Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data

Transcription

1 Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data Olivier Guibé - Anna Mercaldo 2 Abstract In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems { whose prototype is p u div(c(x) u (P ) γ ) + b(x) u λ = µ in, u = 0 on, where is a bounded open subset of IR N, N 2, p is the so called p Laplace operator, < p < N, µ is a Radon measure with bounded variation on, 0 γ p, 0 λ p, c and b belong to the Lorentz spaces L N p,r N (), p r + and LN, () respectively. In particular we prove the existence result under the assumption that γ = λ = p, b L N, () is small enough and c L N p,r (), with r < +. We also prove a stability result for renormalized solutions to a class of noncoercive equations whose prototype is (P ) with b 0. Key Words: Existence, stability, nonlinear elliptic equations, noncoercive problems, measures data, renormalized solutions. Introduction In this paper we consider nonlinear elliptic problems whose prototype is { p u div(c(x) u γ ) + b(x) u λ = µ in, u = 0 on, (.) Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS Université de Rouen, Avenue de l Université, BP.2, 7680 Saint-Étienne du Rouvray, France; Olivier.Guibe@univ-rouen.fr 2 Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo, via Cintia, 8026 Napoli, Italy; mercaldo@unina.it

2 where is a bounded open subset of IR N, N 2, p is the so called p Laplace operator, p is a real number such that < p < N, µ is a Radon measure with bounded variation on, 0 γ p, 0 λ p, c and b belong to the Lorentz spaces L N p,r N (), r + and p LN, (), respectively. We are interested in existence results for renormalized solution to (.). The difficulties which arise in proving existence results for solutions to (.) are due both to the lack of coercivity of the operator and to the right-hand side which is a measure (and not an element of the dual space W,p ()). Existence results for noncoercive elliptic problems are well-known when the datum µ belongs to the dual space W,p (). Indeed the linear case, i. e. p = 2, γ = λ =, has been studied by Stampacchia in [St] (see also [Dr]), the case where the operator has only the term b(x) u λ is studied in [DP], the case where the operator has only the term div(c(x) u γ ) is studied in [B] and finally the case where the operator has the two lower order terms div(c(x) u γ ) and b(x) u λ is studied in [DPo2] (see also [G2] for a different proof). If p is greater than the dimension N of the ambient space, then, by Sobolev embedding theorem and duality arguments, the space of measures with bounded variation on is a subset of W,p (), so that the existence of solutions in W,p 0 () is a consequence of previous results. Thus this explain our restriction on p. However, when p N, it is necessary to change the framework in order to study problem (.), since simple examples (the Laplace operator in a ball, i.e. p = 2, b = 0, c = 0, and µ the Dirac mass in the center) show that, in general, the solution has not to be expected in the energy space W,p 0 (). In the linear case Stampacchia defined a notion of solution of problem (.) by duality ([St]), for which he proved the existence and uniqueness under the assumption that 0 is not in the spectrum of the operator, condition which is satisfied if, for example, c N or b L p L () N () is small enough. He also proved that such a solution satisfies the equation in distributional sense and it belongs to W,q 0 () for every q < N/(N ). The techniques used by Stampacchia heavily relies on a duality argument, so that they can not be extended to the general nonlinear case, except in the case where p = 2, the operator has not lower order terms and it is Lipschitz continuous with respect to the gradient ([M2]). The nonlinear case was firstly studied in [BG], [BG2], then the effect of lower order terms were analyzed in [D] (where a term b(x) u λ is considered) and in [DPo] (where both terms div(c(x) u γ ) and b(x) u λ are considered); in all these papers the existence of a solution which belongs to W,q 0 () for every q < N(p )/(N ) and satisfies the equation in the distributional sense is proved when p > 2. The N hypothesis on p is motivated by the fact that, if p 2 N(p ), then. On the N N other hand a classical counterexample ([S], see also [P]) shows that in general such a 2

3 solution is not unique. This implies that, in order to obtain the existence and uniqueness of a solution for p close to, i.e. p 2, it is necessary to go out of the framework of classical N Sobolev spaces. For this reason two equivalent notions of solutions have been introduced: the notion of entropy solution in [BBGGPV], [BGO] and the notion of renormalized solution in [LM], [M], for which the existence and uniqueness have been proved in the case where the datum µ belongs to L () or to L () + W,p (). In [DMOP] the notion of renormalized solution has been extended to the case of general measure with bounded total variation and existence (and partial uniqueness results) is proved. In such papers operators without lower order terms are considered. Both difficulties (right-hand side measure and lower order terms, which produce a lack of coercivity) have been faced in [B], [BGu], [BGu2] and [BMMP3]. In [B] the existence of entropy solutions is proved when the datum µ belongs to L () and the operator has a lower order term of the type div (c(x) u γ ); in [BGu] and [BGu2] the existence of a renormalized solution is proved in the same case. Finally in [BMMP3] the existence of a renormalized solution is proved when the datum µ is a general measure with bounded total variation and the operator has only a lower order term of the type b(x) u λ. In such papers no assumptions on the smallness of the coefficients are made and therefore the operators are in general noncoercive. Uniqueness results for renormalized solution are proved in [BMMP2], when the datum µ is a measure in L () + W,p () and the operator has a lower order term of the type b(x) u λ (see [BMMP4] for the case where µ belongs to W,p ()) and in [BGu], [BGu2] when µ is a measure in L () and a lower order term of the type div(c(x) u γ ) is considered (see also [G] for further uniqueness results). In the present paper and in [GM], we prove the existence of renormalized solutions for the problems whose prototype is (.), where both the two lower terms div(c(x) u γ ) and b(x) u λ appear and where 0 γ p, 0 λ p, c belongs to the Lorentz space L N p,r N (), r +, b belongs to Lorentz space p L N, () and µ is a Radon measure with bounded variation on. In both papers we do not make any coercivity assumption on the operator: we assume that the norm of one of the two coefficients is small when γ = λ = p, while no smallness of such norms is required when γ or λ are less than p. In the present paper we consider nonlinear elliptic problems whose model is (.) and we prove an existence result in the case where µ is a Radon measure with bounded variation on, γ = p, λ = p, b L N, () is small enough and c N L p,r (), N r < + is large. The case γ = p and λ < p and the case γ < p and p λ < p are also studied. The counterpart of the existence result proved in the present paper can be found in [GM], where in particular we prove the existence of a renormalized solution for the 3

4 problem (.) in the case where µ is a Radon measure with bounded variation on, N γ = λ = p, b L N, () is large and c N L p,r, r < + is small enough. () p Let us now explain the idea of the proof of the existence result in the present paper. The first difficulty is to obtain some a priori estimate for u p. This is done by proving uniform estimates of the level sets of u (cf. [B], [BGu], [BGu2]), which allow to obtain an estimate of T k (u) of the type T k (u) p (L p ()) km + L for N every k > 0. Such estimate of T k (u) then imply u p (L N, ()) c, thanks N to a generalization of a result of [BBGGPV], proved in [BMMP3]. Finally we use an extension of the stability result proved in [DMOP] (see also [MP] and [M]). It allows us to handle the term div(c(x) u γ ), which in general does not belong to the dual space W,p (). Such a result could be proved by using the same arguments of [DMOP], but actually the proof which we give here is slightly different. Finally we explicitly remark that, as for the existence result proved in the present paper, the main difficulty in proving the existence result of [GM] is to obtain some a priori estimate for u p. Such a priori estimates are obtain by using a different method. 2 Definitions and statement of existence result In this section we recall some well-known results about the decomposition and convergence of measures (cf. [DMOP]) and some properties of Lorentz spaces (see e.g. [Lo], [H], [O]), which we will use in the following. Then we give the definition of a renormalized solution to nonlinear elliptic problems whose right-hand side is a Radon measure (cf. [DMOP]) and we state our existence result. 2. Preliminaries about measures In this paper is a bounded open subset of IR N, N 2, and p is a real number, < p < N, with p defined by /p + /p =. We denote by M b () the space of Radon measures on with bounded variation and by C 0 b () the space of bounded, continuous functions on. Moreover µ + and µ denote the positive and the negative parts of the measure µ, respectively. Definition 2. We say that a sequence {µ ε } of measures in M b () converges in the narrow topology to a measure µ in M b () if for every ϕ C 0 b (). lim ϕdµ ε = ϕdµ, (2.) ε 0 4

5 Remark 2.2 We recall that, if µ ε is a nonnegative measure in M b (), then {µ ε } converges in the narrow topology to a measure µ if and only if µ ε () converges to µ() and (2.) holds true for every ϕ C 0 (). It follows that if µ ε is a nonnegative measure, µ ε converges in the narrow topology to µ if and only if (2.) holds true for any ϕ C ( ). We denote by cap p (B, ) the standard capacity defined from W,p 0 () of a Borel set B and we define M 0 () as the set of the measures µ in M b () which are absolutely continuous with respect to the p-capacity, i.e. which satisfy µ(b) = 0 for every Borel set B such that cap p (B, ) = 0. We define M s () as the set of all the measures µ in M b () which are singular with respect to the p-capacity, i.e. which are concentrated in a set E such that cap p (E, ) = 0. The following result allows to split every measure in M b () with respect to the p-capacity ([FST], Lemma 2.). Proposition 2.3 For every measure µ in M b () there exists an unique pair of measures (µ 0, µ s ), with µ 0 M 0 () and µ s M s (), such that µ = µ 0 + µ s. The measures µ 0 and µ s will be called the absolutely continuous part and the singular part of µ with respect to the p-capacity. Actually, for what concerns µ 0 one has the following decomposition result ([BGO], Theorem 2.) Proposition 2.4 Let µ 0 be a measure in M b (). Then µ 0 belongs to M 0 () if and only if it belongs to L () + W,p (). Thus if µ 0 belongs to M 0 (), there exists f in L () and g in (L p ()) N such that µ 0 = f div(g), in the sense of distributions. Moreover every function v W,p 0 () is measurable with respect to µ 0 and belongs to L (, µ 0 ) if v further belongs to L (), and one has vdµ 0 = fv + g v, v W,p 0 () L (). As a consequence of the previous results and the Hahn decomposition Theorem we get the following result Proposition 2.5 Every measure µ in M b () can be decomposed as follows µ = µ 0 + µ s = f div(g) + µ + s µ s, where µ 0 is a measure in M 0 (), hence can be written as f div(g), with f L () and g (L p ()) N, and where µ + s and µ s (the positive and the negative parts of µ s ) are two nonnegative measures in M s (), which are concentrated on two disjoint subsets E + and E of zero p-capacity. 5

6 Finally we recall the following result which will be used several times in Section 4 to prove the stability result. It is a consequence of Egorov theorem. Proposition 2.6 Let be a bounded open subset of IR N. Assume that ρ ε is a sequence of L () functions converging to ρ weakly in L () and assume that σ ε is a sequence of L () functions which is bounded is L () and converges to σ almost everywhere in. Then lim ρ ε σ ε = ρσ. ε Preliminaries about Lorentz spaces For < q < and < s < the Lorentz space L q,s () is the space of Lebesgue measurable functions such that f L q,s () = ( 0 [f (t)t q ] s dt t ) /s < +, (2.2) endowed with the norm defined by (2.2). Here f denotes the decreasing rearrangement of f, i.e. the decreasing function defined by f (t) = inf{s 0 : meas {x : f(x) > s} < t}, t [0, ]. For references about rearrangements see, for example, [CR], [K]. For < r <, the Lorentz space L r, () is the space of Lebesgue measurable functions such that f L r, () = sup t [meas {x : f(x) > t}] /r < +, (2.3) t>0 endowed with the norm defined by (2.3). Lorentz spaces are intermediate spaces between the Lebesgue spaces, in the sense that, for every < s < r <, we have L r, () L r,r () = L r () L r, () L s, (). (2.4) The space L r, () is the dual space of L r, (), where + =, and we have the r r generalized Hölder inequality f L r, (), g L r, (), fg f L r, () g L r, (). (2.5) 6

7 More generally, if < p < and q, we get f L p,q (), g L p 2,q 2 (), fg L p,q () f L p,q () g L p 2,q 2 (), (2.6) = p p + p 2, = q q + q 2. Improvements of classical Sobolev inequalities involving Lorentz spaces are proved, for example, in [ALT]. In the present paper we will only use the following generalized Sobolev inequality: a positive constant S N,p depending only on p and N exists such that v L p,p () S N,p v W,p 0 (), (2.7) for every v W,p 0 (). 2.3 Definition of renormalized solution and statement of existence result In the present paper we consider a nonlinear elliptic problem which can formally be written as { div(a(x, u, u) + K(x, u)) + H(x, u, u) + G(x, u) = µ div(f ) in, u = 0 on, (2.8) where a : IR IR N IR N and K : IR IR N are Carathéodory functions satisfying a(x, s, ξ)ξ α ξ p, α > 0, (2.9) a(x, s, ξ) c [ ξ p + s p + a 0 (x) ], a 0 (x) L p (), c > 0, (2.0) (a(x, s, ξ) a(x, s, η), ξ η) > 0, ξ η, (2.) K(x, s) c 0 (x) s γ + c (x), 0 γ p, c 0 L N p,r (), N r +, p c L p (), (2.2) for almost every x and for every s IR, ξ IR N, η IR N. Moreover H : IR IR N IR and G : IR IR are Carathéodory functions satisfying H(x, s, ξ) b 0 (x) ξ λ + b (x), 0 λ p, b 0 L N, (), b L (), (2.3) 7

8 G(x, s)s 0, (2.4) G(x, s) d (x) s t + d 2 (x), d L z, (), d 2 L (), for almost every x and for every s IR and ξ IR N, where 0 t < N(p ) N p N(p ), z = N p t Finally µ is a measure in M b () that is decomposed in according to Proposition 2.3, and (2.5) and z + z =. (2.6) µ = f div(g) + µ + s µ s, (2.7) F ( L p () ) N. (2.8) Remark 2.7 Observe that, by (2.4) if the function c 0 belongs to the Lebesgue space L t () for some t N and the function b p 0 belongs to the Lebesgue space L q () for some q > N, then the conditions c 0 L N p,r N (), r +, and b p 0 L N, () (as requested in hypotheses (2.2) and (2.3)) are satisfied. In the present paper we consider renormalized solution to the problem (2.8). Before giving the definition of such a notion of solution, we need a few notation and definitions. For k > 0, denote by T k : IR IR the usual truncation at level k, that is { s s k, T k (s) = k sign(s) s > k. Consider a measurable function u : IR which is finite almost everywhere and satisfies T k (u) W,p 0 () for every k > 0. Then there exists (see e.g. [BBGGPV], Lemma 2.) an unique measurable function v : IR N, finite almost everywhere, such that T k (u) = vχ { u k} almost everywhere in, k > 0. (2.9) We define the gradient u of u as this function v, and denote u = v. Note that the previous definition does not coincide with the definition of the distributional gradient. However if v (L loc()) N, then u W, loc () and v is the distributional gradient of u. In contrast there are examples of functions u L loc() (and thus such that the gradient of u in the distributional sense is not defined) for which the gradient u is defined in the previous sense (see Remarks 2.0 and 2., Lemma 2.2 and Example 2.6 in [DMOP]). 8

9 Definition 2.8 We say that a function u : IR, measurable on, almost everywhere finite, is a renormalized solution of (2.8) if it satisfies the following conditions T k (u) W,p 0 (), k > 0; (2.20) u p L N N p, (); (2.2) u p belongs to L N, (), (2.22) where u is the gradient introduced in (2.9); lim a(x, u, u) u ϕ = ϕdµ + n + s, n n<u<2n (2.23) lim a(x, u, u) u ϕ = ϕdµ n + s, n 2n<u< n (2.24) for every ϕ C 0 b (); and finally lim K(x, u) u = 0; (2.25) n + n n< u <2n a(x, u, u) u h (u)v + a(x, u, u) v h(u) + K(x, u) u h (u)v + K(x, u) v h(u) + H(x, u, u)h(u)v + G(x, u)h(u)v = fh(u)v + (g + F ) u h (u)v + (g + F ) v h(u), (2.26) for every v W,p () L (), for all h W, (IR) with compact support in IR, which are such that h(u)v W,p 0 (). Since h(u)v W,p 0 () and since supp(h) [ 2n, 2n] (for a suitable n > 0 depending on h), we can rewrite (2.26) as follows a(x, T 2n (u), T 2n (u)) T 2n (u) h (u)v + a(x, T 2n (u), T 2n (u)) v h(u) + K(x, T 2n (u)) T 2n (u) h (u)v + K(x, T 2n (u)) v h(u) (2.27) + H(x, T 2n (u), T 2n (u))h(u)v + G(x, T 2n (u))h(u)v = fh(u)v + (g + F ) T 2n (u) h (u)v + (g + F ) v h(u). Let us observe that every integral in (2.27) is well defined thanks to the fact that T k (u) W,p 0 () for every k > 0, h has compact support and the assumptions (2.9)- (2.8). 9

10 Remark 2.9 Observe that every renormalized solution u of (2.8) is such that a(x, u, u) L N, (), K(x, u) L N,r (), N p r +, G(x, u) L () and H(x, u, u) L (), thanks to the conditions (2.2) and (2.22), and the growth conditions (2.0), (2.2), (2.3) and (2.5) on a, K, H and G respectively. Observe also that, since p < N, then L p () L N,r (), r + and therefore the term K(x, u) does not belong in general to (L p ()) N. Moreover u is a solution of (2.8) in the distributional sense, that is u satisfies the following equality a(x, u, u) φ + K(x, u) φ + = φdµ + for all φ C 0 (). Indeed we take φ C 0 () and h = h n defined by H(x, u, u)φ + F φ, 0, s > 2n 2n s h n (s) =, n < s 2n n, s n, in (2.26); then we let n tend to infinity and we obtain (2.28). N p G(x, u)φ (2.28) (2.29) The main result of the present paper is the following existence result (in Section 4 we state and we prove a generalization of the stability result of [DMOP]) Theorem 2.0 Under assumptions (2.9)-(2.8), there exists at least one renormalized solution u of (2.8) if one of the following conditions holds true ) γ = λ = p, c 0 L N p,r (), r < + and b 0 L N, () is small enough; 2) γ = p, λ < p, c 0 L N p,r (), r < + ; 3) γ < p, λ < p, c 0 L N p, (). Remark 2. The counterpart of such existence result, that is the case γ = λ = p, b 0 L N, () and c 0 N p,r L (), r < +, small enough and the case γ < p, λ = p with c 0 L N p, () and b 0 L N, () (without smallness hypothesis on the norm) are investigated in [GM]. 0

11 Remark 2.2 We assume that γ = p and c 0 belongs to L N p,r (), r < + in condition ) and 2), while we assume that γ < p and c 0 belongs to L N p, () in condition 3). Actually we prove Theorem 2.0, by using the stability result (Theorem 4.) which holds true under the assumption that γ < p and c 0 L N p, () or under the assumption that γ = p and c 0 L N p,r (), r < +. The proof of Theorem 2.0 is made by several steps. We begin by approximating the data of the problem (2.8). Then we obtain the a priori estimate for the gradients of the solutions u ε to the approximate problems. We prove them in Section 3 below, Theorem 3.2. The last step in the proof of Theorem 2.0 consists to pass to the limit in the approximate problems. This is done by reconduce the problem to apply the Theorem 4., which is an extension of the stability result proved in [DMOP] when K(x, s) 0; our result allows to deal with the term div(k ε (x, u ε )) which is not in general bounded in W,p (). 3 Approximate problems and a priori estimates We begin this Section by approximating the data of the problem (2.8). Then we prove Thorem 3.2 below which gives the a priori estimates for the gradients of the solutions u ε to the approximate problems. Let us explain our method in the most delicate case, that is γ = λ = p. We firstly prove an estimate of the level sets of the function u ε, which allows us to choose a level set { u ε > σ} in such a way that c 0 N L p,r is small enough. This allows us to obtain an estimate for ( uε >σ) T k (u ε ) (L p ()) N which implies the estimate of u ε p thanks to the generalization of the result of [BBGGPV] proved in [BMMP3] and stated below. Lemma 3. Assume that is an open subset of IR N with finite measure and that < p < N. Let u be a measurable function satisfying T k (u) W,p 0 (), for every positive k, and such that T k (u) p Mk + L, k > 0, (3.) where M and L are given constants. Then u p belongs to L p p, (), u p belongs to L N, () and [ ] u p C(N, p) M + L p p L p, (3.2) p, () [ ] u p L N, () C(N, p) M + N p L p, (3.3) where C(N, p) is a constant depending only on N and p and where p = p N.

12 Let us introduce the approximate problems. The Radon measure with bounded variation µ can be decomposed as µ = f div(g) + µ + s µ s, where f L (), g ( L p () ) N and µ + s and µ s (the positive and the negative parts of µ s ) are two nonnegative measures in M b () which are concentrated on two disjoint subsets E + and E of zero p-capacity, according to Proposition 2.5. As in [DMOP] (cf. [BMMP3]), we approximate the measure µ by a sequence µ ε defined as µ ε = f ε div(g) + λ ε λ ε, where { fε is a sequence of L p () functions that converges to f weakly in L (), { λ ε is a sequence of nonnegative functions in L p () that converges to µ + s in the narrow topology of measures, (3.4) (3.5) and { λ ε is a sequence of nonnegative functions in L p () that converges to µ s in the narrow topology of measures. (3.6) Observe that µ ε belongs to W,p (). Let us denote by K ε (x, s) = K(x, T /ε (s)), (3.7) Therefore, by assumptions (2.2)-(2.5), we have H ε (x, s, ξ) = T /ε (H(x, s, ξ)), (3.8) G ε (x, s) = T /ε (G(x, s)). (3.9) K ε (x, s) K(x, s) c 0 (x) s γ + c (x), (3.0) K ε (x, s) c 0 (x) ε γ + c (x), (3.) H ε (x, s, ξ) H(x, s, ξ) b 0 (x) ξ λ + b (x), (3.2) H ε (x, s, ξ) ε, (3.3) G ε (x, s)s 0, (3.4) G ε (x, s) G(x, s) d (x) s r + d 2 (x), (3.5) G ε (x, s) ε. (3.6) 2

13 i.e. u ε = 0. Let u ε W,p 0 () be a weak solution of the following problem { div(a(x, uε, u ε ) + K ε (x, u ε )) + H ε (x, u ε, u ε ) + G ε (x, u ε ) = µ ε div(f ) in u ε W,p 0 () a(x, u ε, u ε ) v + K ε (x, u ε ) v + H ε (x, u ε, u ε )v + G ε (x, u ε )v = f ε v + (g + F ) v + λ ε v λ ε v, on, (3.7) (3.8) v W,p 0 (). The existence of a solution u ε of (3.8) is a well-known result (see e.g. [L], [DPo2], [G2]). The main result of this Section is Theorem 3.2 below which gives an a priori estimate for u ε p. Theorem 3.2 Under the hypotheses of Theorem 2.0, every solution u ε of (3.8) satisfies u ε p L N, () c, (3.9) u ε p N, L N p c, (3.20) () where c is a positive constant which depends only on p,, N, α, b 0 L N, (), b L (), c 0 N L p,r, c L p (), g (L p ()), F N (L p ()), sup ( f N ε L () (), sup λ ε () + λ ε () ) ε ε and on (c 0 ) the decreasing rearrangement of c 0. Proof of Theorem 3.2 Let us begin by proving Theorem 3.2 when assumption ) in Theorem 2.0 is satisfied, i.e. γ = λ = p, c L N p,r N (), r < + and b p L N, () is small enough. First step. In this step we prove the estimate of the level sets of the functions u ε given by (3.39) below. It is performed through a log-type estimate on u ε (cf. [B], [BOP], [BGu], [BGu2]). Define the function ψ : IR IR by ψ(s) = s 0 dr, (3.2) (A p /p + r ) p 3

14 where A is a positive constant which will be specified later. Observe that the following property of ψ(s) holds true ψ(s), A p s IR. (3.22) Observe also that ψ(s) is a Lipschitz function such that ψ(0) = 0. Therefore, since u ε W,p 0 (), the function ψ(u ε ) belongs to W,p 0 (). This allows us to use ψ(u ε ) as test function in (3.7). Then we get a(x,u ε, u ε ) u ε ψ (u ε ) + K ε (x, u ε ) u ε ψ (u ε ) + H ε (x, u ε, u ε )ψ(u ε ) + G ε (x, u ε )ψ(u ε ) (3.23) = f ε ψ(u ε ) + (g + F ) u ε ψ (u ε ) + λ ε ψ(u ε ) λ ε ψ(u ε ). Now we evaluate the various integrals in (3.23). By the definition (3.2) of ψ(s) and ellipticity condition (2.9), we obtain a(x, u ε, u ε ) u ε ψ u ε p (u ε ) α (A p /p + u ε ). (3.24) p Let us now estimate K ε (x, u ε ) (u ε )ψ u ε. By the growth condition (3.0) on K ε and Young s inequality, since and (A p /p + u ε ) p A p we get K ε (x, u ε ) u ε ψ (u ε ) c 0 ψ (u ε ) u ε p u ε + = c 0 u ε p u ε (A p /p + u ε ) + p 3p /p p α p /p c 0 p + 3p /p p α p /p 3p /p p α p /p L p () + α 3p c p (A p /p + u ε ) p + α 3p c ψ (u ε ) u ε c u ε (A p /p + u ε ) p u ε p (A p /p + u ε ) p ( c 0 p L p () + A p c p L p () u ε p (A p /p + u ε ) p + 2α 3p ) Moreover, since we assume that p < N, then p < N p that L N p,r () L p () and by inequality (2.6), we get c 0 L p () L pn (p )(N p),t () 4 u ε p (A p /p + u ε ) p.. This implies, since N c 0 L N p,r (), u ε (A p /p + u ε ) r, p

15 i.e. c 0 L p () where t is defined by = +. p t r Therefore K ε (x, u ε ) u ε ψ (u ε ) ( /p 3p Np p α p /p (p )(N p)t + 2α 3p u ε p (A p /p + u ε ) p. Np (p )(N p)t Np c 0 N (p )(N p)t L p,r, () ) p (N p)t N c0 p L N p,r () + A p c p L p () (3.25) Let us now estimate H ε (x, u ε, u ε )ψ(u ε ). By the definition (3.2) of ψ(s), the growth assumption (3.2) on H ε, the property (3.22) of ψ(s) and the generalized Hölder inequality (2.5), we have H ε (x, u ε, u ε )ψ(u ε ) b 0 u ε p ψ(u ε ) + b ψ(u ε ) [ ] (3.26) b 0 u ε p + b A p [ b0 A L p N, () u ε p L N, () + b ] L (). Moreover, by the sign condition (3.4) on G ε, we get G ε (x, u ε )ψ(u ε ) 0. (3.27) Finally, since (A p /p + u ε ) p A p, we have (g + F ) u ε ψ (u ε ) = and, by (3.22), we also get (g + F ) u ε (A p /p + u ε ) p 3p /p p α p /p A p g + F p (L p ()) N + α 3p u ε p (A p /p + u ε ) p (3.28) f ε ψ(u ε ) A p f ε L (), (3.29) 5

16 λ ε ψ(u ε ) λ A p ε (), (3.30) λ ε ψ(u ε ) λ A p ε (). (3.3) Combining (3.23)-(3.3), we get where Define u ε p (A p /p + u ε ) p ( ) /p p 3p Np α p /p+ (p )(N p)t { + p b 0 αa L p N, () u ε p L N, () (N p)t N c0 p /p + 3p p α ( c p /p p + g + F L p p ) + M () (L p ()) N 0 M 0 = b L () + sup ε { A = + p b 0 α L N, () u ε p L N, () Observe that A > and therefore A p L N p,r () }, (3.32) [ f ε L () + sup λ ε () + λ ε () ]. (3.33) ε /p + 3p p α ( c p /p p + g + F L p p ) + M () (L p ()) N 0 u ε p ( (A p /p + u ε ) 3p /p Np p α p /p+ (p )(N p)t. This implies that A ) p (N p)t N c0 p }. L N p,r () (3.34) +. (3.35) On the other hand, by Poincaré inequality, we get u ε p (A p /p + u ε ) = log(a p /p + u p ε ) p ( = log + u ) ε p A p /p [ ( c(n, p) log + u )] p ε. A p /p (3.36) 6

17 Therefore for any η > 0, we have Denote meas { u ε ηa p /p } = C = [log( + η)] p [log( + η)] p [log( + η)] p ( /p 3p Np c(n, p) α p /p+ (p )(N p)t Combining (3.35) -(3.37), we get { u ε ηa p /p } [ { u ε ηa p /p } [ ( ) p meas { u ε ηa p /p } for any η > 0, or, equivalently, for any ν > 0 log [log( + η)] p log ( + u ε A p /p (N p)t N c0 p C [log( + η)] p, + u ε )] p A p /p (3.37) )] p L N p,r () +. (3.38) meas { u ε exp(c ν)a p /p } ν p. (3.39) Second step. Using in (3.8) the test function T k (u ε ), we obtain a(x, u ε, u ε ) T k (u ε ) + K ε (x, u ε ) T k (u ε ) + H ε (x, u ε, u ε )T k (u ε ) + G ε (x, u ε )T k (u ε ) = f ε T k (u ε ) + (g + F ) T k (u ε ) + λ ε T k (u ε ) λ ε T k (u ε ). Now we evaluate the various terms in (3.40). By ellipticity condition (2.9), we obtain Let us now estimate a(x, u ε, u ε ) T k (u ε )= α K ε (x, u ε ) T k (u ε ). { u ε k} a(x, u ε, u ε ) u ε T k (u ε ) p. (3.40) (3.4) 7

18 Denote σ = exp(c ν)a p /p, (3.42) where ν is a positive constant which will be specified later. Since by the generalized Sobolev inequality (2.7), T k (u ε ) belongs to L p,p (), then T k (u ε ) belongs also to L p,t () for any p t +. Moreover, by the growth condition (3.0) on K ε, the generalized Hölder inequality (2.5) and the Young inequality, we get K ε (x, u ε ) T k (u ε ) c 0 u ε p T k (u ε ) + c T k (u ε ) = c 0 T k (u ε ) p T k (u ε ) + c 0 T k (u ε ) p T k (u ε ) + c T k (u ε ) u ε σ u ε <σ c 0 N T L p,r ( uε σ) k(u ε ) p T L p,t () k(u ε ) p (L p ()) N + 3p /p p α p /p σp c 0 p L p () + α 3p T k(u ε ) p (L p ()) N + 3p /p p α p /p c p L p () + α 3p T k(u ε ) p (L p ()) N. C(N, p, ) c 0 N T L p,r ( uε σ) k(u ε ) p (L p ()) + 2α N 3p T k(u ε ) p (L p ()) N /p ( + 3p p α p /p σ p c 0 p + c ) L p () p L p (), (3.43) where t is choosed such that + p + = and where C(N, p, ) is a positive r t p constant which depend only on N, p and. Let us now estimate H ε (x, u ε, u ε )T k (u ε ). By the growth assumption (3.2) on H ε and the generalized Hölder inequality (2.5), we have H ε (x, u ε, u ε )T k (u ε ) k [ b 0 L N, () u ε p L N, () + b ] (3.44) L (). Moreover, by the sign condition (3.4) on G ε, we get Finally G ε (x, u ε )T k (u ε ) 0. (3.45) f ε T k (u ε ) k f ε L (), (3.46) 8

19 (g + F ) T k (u ε ) α 3p T k(u ε ) p (L p ()) N + 3p /p Combining (3.40)-(3.49), for any k > 0, we get p α p /p g + F p (L p ()) N, (3.47) λ ε T k (u ε ) kλ ε (), (3.48) λ ε T k (u ε ) kλ ε (). (3.49) T k (u ε ) p (L p ()) N p α C(N, p, ) c 0 L N p,r ( uε σ) T k(u ε ) p (L p ()) N + p α k [ b 0 L N, () u ε p L N, () + M 0 /p ( + 3p α p /p+ σ p c 0 p + c ) L p () p + g + F L p p () (L p ()), N where M 0 is defined by (3.33). On the other hand, since (3.39) holds true, we can choose ν = ν in such a way that ] meas { u ε exp(c ν)a p /p } ν p < τ, for some τ > 0, implies Observe that such ν is independent on ε. Therefore we obtain where p α C(N, p, ) c 0 L N p,r ( uε exp(c ν)) < 2. T k (u ε ) p (L p ()) N M k + L, k > 0, (3.50) M = 2p α [ b0 L N, () u ε p L N, () + M ] 0, L /p = 2 3p α p /p+ (σp c 0 p + c L p () p and M 0 is defined by (3.33). L p () + g + F p (L p ()) N ) 9

20 By Lemma 3., the definition (3.42) of σ and the definition (3.34) of A, we get u ε p L N, () C(N, p) [ M + N p (L ) p ] { C(N, p) α b 0 L N, () u ε p L N, () + 2p α M 0 [ + N p 3 p +/p ( 2 /p exp( pc ν )]} )A c α p 0 L p () + c L p () + g + F (L p ()), N and then 2 p u ε p L N, () C + p C(N, p) (2 + N p +/p 3p 2 /p α α b 0 L N, () u ε p L N, (), exp( pc ν ) ) c p 0 L p () (3.5) with C = C(N, p)2 p α M 0+C(N, p) N ( [ p 3 p +/p 2 /p α exp( pc ν ) c p 0 L p () + 3p /p p α p /p ( c L p () + g + F (L p ()) N ) + M 0 ) + ] c p + g + F L p p () (L p ()) N Since we assume that b 0 L N, () is small enough, or more exactly, b 0 L N, () < α ( p C(N, p) 2 + N p 3p +/p 2 /p α we obtain (3.9). Now we prove (3.20). By (3.50) and Lemma 3., we obtain [ ] u ε L N, () C(N, p) M + p (L ) p { = C(N, p) 2 p α [ + /p p 2 3p α p p + (σp c 0 p + c L p () p [ b0 L N, () u ε p L N, () + M 0] 20 L p () ), exp( pc ν ) c p 0 L p () ] /p + g + F p ) (L p ()) N.

21 By a calculation similar to that made for estimate (3.5) of u ε p, we obtain (3.20) and therefore the conclusion of the proof of Theorem 3.2 under the assumption ) of Theorem 2.0. We now prove Theorem 3.2 when assumption 2) in Theorem 2.0 is satisfied, i.e. γ = p, λ < p and c 0 L N p,r N (), r <. p Observe that the assumptions on γ and c 0 are exactly the same of the previous case, while we now assume that λ < p (and not more λ = p and b 0 L N, () small). Therefore the estimates of the various integrals in (3.23) and in (3.40) are exactly the same, except the estimate (3.26) of H ε (x, u ε, u ε )ψ(u ε ) and (3.44) of H ε (x, u ε, u ε )T k (u ε ). Let us explain the method to replace the estimate (3.26) in this case. By a calculation similar to (3.26), since λ < p, we obtain H ε (x, u ε, u ε )ψ(u ε ) ] [ A L θ, () b 0 L N,() u ε p λ p + b p L N, () L () (3.52) where θ is defined by + = and where N { A = + p α λ (p )N + θ L θ, () b 0 L N, () u ε p λ p L N, () /p + 3p p α ( c p /p p + g + F L p p ) + M () (L p ()) N 0 Let us explain the method to replace the estimate (3.44). Since λ < p, we obtain [ ] H ε (x, u ε, u ε )T k (u ε ) k L () b θ, 0 L N, () u ε p λ p + b L N, () L (). (3.53) By replacing estimate (3.26) by (3.52) and the estimate (3.44) by (3.53), we argue as in the previous case and we obtain u ε p L N, () C + p C(N, p) (2 + N α p 3p exp(pc ν α p ) c 0 L p () ) } b 0 L N, () u ε p λ p L N, (). Since λ < p, we obtain the estimate of u ε p L N, () (without any assumption on the smallness of b 0 L N, ()). Similarly we get (3.20). Finally let us now prove Theorem 3.2 when assumption 3) in Theorem 2.0 is satisfied, i.e. γ < p, λ < p, c 0 L N p, (). 2

22 where Using T k (u ε ) as test function, since γ < p we easily obtain T k (u ε ) p (L p ()) N M k + L, ] M = C(α, γ, p) [ L θ, () b 0 L N,() u ε p λ p + M L N, () 0, [ ] L = C (α, γ, p) By Lemma 3., we get: c 0 p (p ) γ L p N, () + c p + g + F L p p () (L p ()) N u ε p L N, () C(N, p) [ M + N p ( L ) p ], u ε L N, () C(N, p) [ M + p ( L ) p ], and by definition of M, since λ < p we obtain (3.9) and (3.20). To conclude the proof of Theorem 2.0, we have to pass to the limit in the approximate problem (3.7). As in [BMMP3], the idea is to prove that u ε satisfies { div(a(x, uε, u ε ) + K ε (x, u ε )) = f ε Ψ ε div(g) + div(f ) + λ ε λ ε in D (),. u ε W,p 0 (), where Ψ ε = H ε (x, u ε, u ε ) + G ε (x, u ε ) H(x, u, u) + G(x, u) in L () strongly, and to apply a stability result which allows to pass to the limit in (3.55). We devote next Section to prove the stability result, which we need. (3.54) (3.55) 4 A generalization of the stability results of [DMOP] In the present Section we prove a generalization of the stability result given in [DMOP] (see also [MP] and [M]). The main feature of our result is due to the term div(k ε (x, u)), where K ε (x, u) belongs to (L N, ()) N. Since p < N, we have (L p ()) N (L N, ()) N, which implies that K ε (x, u) does not in general belong to (L p ()) N and therefore the term div(k ε (x, u)) is not in general an element of the dual space W,p (). We explicitly remark that the proof of stability result in [DMOP] (Theorem 3.4) can be adapted to our case, however here we use a slightly different method. 22

23 In the present Section we consider a nonlinear elliptic problem which can be formally written as { div(a(x, u, u) + Kε (x, u)) = µ ε in, (4.) u = 0 on. where ε belongs to a sequence of positive numbers that converges to zero and the function a : IR IR N IR N is a Carathéodory function which satisfies assumptions (2.9), (2.0) and (2.) Moreover K ε : IR IR N is a Carathéodory function which satisfies the growth condition (2.2), i.e. K ε (x, s) c 0 (x) s γ + c (x), (4.2) for almost every x and for every s IR, where c 0 and c satisfy the assumptions in (2.2). Denoted by K : IR IR N a Carathéodory function, we also assume that K ε (x, s ε ) K(x, s) for every sequence s ε IR such that (4.3) s ε s almost everywhere in. Finally we assume that µ ε is a measure in M b () which can be decomposed as µ ε = f ε div(g ε ) + λ ε λ ε, µ = f div(g) + µ + s µ s, where, denoted by µ the measure in M b () given by (2.7), we assume that f L () µ s = µ + s µ s is a measure in M s () with positive and negative parts µ + s and µ s respectively; (4.4) { fε is a sequence of functions in L () such that f ε f in L () weakly; g ε is a sequence of functions in (L p ()) N such that g ε g in (L p ()) N strongly; { λ ε is a non negative measure in M b () such that λ ε µ + s in the narrow topology; { λ ε is a non negative measure in M b () such that λ ε µ s in the narrow topology. (4.5) (4.6) (4.7) (4.8) Observe that, according to Proposition 2.5, we can decompose λ ε and λ ε in the following way 23

24 λ ε = λ ε,0 + λ ε,s, λ ε = λ ε,0 + λ ε,s with λ ε,0, λ ε,0 M 0 (), λ ε,0, λ ε,0 0 and λ ε,s, λ ε,s M s (), λ ε,s, λ ε,s 0. On the other hand, using Propositions 2.4 and 2.5, the measure µ ε decomposed as µ ε = µ ε,0 + µ ε,s = µ ε,0 + µ + ε,s µ ε,s, can be where µ ε,0 is a measure in M 0 () and where µ + ε,s and µ ε,s (the positive and the negative parts of µ ε,s ) are two nonnegative measures in M s (), which are concentrated on two disjoint subsets E + ε and E ε of zero p-capacity. Therefore we can conclude, by the definition of µ ε, that and We have (cf. [DMOP], Remark 3.5) We prove the following theorem µ ε,0 = f ε + div(g ε ) + λ ε,0 λ ε,0, (4.9) µ ε,s = λ ε,s λ ε,s. 0 µ + ε,s λ ε,s 0 µ ε,s λ ε,s. (4.0) Theorem 4. Assume that (2.9), (2.0), (2.), (4.2), (4.3), (4.4) (4.8) hold true with γ = p and c 0 L N p,r N (), r < +. Let u p ε be a renormalized solution of (4.). Then, up to a subsequence still indexed by ε, u ε converges almost everywhere to a renormalized solution u to the problem { div(a(x, u, u) + K(x, u)) = µ in, u = 0 on, (4.) and moreover T k (u ε ) T k (u) in W,p 0 () strongly, k > 0. The same conclusions hold true if we assume that (4.2) holds true with γ < p and c 0 L N p, (). Remark 4.2 The stability result given by Theorem 4. coincides with the stability result proved in [DMOP] (Theorem 3.4) (see also [M]) when K ε = 0. Therefore we prove an extension of such a result. We explicitly remark that our proof is slightly different. As in [DMOP], our stability result implies an existence result for renormalized solution to the problem (4.); such a result extends the existence result proved in [BGu2] in the case where µ belongs to L (). 24

25 Remark 4.3 Observe that, by the growth assumption (4.2) and the convergence assumption (4.3) on K ε, we deduce K(x, s) c 0 (x) s γ + c (x), (4.2) for almost every x and for every s IR. Observe also that, since a renormalized solution u ε to problem (4.) and a renormalized solution u to the problem (4.) satisfy the conditions u ε p L N N p, () and u p L N N p, (), by growth assumptions (4.2) on K ε and growth condition (4.2) on K, we deduce that K ε (x, u ε ) L N N,r () and K(x, u) L N N,r (), N p r +. On the other hand, since p < N, L p () L N N, (). This implies that, in general, K ε (x, u ε ) and K(x, u) does not belong to (L p ()) N and therefore the terms div(k ε (x, u ε )) and div(k(x, u)) are in general not elements of the dual space W,p (). Remark 4.4 Observe that Theorem 4. holds true under the same assumption, if we replace the right-hand side by a more general datum µ div(f ), with F (L p ()) N. Indeed K ε (x, s) (resp. K(x, s)) can be replaced by K ε (x, s) F (x) (resp. K(x, s) F ) which verifies conditions (4.2) and (4.3) (with c replaced by c + F ). Proof of Theorem 4. We begin by proving Theorem 4. under the assumptions that γ = p, c 0 L N p,r N (), r <. Observe that from now on, such assumptions will p be used only to obtain a priori estimates for u ε p and u ε p in the preliminary step below and to prove (4.24) and (4.25) in the second step below. Preliminary step. We begin by proving the following a priori estimate for the renormalized solution u ε u ε p N c, (4.3) L p, () u ε p L N, () c, (4.4) where c is a positive constant depending only on the data, which does not depend on ε. This is done by adapting to the case of renormalized solution u ε the techniques used in the proof of Theorem 3.2 above which allows us to prove the same a priori estimate for the weak solution of problem (3.7). Since the arguments are very similar to that of Theorem 3.2, we do not give the details, but we only give a sketch of the proof. 25

26 Since u ε is a renormalized solution, we have a(x, u ε, u ε ) u ε h (u ε )v + a(x, u ε, u ε ) v h(u ε ) + K ε (x, u ε ) u ε h (u ε )v + K ε (x, u ε ) v h(u ε ) = f ε h(u ε )v + g ε u ε h (u ε )v + g ε v h(u ε ) + λ ε,0h(u ε )v + λ ε,0h(u ε )v, (4.5) for every v W,p () L (), for all h W, (IR) with compact support in IR, which are such that h(u ε )v W,p 0 (). Firstly we use in (4.5) the test function h n (u ε )ψ(t 2n (u ε )) where h n is defined by (2.29) and ψ is defined by (3.2) (with A ). Then by calculations similar to that of first step of the proof of Theorem 3.2, we get the following estimate for the level sets of u ε, meas{ u ε exp(c η)}, η > 0, (4.6) ηp where C is defined by (3.38). Secondly we use in (4.5) the test function h n (u ε )T k (u ε ) for every k > 0, where h n (s) is defined by (2.29) and, using calculations similar to that of second step of the proof of Theorem 3.2, we get T k (u ε ) p (L p ()) N Mk + L, k > 0 (4.7) for some M > 0 and L > 0. This implies, by Lemma 3., (4.3) and (4.4). Estimate (4.7) and the growth assumptions on K ε, since the operator a is strictly monotone, allow us to use standard techniques (see, e.g., [BMu] and [DMOP]) to say that there exists a measurable function u : IR, finite almost everywhere in and such that, up to a subsequence still indexed by ε, T k (u ε ) T k (u) in W,p 0 () weakly, k > 0, (4.8) u ε u almost everywhere in, (4.9) u ε u almost everywhere in, (4.20) as ε goes to 0. First step. In this step we prove that the function u is solution of (4.) in the sense of distributions. By assumption (4.3) and (4.9), it follows that K ε (x, u ε ) converges to K(x, u) almost everywhere in. Moreover, the growth assumption (4.2) on K ε and the estimate 26

27 (4.3) of u ε p imply that K ε (x, u ε ) is bounded in L N, (). Therefore Lebesgue convergence theorem gives K ε (x, u ε ) K(x, u) in L r () strongly, r < N/(N ). (4.2) In a similar way by (2.0) on a, (4.3), (4.4), (4.9) and (4.20) and Lebesgue convergence theorem, we get a(x, u ε, u ε ) a(x, u, u) in L r () strongly r < N/(N ). (4.22) Since u ε is a renormalized solution, it is also a solution in the sense of distribution of (4.), that is a(x, u ε, u ε ) φ + K(x, u ε ) φ = φdµ ε, (4.23) for all φ C0 (), (cf. Remark 2.9 above). Therefore, using the convergences in (4.2) and (4.22) and the assumptions (4.4) - (4.8), we easily can pass to the limit in (4.23) and obtain that u is solution in the sense of distribution of (4.). Second step. In this step we prove that and lim lim sup K ε (x, u ε ) u ε = 0, (4.24) n ε 0 n {n< u ε <2n} lim sup n K(x, u) u = 0. (4.25) n {n< u <2n} Since by the generalized Sobolev inequality (2.7), T k (u ε ) L p,t () for any p t +, by the growth condition (3.0) on K ε, the generalized Hölder inequality (2.5) and (4.7), we get n n< u ε <2n K ε (x, u ε ) u ε c 0 u ε p u ε + n n< u ε <2n n c 0 T 2n (u ε ) p T 2n (u ε ) + n n< u ε <2n n c n T 2n(u ε ) p (L p ()) c N 0 N L p,r ({n< uε <2n}) + c n /p L p () n T 2n(u /p ε ) (L p ()) N n< u ε <2n n< u ε <2n c u ε c T 2n (u ε ) c c 0 L N p,r ({n< uε <2n}) + c n /p c L p (). (4.26) Moreover ( {n< uε <2n} c 0 N = L p,r [(c 0 ) (t)t p N ] r dt ) /r ({n< uε <2n}) 0 t 27

28 tends to 0 when firstly ε goes to zero and then n goes to infinity (indeed u ε converges to u a.e. in and u is finite a.e. in ). Therefore we obtain (4.24) by using Fatou Lemma in (4.26). In a similar way, we get also (4.25). Third step. In this step we prove a slightly different version of Lemma 6. of [DMOP], i.e. we prove that and lim sup lim sup a(x, u ε, u ε ) u ε ϕ ϕdµ + s (4.27) n ε 0 n {n<u ε<2n} lim sup lim sup a(x, u ε, u ε ) u ε ϕ ϕdµ s, (4.28) n ε 0 n { 2n<u ε< n} for any ϕ C ( ) with ϕ 0. We begin by proving (4.27). For any n, let us define the s n : IR IR and h η : IR IR by s n (r) = T 2n(r) T n (r), n h η (r) = s η (r). Denote v + (x) = max{0, v(x)} and v (x) = max{0, v(x)} for almost every x. Using the test function h η (u ε )s n (u + ε )ϕ in (4.5) and letting η goes to infinity, we have a(x, u ε, u ε ) ϕs n (u + ε ) + a(x, u ε, u ε ) u ε ϕ n {n<u ε<2n} + K ε (x, u ε ) ϕs n (u + ε ) + K ε (x, u ε ) u ε ϕ n {n<u ε<2n} = f ε s n (u + ε )ϕ + g ε ϕs n (u + ε ) + g ε u ε ϕ n {n<u ε<2n} + s n (u + ε )ϕdλ ε,0 + ϕdµ + ε,s s n (u + ε )ϕdλ ε,0, (4.29) for any ϕ C ( ) nonnegative. Now we pass to the limit in the various terms in (4.29), first as ε 0 and then as n. Since s n (u + ε ) is bounded by, we have s n (u + ε ) s n (u + ) almost everywhere and weakly- in L (). (4.30) Therefore, from (4.2) and (4.22), it follows that a(x, u ε, u ε ) ϕs n (u + ε ) = a(x, u, u) ϕs n (u + ), lim ε 0 28

29 and lim K ε (x, u ε ) ϕs n (u + ε 0 ε ) = K(x, u) ϕs n (u + ). Since both the functions a(x, u, u) and K(x, u) belong to (L r ()) N for r < N/(N ), s n (u + ) is bounded by, s n (u + ) 0 almost everywhere in and ϕ C ( ), Lebesgue convergence theorem implies that lim lim a(x, u ε, u ε ) ϕs n (u + n ε 0 ε ) = lim a(x, u, u) ϕs n (u + ) = 0, (4.3) n + lim lim n ε 0 K ε (x, u ε ) ϕs n (u + ε ) = Moreover, from (4.24), we get lim n + s n (u + )K(x, u) ϕ = 0. (4.32) lim lim sup K ε (x, u ε ) u ε ϕ = 0. (4.33) n ε 0 n {n< u ε <2n} Since f ε converges weakly to f in L () and since s n (u + ε ) converges to s n (u + ) almost everywhere while s n (u + ε ) a.e., Proposition 2.6 gives lim lim f ε s n (u + n ε 0 ε )ϕ = lim fs n (u + )ϕ = 0. (4.34) n Furthermore, by (4.6), since s n (u + ε ) converges to s n (u + ) almost everywhere, we get lim n lim sup g ε ϕs n (u + ε ) = lim g ϕs n (u + ) = 0. (4.35) ε 0 n On the other hand, Hölder s inequality implies n g ε u ε ϕ {n<u ε<2n} ϕ L () g ε n /p (L p ()) N from which it follows (using (4.7)) ( ) /p u ε p, n {n<u ε<2n} lim lim sup g ε u ε ϕ = 0. (4.36) n ε 0 n {n<u ε<2n} Since 0 µ + ε,s λ ε,s (see (4.0)), ϕ 0, 0 s n (u + ε ) and since λ ε,0 is a positive measure, we have s n (u + ε )ϕdλ ε,0 + ϕdλ ε. ϕdµ + ε,s ϕdλ ε, ϕdλ ε,s s n (u + ε )ϕdλ ε,0 (4.37)

30 Combining (4.29) (4.37), we get for every ϕ C ( ) with ϕ 0 a(x, u ε, u ε ) u ε ϕ ω(ε, n) + ϕdλ ε, n {n<u ε<2n} where ω(ε, n) denotes a function such that n lim lim sup ω(ε, n) = 0. ε 0 Since λ ε µ + s in the narrow topology, we obtain (4.27). Using s n (u ε )ϕ and similar arguments we have also (4.28). Fourth step. In this step we prove that the limit function u is a renormalized solution to the problem (4.). We begin by proving that for every v W,p () L () and for every h W, (IR) with compact support in IR, which are such that h(u)v W,p 0 (), we have a(x, u, u) u h (u)v + a(x, u, u) v h(u) + K(x, u) u h (u)v + K(x, u) v h(u) (4.38) = fh(u)v + g u h (u)v + g v h(u). For any δ > 0, let us consider two cut-off functions ψ δ + and ψ δ belonging to C 0 () such that 0 ψ δ +, 0 ψ δ, supp(ψ+ δ ) supp(ψ δ ) = ; (4.39) lim ψ δ + δ 0 p = lim ψδ p = 0; (4.40) δ 0 ψ δ + dµ s = 0, lim ψδ dµ+ s = 0; (4.4) lim δ 0 lim lim δ 0 ε 0 lim δ 0 δ 0 ( ψ + δ )dµ+ s = 0, lim lim lim δ 0 ε 0 δ 0 ψδ dλ ε = 0, lim lim δ 0 ε 0 ( ψ + δ )dλ ε = 0, lim δ 0 lim ε 0 ( ψ δ )dµ s = 0; (4.42) ψ + δ dλ ε = 0; (4.43) ( ψ δ )dλ ε = 0. (4.44) The existence of such cut-off functions is proved in Lemma 5. of [DMOP]. Moreover let h be an element of W, (IR) with compact support and let v W,p () L () such that h(u)v belongs to W,p 0 (). 30

31 Using the test function h n (u ε )h(u)v( ψ + δ ψ δ ) in (4.5), where h n is defined by (2.29), we get h ε(u ε )h(u)v( ψ + δ ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] u ε + h (u)h n (u ε )v( ψ δ + ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] u + h(u)h n (u ε )( ψ δ + ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] v + h(u)h n (u ε )v[a(x, u ε, u ε ) + K ε (x, u ε )] ( ψ δ + ψ δ ) = f ε h n (u ε )h(u)v( ψ δ + ψ δ ) + g ε [h n (u ε )h(u)v( ψ δ + ψ δ )] + h n (u ε )h(u)v( ψ δ + ψ δ )dλ ε,0 h n (u ε )h(u)v( ψ δ + ψ δ )dλ ε,0. (4.45) We now pass to the limit in (4.45) first as ε 0, then as n and finally as δ 0. From (4.24), we have lim n lim sup ε 0 h n(u ε )h(u)v( ψ δ + ψ δ )K ε(x, u ε ) u ε = 0. (4.46) By (4.27), since ( ψ + δ 0 lim sup lim sup n ε 0 n h L (IR) v L () ψ δ )a(x, u ε, u ε ) u ε is positive, one has {n< u ε <2n} ( h(u)v ( ψ δ + ψ δ )a(x, u ε, u ε ) u ε ( ψ δ + ψ δ )dµ+ s + ( ψ δ + ψ δ )dµ s ), (4.47) and therefore (4.4) and (4.42) lead to lim δ 0 lim sup lim sup h(u)v ( ψ δ + n ε 0 n ψ δ )a(x, u ε, u ε ) u ε = 0. (4.48) {n< u ε <2n} By the weak convergence of T 2n (u ε ) in W,p 0 (), we deduce that a(x, T 2n (u ε ), T 2n (u ε )) converges to a(x, T 2n (u), T 2n (u)) weakly in (L p ()) N. Moreover, since h n (u ε ) is bounded by and converges to h n (u) a.e., this implies that lim h (u)h n (u ε )v( ψ ε 0 δ + ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] u = h (u)h n (u)v( ψ δ + ψ δ )[a(x, u, u) + K(x, u)] u On the other hand supp(h) is compact, that is supp(h) ] M, M[ for some M > 0. Therefore a(x, u, u) in the last integral can be replaced by a(x, T M (u), T M (u)) which 3

32 belongs to (L p ()) N. Moreover, by (4.40), ψ + δ and ψ δ converge to 0 strongly in W,p () as δ tends to zero. Thus, we obtain that lim lim lim δ 0 n ε 0 = h (u)h n (u ε )v( ψ + δ h (u)v[a(x, u, u) + K(x, u)] u. Similar arguments yield that lim lim δ 0 n ε 0 lim = and lim lim lim δ 0 n ε 0 h(u)h n (u ε )( ψ + δ h(u)[a(x, u, u) + K(x, u)] v ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] u ψ δ )[a(x, u ε, u ε ) + K ε (x, u ε )] v (4.49) (4.50) h(u)h n (u ε )v[a(x, u ε, u ε ) + K ε (x, u ε )] ( ψ δ + ψ δ ) = 0. (4.5) Using Proposition 2.6, the point-wise convergence of u ε (4.9), the definition of h n (2.29) and the property (4.40) of ψ δ + and ψδ, we obtain that lim lim lim f ε h n (u ε )h(u)v( ψ + δ 0 n ε 0 δ ψ δ ) = fh(u)v. (4.52) Similarly it can be shown that lim lim lim g ε [h n (u ε )h(u)v( ψ δ 0 n ε 0 δ + Since λ ε,0 is a nonnegative measure, we have ψ δ )] = g [h(u)v]. (4.53) h n (u ε )h(u)v( ψ δ + ψ δ )dλ ε,0 h L (IR) v L () ( ψ δ + ψ δ )dλ ε,0. Therefore the inequality 0 λ ε,0 λ ε, and (4.43) and (4.44), imply lim δ 0 lim sup n In the same way we obtain lim δ 0 lim sup n lim sup ε 0 lim sup ε 0 This conclude the proof of (4.38). h n (u ε )h(u)v( ψ δ + ψ δ )dλ ε,0 = 0. (4.54) h n (u ε )h(u)v( ψ δ + ψ δ )dλ ε,0 = 0. 32

33 Since we have proved that u satisfies (4.25), to conclude the proof that u is a renormalized solution of (4.), it remains to prove that we have lim a(x, u, u) u ϕ = ϕdµ + n + s, (4.55) n n<u<2n lim a(x, u, u) u ϕ = ϕdµ n + s. (4.56) n 2n<u< n for any ϕ belonging to Cb 0 (). Since a(x, u, u) u is positive, Remark 2.2 allows us to prove (4.55) and (4.56) for ϕ C ( ). As a consequence of (4.27), the convergence almost everywhere of u ε to u (4.9) and Fatou lemma, we have lim sup a(x, u, u) u ϕ ϕdµ + s, (4.57) n + n n<u<2n lim sup n + a(x, u, u) u ϕ ϕdµ s, (4.58) n 2n<u< n for any ϕ C ( ) with ϕ 0. On the other hand, since u is solution of (4.) in the sense of distributions, we can use h n (u)ψ as test function in (4.38), where h n is defined by (2.29) and ψ C0 (). Then, letting n goes to +, we obtain lim n h n(u)a(x, u, u) uψ = ψdµ + s + ψdµ s, (4.59) for any ψ C 0 (). Let ϕ C ( ) with ϕ 0. Since 0 ψ + δ, 0 ψ δ, we have 0 ϕ( ψ δ )ψ+ δ ϕ and since ψ + δ, ψ δ C 0 (), we get ϕ( ψ δ )ψ+ δ C 0 (). Recalling that h n(s) = n ( χ {n<s<2n} + χ { 2n<s< n} ) a.e. in IR, from (4.59), we get lim inf n n {n<u<2n} n {n<u<2n} a(x, u, u) uϕ lim inf a(x, u, u) uϕ( ψ n δ )ψ+ δ ( ) n lim h n(u)a(x, u, u) uϕ( ψδ )ψ+ δ = ϕ( ψδ )ψ+ δ dµ+ s ϕ( ψδ )ψ+ δ dµ s = ϕψ δ + dµ+ s ϕψδ ψ+ δ dµ+ s ϕ( ψδ )ψ+ δ dµ s. Now we pass to the limit, as δ 0 in the above inequality. By (4.4) and (4.42) we obtain lim inf a(x, u, u) uϕ ϕdµ + n s n {n<u<2n} 33