ((w − wn )+ )dx {x∈Ω:wn ≤w≤k} +2 θ(un )∇wn ∇w φ((w − wn )+ )dx. {x∈Ω:wn ≤w≤k}

For every k > 0, we consider A = {x ∈ Ω : wn (x) ≤ w(x) ≤ k} and M = maxs∈A θ(s). Recall that θ is continuous increasing function given in (1.5) and that wn = Tk (un ) and w = Tk (u). Since φ((w − wn )+ ) converges to zero in Lp (Ω), ∀p > 1, and thanks to an argument by duality, we have that g(x, un , ∇un )φ((w − wn )+ ) dx (3.17) Ω |∇((wn − w)− )2 |φ((w − wn )+ )|dx. ≤M {x∈Ω:wn ≤w≤k}

12


From (3.16) and (3.17), we conclude that |∇((wn − w)− )2 |(αφ − M |φ|)((w − wn )+ ) dx = o(1). Ω

Choosing T in such a way that φ satisfies (αφ − M |φ|) ≥ 12 , it follows that for every k > 0, lim (wn+ − w+ )− W 1,2 (Ω) = 0.

n→∞

0

(3.18)

From (3.13), (3.14), and (3.18) we conclude that for every fixed k > 0 we have that Tk (un ) → Tk (u) strongly in W01,2 (Ω).

Proposition 3.4. The sequence g(x, un , ∇un ) converges to g(x, u, ∇u) in L1 (Ω). Proof. Since g(x, un , ∇un ) → g(x, u, ∇u) almost everywhere, we prove that g(x, un , ∇un ) is uniformly equi-integrable. For any σ ∈ R+ , |g(x, un , ∇un )| dx E 2 θ(σ)|∇Tσ (un )| dx + |g(x, un , ∇un )| dx. ≤ {x∈E:|un (x)|≥σ}

E

Hence, choosing E with |E| small enough and applying Lemma 3.2 we obtain the conditions to apply Vitali’s theorem, and we conclude. Remark 3.5. In the same way, we can prove that |∇un |2 converges to |∇u|2 strongly in L1 (Ω). Since |∇un |2 converges to |∇u|2 almost everywhere, we prove that |∇un |2 is uniformly equi-integrable. For any σ ∈ R+ , 2 2 |∇un | dx = |∇Tσ (un )| dx + |∇un |2 dx. E

E

{x∈E:|un (x)|≥σ}

Hence, choosing E with |E| small enough and applying Lemma 3.2 we get the main conditions to apply Vitali’s Theorem, and then the result follows. Thus, we can now pass to the limit in (3.7) to obtain that u is a solution to (1.1) in a suitable sense. Assume ϕ ∈ W01,2 (Ω) ∩ L∞ (Ω); then using the above regularity results we can use Tk [u − ϕ] as test function in (1.1). More precisely, we have proved the next existence result.


13

Theorem 3.6. Assume (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7), and let h(x) be an admissible weight. There exists a solution u to problem (1.1) in the following sense: ⎧ ⎪ u ∈ W01,2 (Ω), g(x, u, ∇u) ∈ L1 (Ω), ⎪ ⎪ ⎪ ⎨ a(x, u)∇u∇Tk [u − ϕ] + g(x, u, ∇u) Tk [u − ϕ] (3.19) Ω Ω ⎪ ⎪ ⎪ 1,2 ∞ ⎪ f Tk [u − ϕ], ∀ϕ ∈ W0 (Ω) ∩ L (Ω), ∀k > 0. ⎩ = Ω

Remark 3.7. Notice that in this case we do not know in general if a(x, u)|∇u|2 ∈ L1 (Ω) nor that ug(x, u, ∇u) ∈ L1 (Ω). It is clear that in general, if u is a solution to problem (1.1) in the sense of Theorem 3.6, then we don’t know if u is a distributional solution. To overcome this difficulty we need to add extra conditions on a. More precisely, we have the next result. Theorem 3.8. Let α < 1 and assume that a(x, s) satisfies the following hypothesis: ⎧ 2α ⎨ a (x, s)|ξ|2 ≤ C|g(x, s, ξ)|, ∀|s| ≥ σ, (H) ≡ (3.20) ⎩ 2(1−α) (x, s) ≤ C|s|β , β < 2∗ , ∀|s| ≥ σ. a Then if u ∈ W01,2 (Ω) is a solution to problem (1.1) in the sense of (3.19), we have that u is a solution in the distributional sense. Proof. It is sufficient to prove that a(x, un )|∇un | converges to a(x, u)|∇u| in L1 (Ω). Let E ⊂ Ω be a measurable set, a(x, un )|∇un | dx E a(x, un )|∇Tσ (un )| dx + a(x, un )|∇un | dx. = {x∈E:|un (x)|≥σ}

E

Hence, choosing E with |E| small enough and since Tk (un ) → Tk (u) in W01,2 (Ω), it follows that a(x, un )|∇Tσ (un )| dx → 0 uniformly in n. (3.21) E

Let us analyze the second term on the right hand in the equation (3.21). a(x, un )|∇un | dx {x∈E:|un (x)|≥σ}

14


≤

a (x, un )|∇un | dx + 2α

{x∈E:|un (x)|≥σ}

2

{x∈E:|un (x)|≥σ}

a2(1−α) (x, un ) dx.

Since a(x, s) satisfies hypothesis (H) in (3.20), then using the fact that un → u strongly in Lp (Ω), ∀p < 2∗ , and thanks to Lemma 3.2, it follows that choosing E with |E| small enough, a(x, un )|∇un | dx → 0 uniformly in n. {x∈E:|un (x)|≥σ}

Applying Vitali’s theorem, we conclude the proof.

Remark 3.9. Let us consider the main example given in (1.8); then, applying the previous Theorem 3.8, we obtain that m−1 |∇u|2 , ∀|u| ≥ σ, (1 + |u|m )2α |∇u|2 ≤ C m 2 |u| (1 + |u|m )2(1−α) ≤ C|u|β , β < 2∗ , ∀|u| ≥ σ. 1,2 ∗ Let α = m−1 2m < 1; then if 1 < m < 2 − 1 and u ∈ W0 (Ω) is a solution to problem (1.8) in the entropy sense, we have that u is a solution in the distributional sense.

3.1. Regularity of solutions. According to the previous section, there exists a solution u ∈ W01,2 (Ω) to problem (1.1) for all f ∈ L1 (Ω). In this section, we report the extra summability of u given by extra summability of f , under the next stronger hypothesis on h. It is worth pointing out that the one order term provides an extra summability, beyond of that given by the classical Stampacchia results. Let consider the problem (1.1) with h satisfying the condition |∇φ|2 dx 1 Ω h(x) ≥ 0, h(x) ∈ L (Ω), λ1 (h) = inf > 0. (3.22) 2 φ∈W01,2 (Ω)\{0} Ω h|φ| dx In particular, notice that condition (3.22) implies condition (2.1). Consider f ∈ Lm (Ω), and let’s distinguish two cases according to duality of f : m ≥ 2N 2N N +2 and 1 < m < N +2 . The proofs are similar to those in [1] (see also [11]) and thus we will skip them. Theorem 3.10. (1) Assume that f ∈ Lm (Ω) with m ≥ N2N +2 and that hypothesis (3.22) 1,2 holds. Let u ∈ W0 (Ω) be a solution to problem (1.1). Then, |u|q |u|p |∇u|2 dx < ∞, for all 1 ≤ p < 2, and dx < ∞, ∀q < 4. 2 Ω Ω |x|


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(2) Assume that f ∈ Lm (Ω) with 1 < m < N2N +2 , and suppose that 1,2 hypothesis (2.1) holds for h. Let u ∈ W0 (Ω) be a solution to problem (1.1); then 2∗ 2(m − 1) 2N up |∇u|2 dx < ∞, ∀p < p¯ = , 2∗ = . ∗ 2m − 2 (m − 1) N −2 Ω Theorem 3.11. Assume that f ∈ Lm (Ω) with m ≥ be a solution to problem (1.1) with h(x) = |x|1 2 .

2N N +2 .

Let u ∈ W01,2 (Ω)

1,2 N p i) If 2 , then u ∈ W0 (Ω) for all p > 1; in particular, we have m≥ up dx < ∞, ∀p ≥ 1 and α < N. α Ω |x| m(N −2) N 1 2∗ ii) If N2N +2 ≤ m < 2 , then by setting r = 2 2∗ + 1 ≡ N −2m ≥ 2 we m−

have up ∈ W01,2 (Ω), for all p ≤ r.

2

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[10] L. Boccardo, L. Orsina, and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete and Continuous Dynamical System, 16 (2006), 513–523. [11] T. Leonori, Large solutions for a class of nonlinear el liptic equations with gradient terms, Adv. Nonlinear Stud., 7 (2007), 237–270. [12] J. Leray and J.L. Lions, Quelques rèsultats de Visik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97–107. [13] J. Maly and W. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, 51 (1997). [14] E.H. Lieb and M. Loos, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, R.I., 2001. [15] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre a ` coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189–258.