Ricerche mat. (2014) 63 (Suppl 1):S41–S56 DOI 10.1007/s11587-014-0198-4

Continuous dependence on the data for nonlinear elliptic equations with a lower order term M. F. Betta · A. Mercaldo · R. Volpicelli

Received: 23 June 2014 / Revised: 13 July 2014 / Published online: 1 August 2014 © Universitá degli Studi di Napoli "Federico II" 2014

Abstract We consider a class of Dirichlet boundary value problems for nonlinear elliptic equations with a first order term and we prove continuous dependence on the data for solutions to problems in such a class. We deduce uniqueness results which in a case improve known results. Keywords equations

Continuous dependence on the data · Uniqueness · Nonlinear elliptic

Mathematical Subject Classification

35J25 · 35J60

1 Introduction In the present paper we prove continuous dependence on the data for solutions to homogeneous Dirichlet problem of the type 

−div (a (x, ∇u)) = H (x, ∇u) + f in  u=0 on ∂,

(1.1)

Communicated by Salvatore Rionero. M. F. Betta Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola C4, 80143 Naples, Italy e-mail: [email protected] A. Mercaldo · R. Volpicelli (B) Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo, Via Cintia, 80126 Naples, Italy e-mail: [email protected] A. Mercaldo e-mail: [email protected]

123

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M. F. Betta et al.

where  is a bounded open subset of R N , N ≥ 2. We assume that a :  × RN → RN and H :  × RN → R are Carathéodory functions which satisfy the ellipticity condition a (x, z) · z ≥ |z| p ,

(1.2)

     a (x, z) − a x, z  · z − z  > 0, z = z  ,

(1.3)

the monotonicity condition

and the growth conditions |a (x, z)| ≤ a0 |z| p−1 + a1 , a0 , a1 > 0, |H (x, z)| ≤ h |z|q , h > 0

(1.4) (1.5)

with 1 < p < N , p − 1 < q < p, for almost every x ∈ R N , for every z, z  ∈ R N . Moreover we assume that f belongs to a suitable Lebesgue space. The first difficulty which arises in studying such a problem is to give a meaning to the notion of solution. When the right-hand side of (1.1) is an element of the dual  space W −1, p (), weak solutions are well-defined. When this case does not occur a different notion of solution has to be considered. We adopt “solution obtained as limit of approximations” ([13], see also [14]). In Sect. 2 we describe this type of solution; here we just recall that a “solution obtained as limit of approximations” u to (1.1) is the limit of a sequence of bounded weak solutions to the so-called approximated problems whose data are regular enough and approach the datum f in some sense. Equivalent notion of solution can be found in litterature such as renormalized solution [22,25] or entropy solution [7]. Existence of solutions to problem (1.1) have been proved by many authors; we only quote a few contributions and we refer to the references therein: [1,3,17–20,23]. All these results require a smallness condition on the datum f and they have been proved by distinguishing three intervals of values of q, i.e. p−1
N ( p − 1) , N −1

N ( p − 1) p ≤q < p−1+ , N −1 N p ≤ q < p. p−1+ N

123

(1.6) (1.7) (1.8)

Continuous dependence on the data

S43

In [3] existence results have been proved under sharp assumptions on the summability of the datum f and its smallness condition. In the present note we assume that q belongs only to the intervals (1.6) or (1.7) and we use existence results of [3] when f belongs to Lebesgue spaces. Precisely when q satisfies (1.6), the existence of a “solution obtained as limit of approximations” to (1.1) is given in [3, Theorem 5.2] under the assumption that f ∈ L 1 () and satisfies the smallness condition 1/N

f L 1 < K 1 =

q

(N ω N ) q− p+1 ||

q−N (q− p+1) N (q− p+1)



q − N (q − p + 1) N (q − p + 1)h



p−1 q− p+1

.

(1.9)

Analogously when q satisfies (1.7), with q > NN( p−1) −1 , the existence of a “solution obtained as limit of approximations” to (1.1) is proved in [3, Theorem 5.5] under the assumption that f ∈ L f

L

N (q− p+1) q

N (q− p+1) q

() and satisfies the smallness condition

  q   ( p−1) p − 1 q− p+1 q − p+1 1/N N (q − p + 1) − q q− p+1 ωN < K0 = . q q − p+1 hq (1.10)

A similar result holds true also in the limit case q = NN( p−1) −1 (see [3, Theorem 5.6]) when we assume that, for a constant M > ||, the datum f satisfies the condition  0

||

f ∗ (s) log N −1 (M/s) ds < K 2 =

ω N (N − 1) N −1 N N . h N −1

(1.11)

Here f ∗ is the decreasing rearrangement of f , i.e. f ∗ (s) = sup {t ≥ 0 : μ(t) > s} , s ∈ [0, ||) , where μ(t) = |{x ∈  : | f (x)| > t}| , t ≥ 0, is the distribution function of f . The purpose of this note is to prove the continuous dependence on the data which obviously implies uniqueness results for problem (1.1). These results are available under more restrictive structural assumptions on the operator: we change the monotonicity condition (1.3) in the following “strong monotonicity” condition, i.e.    2      a (x, z) − a x, z  · z − z  ≥ α(ε + |z| + z  ) p−2 z − z   ,

α>0

(1.12)

where ε is nonnegative and strictly positive if p > 2, and we assume the locally Lipschitz condition on H , i.e.

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         H (x, z) − H x, z   ≤ β η + |z| + z   q−1 z − z   , β > 0,

(1.13)

where η is nonnegative and strictly positive if 1 < p ≤ 2. Moreover, as pointed out, we assume that q belongs to the interval (1.6) or (1.7). Actually in order to prove uniqueness, one has to assume q ≤ Np + p −1, since, even if existence results for weak solutions have been proved when q belongs to the interval in (1.8), (see [3,20]) classical counter-examples show that uniqueness of these solutions fails when q > p − 1 + Np . However, our method does not allow to prove continuous dependence on the data when q = Np + p − 1, that is we have not a uniqueness result for such a value of q. Uniqueness result for this value of q have been proved in [9] and [26], by proving a comparison principle. In contrast our results improve the known uniqueness results (cf. [2,9,11,26] and the references therein) available in letterature when p − 1 < q < p − 1 + Np . Now we can state our main results. In the following we denote by λ∗ , for 1 < λ < N , λ . the Sobolev exponent of λ, i.e. λ∗ = NN−λ Theorem 1.1 Let N ≥ 2 and p such that ⎧ 2N ⎪ ⎨ < p < 2, if N = 2 N +1 ⎪ ⎩ 2N < p ≤ 2, if N ≥ 3. N +1

(1.14)

Assume (1.2), (1.4), (1.5), (1.12) and (1.13) with p−1
N ( p − 1) . N −1

Let u, v be “solutions obtained as limit of approximations” to problem (1.1) with data f, g ∈ L 1 () such that f L 1 < K 1,

g L 1 < K 1 ,

with K 1 defined in (1.9). Then we have u − v L γ ≤ C f − g L 1 ,

(1.15)

for every γ < NN( p−1) − p and the constant C depends on N , ||, p, q, α, β, η , h, f L 1 and g L 1 . Moreover C depends on f L 1 and g L 1 in such a way that it is bounded when f, g vary in sets which are bounded and equi-integrable in L 1 (). In particular problem (1.1) has a unique “solution obtained as limit of approximations”. Theorem 1.2 Let N ≥ 2 and p such that 

123

1 < p < 2, if N = 2 1 < p ≤ 2, if N ≥ 3.

(1.16)

Continuous dependence on the data

S45

Assume (1.2), (1.4), (1.5), (1.12) and (1.13) with p N ( p − 1)
N (q − p + 1) N (2 − p) max , q p

< m < ( p ∗ ) .

Let u, v be “solutions obtained as limit of approximations” to problem (1.1) with data f, g such that f N (q− p+1) < K 0 , g N (q− p+1) < K 0 (1.17) L

q

L

q

with K 0 defined in (1.10). Then we have u − v L γ ∗ ≤ C f − g L m ,

(1.18)

for every γ < m ∗ ( p − 1), where the constant C depends on N , ||, p, q, α, β, η, h, f L m and g L m . Moreover C depends on f L m and g L m in such a way that it is bounded when f, g vary in sets which are bounded and equi-integrable in L m (). In particular problem (1.1) has a unique “solution obtained as limit of approximations”. Theorem 1.3 Let N ≥ 2 and p such that 

1 < p < 2, 1 < p ≤ 2,

if N = 2 if N ≥ 3.

(1.19)

Assume (1.2), (1.4), (1.5), (1.12) and (1.13) with q=

N ( p − 1) N −1

and suppose that f, g ∈ L m (), with

 N (2 − p) < m < ( p ∗ ) . max 1, p Let u, v be “solutions obtained as limit of approximations” to problem (1.1) with data f, g such that, for a constant M > ||, 

|| 0

f ∗ (s) log N −1 (M/s) ds < K 2



||

g ∗ (s) log N −1 (M/s) ds < K 2 . (1.20)

0

where K 2 is defined in (1.11). Then we have u − v L γ ∗ ≤ C f − g L m ,

(1.21)

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for every γ < m ∗ ( p − 1), where the constant C depends on N , ||, p, q, α, β, η, h, f L m and g L m . Moreover C depends on f L m and g L m in such a way that it is bounded when f, g vary in sets which are bounded and equi-integrable in L m (). In particular problem (1.1) has a unique “solution obtained as limit of approximations”. Observe that the first result gives uniqueness under the sharp assumptions of the existence result when the data are in L 1 () and improves the uniqueness result in [12] (see also [9,21,26]) since we have uniqueness for a larger interval of values of p, in the case p ≤ 2. Theorems 1.2 and 1.3, according to known uniqueness results, give uniqueness for “solution obtained as limit of approximations” to problem (1.1) with data f having a larger summability with respect to the summability of f which assures the existence; such an assumption gives larger summability of the gradient of a solution (see Sect. 2). Continuous dependence on the data also holds true for p > 2. We state the results in Sect. 3, but in this case we have not improvements of the known uniqueness results. Our approach is based on classical methods of symmetrization due to [27], which allow to prove a pointwise estimate for the difference of two solutions to problem (1.1) by using isoperimetric inequality and classical properties of rearrangement, as made in [4,5,10]. 2 Preliminary results In this section we recall a few preliminary results and we begin by explaining what we mean for “solution obtained as limit of approximations” to problem (1.1). As pointed out in the Introduction the definition of weak solution does not fit the case when p − 1 < q < p − 1 + Np , since in general the right-hand side of (1.1) is  not more an element of the dual space W −1, p () and we refer to “solution obtained as limit of approximations”. If f belongs to the Lebesgue space L m (), for some m ≥ 1, a measurable function u :  → R is called “solution obtained as limit of approximations” to problem (1.1) (see [13,14]) if 1, p

(i) For every k > 0, Tk (u) ∈ W0 () ; (ii) A sequence of functions f n ∈ C0∞ () exists such that f n −→ f strongly in L m ()

(2.1)

and a sequence of weak solutions u n ∈ W0 () ∩ L ∞ () to the approximated problems  −div (a (x, ∇u n )) = Tn (H (x, ∇u n )) + f n in  (2.2) on ∂, un = 0 1, p

satisfies u n → u a.e. in . ∇u n → ∇u a.e. in  ,

123

(2.3) (2.4)

Continuous dependence on the data

S47

(iii) u is a solution in the sense of distributions to problem (1.1), i.e.    a(x, ∇u)∇φ = H (x, ∇u)φ + f φ, ∀φ ∈ C0∞ (). 





(2.5)

Here Tn denotes the usual truncation at level n > 0, i.e. defined, for a given n > 0, as  Tn (s) =

s n signs

if |s| ≤ n if |s| > n,

s ∈ R.

We explicitely remark that the gradient ∇u is defined according to Lemma 2.1 in [7] which states the existence of a measurable function v :  → R N such that ∇Tk (u) = vχ{|u|≤k} a.e. in , for every k > 0. We define the gradient ∇u as N  1 1,1 () , then u belongs to Wloc () and v is the this function v. If v belongs to L loc distributional gradient of u (see [15]). A few brief comments about the definition of “solution obtained as limit of approximations” are in order. Assuming that f satisfies the sharp smallness condition, if (2.1) holds, one can prove a priori estimates for |∇u n |. These imply the existence of a function u which satisfy (i) and (2.3), (2.4). Then a standard passage to the limit gives (iii). Finally we recall some a priori estimates for the gradients of these solutions to (1.1) when the datum f has a “larger summability”. They have been proved in [9] when the datum is in Lorentz spaces (see also [20]); these estimates say how the summability of the gradient of a “solution obtained as limit of approximations” to (1.1) increases when the summability of the source term increases. Here we just state the result which we use in the following; it concerns weak solutions to approximated problems (2.2) introduced above. Proposition 2.1 Let 1 < p < N . Assume (1.2–1.5) with p−1
p . N

Let u ∈ W0 () ∩ L ∞ () be a solution to (1.1) with f ∈ L ∞ () which satisfies N ( p−1) p (1.9), if p − 1 < q < NN( p−1) −1 or satisfies (1.10), if N −1 < q < p − 1 + N or 1, p

satisfies (1.11), if q =

N ( p−1) N −1 .

Then ∇u L m ∗ ( p−1) ≤ C

for

(2.6)

 N (q − p + 1) < m < ( p ∗ ) . max 1, q

Here C is a positive constant depending on N , p, q, m, h, || , and f L m . Moreover C depends on f L m in such a way that it is bounded when f varies in sets which are bounded and equi-integrable in L m ().

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3 Proofs of main results We begin by proving Theorem 1.1. Proof of Theorem 1.1 Since u, v are “solutions obtained as limit of approximations” to (1.1), there exist f n , gn ∈ C0∞ () functions which strongly converge to f, g in L 1 () 1, p respectively and u n , vn ∈ W0 () ∩ L ∞ () weak solutions to the approximated problems (2.2) with data f n , gn respectively. Moreover (2.3) and (2.4) hold. Set wn = u n − vn and h n = f n − gn . For any fixed t, k > 0 we consider the function

φn,t

⎧ ⎪ ⎨ k sign wn = wn − t sign wn ⎪ ⎩ 0

if |wn | > t + k if t < |wn | ≤ t + k otherwise.

By using φn,t as test function in the weak formulation of problem (2.2) with data f n and gn respectively, subtracting and dividing by k, we have 1 k

 t<|wn |≤t+k

[a(x, ∇u n ) − a(x, ∇vn )] · ∇wn d x 



=

|wn |>t+k

[Tn (H (x, ∇u n ))−Tn (H (x, ∇vn ))] sign wn d x +

|wn |>t+k



h n sign wn d x

1 [Tn (H (x, ∇u n )) − Tn (H (x, ∇vn ))] (wn − t sign wn ) d x k t<|wn |≤t+k  1 h n (wn − t sign wn ) d x. + k t<|wn |≤t+k

+

By (1.12) and (1.13) we get α k

 t<|wn |≤t+k



≤β

|∇wn |2 dx (|∇u n | + |∇vn |)2− p

|wn |>t+k

(η + |∇u n | + |∇vn |)q−1 |∇wn | d x +

 |wn |>t+k

|h n | d x

 β (η + |∇u n | + |∇vn |)q−1 |∇wn ||wn − t sign wn | d x k t<|wn |≤t+k  1 |h n ||wn − t sign wn | d x = I1 + I2 + I3 + I4 . + k t<|wn |≤t+k

+

Let us observe that I3 , I4 go to zero as k vanishes. Moreover we estimate I2 by the Hardy-Littlewood inequality (see, for example, [8], Chapter 2, Proposition 3.6), and we get

123

Continuous dependence on the data

−



d dt

S49

|∇wn |2 |wn |>t

(|∇u n | + |∇vn |)2− p

dx ≤

β α

 (η + |∇u n | + |∇vn |)q−1 |∇wn | d x

|wn |>t  νn (t)

1 + α

0

h ∗n (σ ) dσ.

(3.1)

where νn denotes the distribution function of wn . We now evaluate the first integral on the right-hand side of (3.1). By coarea formula and Schwarz inequality, we obtain  (η + |∇u n | + |∇vn |)q−1 |∇wn | d x |wn |>t

+∞

  d (η + |∇u n | + |∇vn |)q−1 |∇wn | d x dτ dτ |wn |>τ t  21  +∞  d |∇wn |2 − dx ≤ dτ |wn |>τ (|∇u n | + |∇vn |)2− p t 1   2 d 2q− p × − (η + |∇u n | + |∇vn |) dτ. dτ |wn |>τ 

−

=

(3.2)

N ( p−1) Since we assume N2N +1 < p ≤ 2, it results p − 1 ≤ p/2 < N −1 and then in order to estimate the integral on the right-hand side of (3.2) we have to distinguish the cases q > p/2 or q ≤ p/2. Let us begin with the case q > p/2. Following [6] (see also [16,24]) we define a function Fn such that  d (η + |∇u n | + |∇vn |)2q− p d x. (3.3) Fn (νn (t))|νn (t)| = − dt |wn |>t

A significant property tells that Fn is weak limit of functions with the same rearrangement as (η + |∇u n | + |∇vn |)2q− p , therefore any Lebesgue norm of Fn can be estimated from above with the same norm of (η + |∇u n | + |∇vn |)2q− p . Hence (3.2) becomes  (η + |∇u n | + |∇vn |)q−1 |∇wn | d x |wn |>t

+∞ 



≤

d − dτ

t

 |wn |>τ

|∇wn |2 dx (|∇u n | + |∇vn |)2− p

 21

1

1

[Fn (νn (τ ))] 2 |νn (τ )| 2 dτ. (3.4)

1

Set K N = N ω NN ; by using the isoperimetric and Schwarz inequalities, it follows K N νn (t)

1− N1

  21   d d |∇wn |2 ≤− |∇wn | d x ≤ − dx dt |wn |>t dt |wn |>t (|∇u n | + |∇vn |)2− p  1  2 d × − (η + |∇u n | + |∇vn |)2− p d x . (3.5) dt |wn |>t

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Then if we set G n (νn (t))|νn (t)|



d =− dt

|wn |>t

(η + |∇u n | + |∇vn |)2− p d x ,

(3.6)

inequality (3.5) implies  21   1 1 |∇wn |2 d 1≤ dx [G n (νn (t))] 2 |νn (t)| 2 . − 2− p 1− N1 dt (|∇u | + |∇v |) n n |wn |>t K N νn (t) (3.7) Combining (3.4) and (3.7), we have  (η + |∇u n | + |∇vn |)q−1 |∇wn | d x 1

|wn |>t



≤

+∞ 

t

×

d − dτ

 |wn |>τ

|∇wn |2 dx (|∇u n | + |∇vn |)2− p

1 2



1

[Fn (νn (τ ))] [G n (νn (τ ))] 2 |νn (τ )| 1

K N νn (τ )1− N

dτ.

(3.8)

Therefore (3.1) implies 

|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |)     d |∇wn |2 β +∞ − dx ≤ α t dτ |wn |>τ (|∇u n | + |∇vn |)2− p 1 1  1 νn (t) ∗ [Fn (νn (τ ))] 2 [G n (νn (τ ))] 2 |νn (τ )| dτ + h n (σ ) dσ. × 1 α 0 K N νn (τ )1− N

−

d dt

(3.9)

Via Gronwall’s lemma, we obtain 

|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |)   1 1  νn (t)  β [Fn (σ )] 2 [G n (σ )] 2 1 νn (t) ≤ exp dσ h ∗n (r ) dr, 1 α 0 αKN r σ 1− N

d − dt

if it results



||

1

1

[Fn (σ )] 2 [G n (σ )] 2

0

1

σ 1− N

dσ < +∞.

Since it is well known that |∇u n |, |∇vn | ∈ L ρ (), with ρ < Theorem 5.1) then Fn ∈ with 2− p < θ <

123

r L 2q− p ()

(3.10)

with 2q − p < r <

N ( p−1) N −1 . Therefore (3.11) holds if r, θ

N ( p−1) N −1

(3.11) N ( p−1) N −1

(see e.g. [3] θ

and G n ∈ L 2− p ()

are such that

1 N

>

2q− p 2− p 2r + 2θ .

Continuous dependence on the data

S51

This choise of ρ and θ is possible since N ( p−1) N −1 .

(2q− p)(N −1) 2N ( p−1)

>

1 N

(2− p)(N −1) 2N ( p−1)

+

when

From (3.7), (3.10) and (3.11), in a standard way (see [27]), we get the q < following differential inequality −

d G n (s) (wn )∗ (s) ≤ C 2−2/N ds s



s

h ∗n (r ) dr ≤ C

0

G n (s) h n L 1 , s ∈ (0, ||). (3.12) s 2−2/N

Here and in the following C will be a positive constant which can vary from line to line and it depends only on the data of the problem but not on the norm of the source term. Integrating between s and || and using Hölder inequality we obtain 

∗

(wn ) (s) ≤ C G n

θ

L 2− p

h n L 1

||

r

   θ −2+ N2 2− p

 dr

1   θ 2− p

, s ∈ (0, ||).

s

(3.13) Since G n θ is bounded by a constant which does not depend on n and L 2− p   θ   2 −2 + N 2− p + 1 < 0 we have 2

(wn )∗ (s) ≤ C h n L 1 s −1+ N −

2− p θ

, s ∈ (0, ||).

(3.14)

By passing to the limit as n goes to infinity, it follows 2

(u − v)∗ (s) ≤ C h L 1 s −1+ N −

2− p θ

, s ∈ (0, ||).

(3.15)

By (3.15), (1.15) easily follows since for any γ < NN( p−1) − p it is possible to choose   N ( p−1) 2− p 2 2 − p < θ < N −1 such that 1 − N + θ γ < 1. Assume now q ≤ p/2. Proceeding as in the previous case, instead of (3.4) we have  |wn |>t

≤η

(η + |∇u n | + |∇vn |)q−1 |∇wn | d x

2q− p 2

+∞ 

 t

d − dτ

 |wn |>τ

|∇wn |2 dx (|∇u n | + |∇vn |)2− p

 21

1

[νn (τ )] 2

(3.16)

and using (3.7)  |wn |>t

≤η

(η + |∇u n | + |∇vn |)q−1 |∇wn | d x

2q− p 2

+∞ 

 t

d − dτ

 |wn |>τ

|∇wn |2 dx (|∇u n | + |∇vn |)2− p



1

[G n (νn (τ ))] 2 |νn (τ )| 1

K N νn (τ )1− N

dτ.

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M. F. Betta et al.

Then by (3.1) 

|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |) 2q− p    +∞  d |∇wn |2 βη 2 − ≤ d x α dτ |wn |>τ (|∇u n | + |∇vn |)2− p t 1  1 νn (t) ∗ [G n (νn (τ ))] 2 |νn (τ )| dτ + h n (σ ) dσ. × 1 α 0 K N νn (τ )1− N

d − dt

(3.17)

Using Gronwall’s lemma we get 

|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |)  2q− p    νn (t) [G (σ )] 21 βη 2 1 νn (t) n ≤ exp dσ h ∗n (r ) dr. 1− N1 α 0 αKN r σ

d − dt

if it results that



||

(3.18)

1

[G n (σ )] 2

dσ < +∞. (3.19) 1 σ 1− N Then the proof goes on in the same way as before. So we have only to check that 0

1

2θ

if p >

2N N +1 .

(3.19) holds. This is true, since G n2 ∈ L 2− p with 2 − p < θ < 1 N

>

2− p 2θ

since

1 N

>

(2− p)(N −1) 2N ( p−1)

N ( p−1) N −1

and moreover

Now we prove Theorem 1.2 Proof of Theorem 1.2. As in the proof of Theorem 1.1 we arrive to (3.2). Now we have to distinguish the case q > p/2 by q ≤ p/2. When q > p/2 we arrive at 

|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |)   1 1  νn (t)  β [Fn (σ )] 2 [G n (σ )] 2 1 νn (t) exp dσ h ∗n (r ) dr, ≤ 1− N1 α 0 αKN r σ

−

d dt

(3.20)

with Fn and G n defined in (3.3) and (3.6) respectively if 

||

1

1

[Fn (σ )] 2 [G n (σ )] 2 1

σ 1− N

0

By Proposition 2.1, |∇u n |, |∇vn | ∈ L m Gn ∈ L

m ∗ ( p−1) 2− p

123

(). Hence

1 N

>

dσ < +∞.

∗ ( p−1)

(), then Fn ∈ L

2q− p 2− p 2m ∗ ( p−1) + 2m ∗ ( p−1)

(3.21) m ∗ ( p−1) 2q− p

since we assume m >

() and

N ( p−q+1) . q

Continuous dependence on the data

S53

From (3.7), (3.20) and (3.21), in a standard way (see [27]), we get the following differential inequality d G n (s) − (wn )∗ (s) ≤ C 2−2/N ds s



G n (s) h n L m , s ∈ (0, ||). 1−2/N +1/m s 0 (3.22) Integrating between s and ||, by Hölder inequality, we obtain s

h ∗n (r ) dr ≤ C



∗

(wn ) (s) ≤ C h n L m G n L

||

r

m ∗ ( p−1) 2− p

  ∗  m ( p−1)  −1+ N2 − m1 2− p

1    m ∗ ( p−1)

dr

2− p

,

s

s ∈ (0, ||).

(3.23)

Observe that G n m ∗ ( p−1) is bounded by a constant which does not depend on n L 2− p    ∗ ( p−1)  and we have that −1 + N2 − m1 m 2− + 1 < 0 for m < ( p ∗ ) < Np . Therefore p 2

(wn )∗ (s) ≤ C h n L m s N

p − m1 − m ∗2− ( p−1)

, s ∈ (0, ||).

(3.24)

By passing to the limit as n goes to infinity it follows 2

(u − v)∗ (s) ≤ C h L m s N

p − m1 − m ∗2− ( p−1)

, s ∈ (0, ||).

 By (3.25), (1.21) easily follows since − N2 + m1 + m ∗ ( p − 1). Now assume p/2 ≥ q. Instead of (3.20) we have

2− p



m ∗ ( p−1)

(3.25)

γ ∗ < 1 for γ <



|∇wn |2 dx 2− p |wn |>t (|∇u n | + |∇vn |)  2q− p    νn (t) [G (σ )] 21 βη 2 1 νn (t) n exp dσ h ∗n (r ) dr , ≤ 1 α 0 αKN r σ 1− N

−

d dt

if it results:



||

1

This is true, since G n2 ∈ L 2− p 2m ∗ ( p−1) .

1

[G n (σ )] 2 1

σ 1− N

0

(3.26)

2m ∗ ( p−1) 2− p

dσ < +∞.

() and since m >

(3.27) N (2− p) p

we have

1 N

>

Then the proof goes on exactly as in the previous case.

Finally we prove Theorem 1.3 Proof of Theorem 1.3 We can argue as in the proof of Theorem 1.2 once we observe ∗ that assumptions (1.20) implies |∇u n |, |∇vn | ∈ L m ( p−1) ().

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The same arguments allow to prove analogous results when p > 2. In this case we get continuous dependence on the data which does not improve the known uniqueness results. Here, for completeness, we just give the statements of our theorems. Theorem 3.1 Let N ≥ 3 and p such that ⎧ ⎨ 2 < p < 3, 2(N − 2) , ⎩2 < p < N −3

if N = 3 if N ≥ 4.

(3.28)

Assume (1.2), (1.4), (1.5), (1.12) and (1.13) with p−1
p p−1 + . N −1 2

Let u, v be “solutions obtained as limit of approximations” to problem (1.1) with data f, g ∈ L 1 () such that f L 1 < K 1 , g L 1 < K 1 , where K 1 is defined in (1.9). Then we have u − v L γ ∗ ≤ C f − g L 1 ,

(3.29)

for every 1 < γ < NN−1 and the constant C depends on N , ||, p, q, α, h, β, ε, f 1 and g 1 . Moreover C depends on f 1 and g 1 in such a way that it is bounded when f, g vary in sets which are bounded and equi-integrable in L 1 (). In particular problem (1.1) has a unique “solution obtained as limit of approximations”. Theorem 3.2 Let N ≥ 3 and p such that ⎧ ⎨ 2 < p < 3, 2(N − 2) , ⎩2 < p < N −3

if N = 3 if N ≥ 4.

(3.30)

Assume (1.2), (1.4), (1.5), (1.12) and (1.13) with p−1 p p p + ≤q< + . N −1 2 2 N Let u, v be “solutions obtained as limit of approximations” to problem (1.1) with data f, g respectively such that either (i) If

p−1 N −1

+

p 2

≤q<

N ( p−1) N −1 ,

f, g ∈ L 1 () where K 1 is defined in (1.9) or

123

and f L 1 < K 1 , g L 1 < K 1

Continuous dependence on the data

(ii) If q =

S55

N ( p−1) N −1



||

f ∗ (s) log N −1 (M/s) ds < K 2 ,

0



||

g ∗ (s) log N −1 (M/s) ds < K 2

0

for a constant M > ||, where K 2 is defined in (1.11) or p p (iii) If NN( p−1) −1 < q < 2 + N , f, g ∈ L

N (q− p+1) q

()

and f

L

N (q− p+1) q

< K 0 , g

L

N (q− p+1) q

< K0

where K 0 is defined in (1.10) Moreover assume that f, g ∈ L m () with (2q − p)N < m < ( p ∗ ) . p − 2 + 2q

(3.31)

u − v L γ ∗ ≤ C f − g L m ,

(3.32)

Then we have for every 1 < γ < NN−1 and the constant C depends on N , ||, p, q, α, β, ε, h, f L m and g L m . Moreover C depends on f L m and g L m in such a way that it is bounded when f, g vary in sets which are bounded and equi-integrable in L m (). In particular problem (1.1) has a unique “solution obtained as limit of approximations”. References 1. Alvino, A.: Sharp a priori estimates for some nonlinear elliptic problems. Boll. Accademia Gioenia di Scienze Naturali in Catania 46, 2–11 (2013) 2. Alvino, A., Betta, M.F., Mercaldo, A.: Comparison principle for some class of nonlinear elliptic equations. J. Differ. Equ. 12, 3279–3290 (2010) 3. Alvino, A., Ferone, V., Mercaldo, A.: Sharp a-priori estimates for a class of nonlinear elliptic equations with lower order terms. Ann. Mat. Pura Appl. doi:10.1007/s10231-014-0416-4 4. Alvino, A., Mercaldo, A.: Nonlinear elliptic problems with L 1 data: an approach via symmetrization methods. Mediterr. J. Math. 5, 173–185 (2008) 5. Alvino, A., Mercaldo, A.: Nonlinear elliptic equations with lower order terms and symmetrization methods. Boll. Unione Math. Italy 1, 645–662 (2008) 6. Alvino, A., Trombetti, G.: Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri. Ricerche Math. 27, 413–428 (1978) 7. Bénilan, Ph., Boccardo, L., Gallouët, Th., Gariepy, R. , Pierre, M., Vázquez, J.L.: An L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 22, 241–273 (1995) 8. Bennett, C., Sharpley, R.: Interpolation of operators. Academic Press, Boston (1988) 9. Betta, M.F., Di Nardo, R., Mercaldo, A., Perrotta, A.: Gradient estimates and comparison principle for some nonlinear elliptic equations (2014) 10. Betta, M.F., Mercaldo, A.: Uniqueness results for nonlinear elliptic equations via symmetrization methods. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 21, 1–14 (2010)

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