PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake, Saeki-ku Hiroshima, Japan Fukuyama University, Gakuencho Fukuyama, Japan Abstract. We consider Sobolev Dirichlet problems as well as Dirichlet problems in the PWB-method for quasi-linear second order elliptic differential equations on a euclidean domain. We discuss boundedness of solutions of these problems and convergence of solutions under perturbation of the 0-th order term. Introduction In the previous papers [MO1] and [MO2], we developed a potential theory and discussed Dirichlet problems for elliptic quasi-linear equations of the form E A,B ) div A x, ux) ) + B x, ux) ) = 0 on a domain in R N N 2), where A x, ξ): R N R N satisfies weighted structure conditions of p-th order with a weight wx) and Bx, t): R R is nondecreasing in t see Section 1 below for more details). The purpose of the present paper is to investigate how the solution of a Dirichlet problem varies under perturbation of the term B. As in [MO1] and [MO2], we consider the space D p ; µ) of bounded continuous functions with finite p, µ)-dirichlet integrals on dµ = w dx) and its subspace D p 0 ; µ) consisting of f D p ; µ) which are uniformly bounded limits of ϕ n C0 ) such that ϕ n f in L p ; µ). Throughout this paper we assume that is p, µ)-hyperbolic, namely 1 / D p 0 ; µ). Given θ D p ; µ), the solution u of E A,B ) satifying u θ D p 0 ; µ) may be called the solution of the Sobolev Dirichlet problem with the boundary data θ and is denoted by u A,B,θ). We first introduce a class of L 1 -functions on, and using functions in this class, we present a condition on B condition B.4) in Section 3) which assures the boundedness of u A,B,θ) Mathematics Subject Classification: Primary 31C45; Secondary 31B25.

2 208 Fumi-Yuki Maeda and Takayori Ono As our main theorem Theorem 4.1), we shall show that if B n, n = 1, 2,... satisfy B.4) with the same data, they are uniformly bounded in a certain sense and if sup M2 t M 1 B n x, t) Bx, t) 0 in L 1 ), then u A,Bn,θ) u A,B,θ) locally uniformly on, where M 1 and M 2 are constants determined by the data given in B.4) and θ. We shall also give the convergence of u A,Bn,θ) to u A,B,θ) in L p ; µ) Theorem 4.2). In [MO1], we have studied Dirichlet problems in the PWB-method with respect to a Royden type ideal boundary for our equation E A,B ). In Section 5, under the same conditions as in Theorem 4.1, we give local uniform convergence of such Dirichlet solutions. 1. Preliminaries As in [MO1] and [MO2] we assume that A : R N R N and B: R R satisfy the following conditions for 1 < p < and a weight w which is p- admissible in the sense of [HKM]: A.1) x A x, ξ) is measurable on for every ξ R N and ξ A x, ξ) is continuous for a.e. x ; A.2) A x, ξ) ξ α 1 wx) ξ p for all ξ R N and a.e. x with a constant α 1 > 0; A.3) A x, ξ) α 2 wx) ξ p 1 for all ξ R N and a.e. x with a constant α 2 > 0; A.4) A x, ξ 1 ) A x, ξ 2 ) ) ξ ) 1 ξ 2 > 0 whenever ξ1, ξ 2 R N, ξ 1 ξ 2, for a.e. x ; B.1) x Bx, t) is measurable on for every t R and t Bx, t) is continuous for a.e. x ; B.2) For any open set D, there is a constant α 3 D) 0 such that Bx, t) α 3 D)wx) t p 1 + 1) for all t R and a.e. x D ; B.3) t Bx, t) is nondecreasing on R for a.e. x. For the nonnegative measure µ : dµx) = wx) dx and an open set D, we consider the weighted Sobolev spaces H 1,p D; µ), H 1,p 0 D; µ) and H1,p loc D; µ) see [HKM] for details). Let D be an open subset of. Then u H 1,p loc D; µ) is said to be a weak) solution of E A,B ) in D if D A x, u) ϕ dx + D Bx, u)ϕ dx = 0 for all ϕ C0 D). u H 1,p loc D; µ) is said to be a supersolution respectively subsolution) of E A,B ) in D if D A x, u) ϕ dx + D Bx, u)ϕ dx 0 respectively 0)

3 Perturbation theory for nonlinear Dirichlet problems 209 for all nonnegative ϕ C0 D). A continuous solution of E A,B ) in an open set D is called A, B)- harmonic in D. We say that an open set D in is A, B)-regular, if D and for any θ H 1,p loc ; µ) which is continuous at each point of D, there exists a unique h CD) H 1,p D; µ) such that h = θ on D and h is A, B)-harmonic in D. Proposition A [MO1; Theorem 1.4] and [HKM; Theorem 6.31]). Any ball B is A, B)-regular. A function u: D R { } is said to be A, B)-superharmonic in D if it is lower semicontinuous, finite on a dense set in D and, for each open set G D and for h CG) which is A, B)-harmonic in G, u h on G implies u h in G. A, B)-subharmonic functions are similarly defined. Note that a continuous supersolution of E A,B ) is A, B)-superharmonic cf. [MO1; Section 2]). We recall the following two spaces which are defined in [MO1] cf. Introduction): D p ; µ) = { f H 1,p loc ; µ) f Lp ; µ), f is bounded continuous }, D p 0 ; µ) = { f D p ; µ) there exist ϕ n C 0 ) such that ϕ n f a.e., {ϕ n } is uniformly bounded, ϕ n f in L p ; µ) }. Note that H 1,p 0 ; µ) D p ; µ) D p 0 ; µ) and the inclusion becomes equality if is bounded. B.5) Lemma 1.1. Suppose B satisfies Bx, t) dx < for any t R. For u D p ; µ), if u is a solution respectively supersolution, subsolution) of E A,B ), then A x, u) ϕ dx + Bx, u)ϕ dx = 0 respectively 0, 0) for all ϕ D p 0 ; µ) respectively for all nonnegative ϕ D p 0 ; µ)). Proof. Choose ϕ n C 0 ) such that ϕ n ϕ a.e., {ϕ n } is uniformly bounded and ϕ n ϕ in L p ; µ) respectively and further ϕ n 0). Since u is a solution respectively supersolution, subsolution) in, we have 1.1) A x, u) ϕ n dx + Bx, u)ϕ n dx = 0 respectively 0, 0).

4 210 Fumi-Yuki Maeda and Takayori Ono Then by A.3) A x, u) ϕ n dx A x, u) ϕ dx cf. the proof of [HKM; Lemma 3.11]). By B.3) and B.5), Bx, u) dx <. Hence, by Lebesgue s convergence theorem Bx, u)ϕ n dx Therefore, letting n in 1.1), we have A x, u) ϕ dx + Bx, u)ϕ dx. Bx, u)ϕ dx = 0 respectively 0, 0). We have the following variant of the comparison principle cf. [MO1; Proposition 1.1]): Lemma 1.2. Suppose is p, µ)-hyperbolic and B satisfies B.5). For u, v D p ; µ), if u is a supersolution of E A,B ), v is a subsolution of E A,B ) and minu v, 0) D p 0 ; µ), then u v on. Proof. Set η = minu v, 0). Since η D p 0 ; µ) and η 0, by Lemma 1.1 we have A x, u) η dx + Bx, u)η dx 0 and By A.4) and B.3), A x, v) η dx + A x, v) A x, u) ) η dx Bx, v)η dx 0. = A x, v) A x, u) v u) dx {u<v} 0 and Bx, v) Bx, u) ) η dx = {u<v} Bx, v) Bx, u) ) v u) dx 0.

5 Perturbation theory for nonlinear Dirichlet problems 211 Thus and hence ) 0 A x, v) η dx + Bx, v)η dx ) A x, u) η dx + Bx, u)η dx 0, {u<v} A x, v) A x, u) ) v u) dx = 0. Therefore again by A.4), v u = 0 a.e. on {u < v}. Thus η = 0 a.e., so that η = c. Since is p, µ)-hyperbolic, we see that c = 0, and the lemma follows. The next lemma will be used in the proof of Theorem 3.2. Lemma 1.3. Suppose K 1 and K 2 are constants such that K 1 K 2. Given θ D p ; µ), let θ = max minθ, K 2 ), K 1. If u θ D p 0 ; µ) and K 1 u K 2 in, then u θ D p 0 ; µ). Proof. Choose ϕ n C 0 ) such that ϕ n u θ a.e., {ϕ n } is uniformly bounded and ϕ n u θ) in L p ; µ). Setting ϕ n = max minϕ n, u K 1 ), u K 2 ), we have ϕ n max minu θ, u K 1 ), u K 2 = u θ a.e. in. Also, by [HKM; Lemma 1.2.2], ϕ n u θ ) in L p ; µ). Since u K 1 0 and u K 2 0, suppϕ n is compact in for each n. Thus, considering mollified functions, we obtain approximating functions for u θ. We recall the following condition, which has been considered in [MO1] and [MO2] for the discussion of resolutivity and harmonizability. C 1 ) There exist a bounded supersolution of E A,B ) in and a bounded subsolution of E A,B ) in. The following theorem which asserts the existence and uniqueness of the Sobolev Dirichlet problem is shown by [MO2; Theorem 2.2 and Proposition 1.5]. Theorem A. Suppose that is p, µ)-hyperbolic and suppose that conditions C 1 ) and B.5) are satisfied. If θ D p ; µ), then there exists a unique A, B)-harmonic function u A,B,θ) on such that u A,B,θ) θ D p 0 ; µ).

6 212 Fumi-Yuki Maeda and Takayori Ono 2. A class of L 1 -functions on Hereafter, we always assume that is p, µ)-hyperbolic, namely, 1 / D p 0 ; µ). Note that any bounded domain is p, µ)-hyperbolic. We consider the following function spaces: F A ) = { f L 1 ) f/w is locally bounded in and div A x, u) = f has a solution in D p 0 ; µ)} F + A ) = { f F A ) f 0 } and F A ) = { f F A ) f 0 }. For f F A ), the solution of div A x, u) = f in D p 0 ; µ) will be denoted by U f. Obviously, 0 F A ) and U 0 = 0. If f L 1 ), f/w is locally bounded on and C 1 ) is satisfied for E A, f ), then f F and U f = u A, f,0). Proposition 2.1. If f 1, f 2 F A ) and f 1 f 2, then U f 1 U f 2. Proof. Since U f 2 is a supersolution of E A, f1 ), U f 1 is a solution of E A, f1 ) and minu f 2 U f 1, 0) D p 0 ; µ), the proposition follows from Lemma 1.2. Corollary 2.1. U f 0 for f F + A ) and U f 0 for f F A ). We can easily show Proposition 2.2. If A satisfies the homogeneity condition A.5) A x, λξ) = λ λ p 2 A x, ξ) for any λ R, then, λf F A ) for f F A ) and λ R, and U λf = λ λ 2 p)/p 1) U f. Proposition 2.3. If f is measurable and g 1 f g 2 for some g 1 F A ) and g 2 F + A ), then f F A ). Proof. Since U g 1 is a bounded subsolution of E A, f ) and U g 2 is a bounded supersolution of E A, f ), Theorem A asserts the existence of u A, f,0), which is U f. Proposition 2.4. In case is bounded, any measurable function f satisfying 0 fx) βwx) respectively βwx) fx) 0) for some β > 0 belongs to F + A ) respectively F A )). Proof. By the above proposition, it is enough to show that βw F + A ) and βw F A ). We consider A 1 : R N R N R N defined by { A x, ξ), x, A 1 x, ξ) = wx) ξ p 2 ξ, x R N \, and take an open ball B. By Proposition A, there exists u CB) H 1,p B; µ) such that u = 0 on B and u is A 1, βw)-harmonic in B. Then u is bounded and A, βw)-harmonic in. It follows from Theorem A that U βw exists and hence βw F + A ). Similarly, we see that βw F A ).

7 Perturbation theory for nonlinear Dirichlet problems 213 Example 2.1. Let = B0, R) = { x < R} for 0 < R <, wx) = x δ with δ > N and A x, ξ) = x δ ξ p 2 ξ. Let g be a non-negative L 1 -function on [0, R) which is bounded on [0, ϱ] for any 0 < ϱ < R and let fx) = x δ g x ). Then f F + A ) and U f x) = R x 1 r N+δ 1 r If we take gt) 1, i.e., fx) = x δ, then where 1/p + 1/p = 1. 0 gt)t N+δ 1 dt) 1/p 1) dr. U f x) = N + δ) 1/p 1) p 1 R p x p ), We can take unbounded f, e.g., fx) = x δ R x ) α with 0 < α < 1. Example 2.2. Let = R N, wx) = x δ with p < N + δ and A x, ξ) = x δ ξ p 2 ξ. Note that is p, µ)-hyperbolic see [MO1; p. 570]). Let g be a non-negative function on [0, ) which is bounded on [0, ϱ] for any 0 < ϱ < and for which gt)t N+δ 1 dt <. Then fx) = x δ g x ) belongs to F + A ) 0 and U f 1 r 1/p 1) x) = r N+δ 1 gt)t dt) N+δ 1 dr. x 0 Proof of Examples 2.1 and 2.2. Obviously, f 0 and f/ x δ is locally bounded on. Also, it is easily verified that f L 1 ). Set Gr) = r 0 gt)t N+δ 1 dt and ux) = R x r 1 N+δ 1 Gr) ) 1/p 1) dr, where we set R = for Example 2.2. Then Gr) is bounded for 0 < r < R and Gr) c ϱ r N+δ for 0 < r < ϱ, ϱ < R. From these it follows that ux) C 1 ) and ux) = x ) 1/p 1) 1 G x ). x x N+δ 1 In case R <, since ux) is bounded, ux) p x δ dx <. In case R =, since ux) p x δ dx c x pn+δ p)/p 1) dx x 1 x 1 and p < N + δ, we have ux) p x δ dx <. Moreover, if x R, then ux) 0. Thus, setting ϕ n x) = max ux) 1/n, 0 ), we have ϕ n D p ; x δ dx)

8 214 Fumi-Yuki Maeda and Takayori Ono and suppϕ n is compact. Since {ϕ n } is uniformly bounded, ϕ n u in and ϕ n u in L p ; x δ dx), it follows that u D p 0 ; x δ dx). Finally, we have div x δ ux) p 2 ux) ) = div x x N G x ) ) = N x N G x ) + x δ g x ) + N x N G x ) = x δ g x ) = fx), in the weak sense, so that u is a solution of div x δ u p 2 u) = f. 3. Boundedness of solutions of Sobolev Dirichlet problems In addition to B.1), B.2) and B.3), we shall always assume that B satisfies condition B.5). Further we consider the following condition on B : B.4) There exist nonnegative numbers T 1, T 2, functions f 1 F + A ) and f 2 F A ) such that B x, T 1 ) f 1 x) and B + x, T 2 ) f 2 x) a.e. in. Example 3.1. Let ζt) be a nondecreasing continuous function on R such that ζt) c t p 1 for t 1 and ζt 0 ) = 0 for some t 0 R. We set Bx, t) = bx)ζt) with b L 1 ) such that b 0 and b/w is locally bounded on. Then B satisfies B.4) with T 1 = t + 0, T 2 = t 0 and f 1 = f 2 = 0. If B satisfies B.4), then T 1 + U f 1 is a supersolution and T 2 + U f 2 is a subsolution of E A,B ). Thus, condition C 1 ) is satisfied. For θ D p ; µ), we define sup θ = inf { k θ k) + D p 0 ; µ)} and inf θ = sup{ k θ k) D p 0 ; µ)}. Theorem 3.1. Suppose B satisfies B.4). Then for any θ D p ; µ), there exists a unique A, B)-harmonic function u A,B,θ) on such that u A,B,θ) θ D p 0 ; µ). Further it satisfies min T 2, inf θ + U f 2 x) u A,B,θ) x) max T 1, sup θ + U f 1 x) on. Proof. Since condition B.4) implies condition C 1 ), the existence and the uniqueness follow from Theorem A, we show only the inequalities. Fix ε > 0 and let v = maxt 1, sup θ)+u f 1 +ε. Since Bx, v) Bx, T 1 ) f 1 by B.4), v is a supersolution of E A,B ). Also, since v sup θ+ε, we see that maxθ v, 0) D p 0 ; µ), and hence that minv u A,B,θ), 0) D p 0 ; µ). Hence by Lemma 1.2, v u A,B,θ) in. Since ε > 0 is arbitrary, maxt 1, sup θ) + U f 1 u A,B,θ) in. The other inequality follows similarly.

9 on. Perturbation theory for nonlinear Dirichlet problems 215 In view of Example 3.1, we can state Corollary 3.1. Let Bx, t) = bx)ζt) be as in Example 3.1. Then min t 0, inf θ u A,B,θ) x) max t + 0, sup θ In view of Propositions 2.4 and 2.2, we also have Corollary 3.2. Suppose is bounded, A satisfies A.5) in Proposition 2.2) and Bx, 0) βwx) a.e. in. Then min inf θ, 0 β 1/p 1) U w x) u A,B,θ) x) max sup θ, 0 + β 1/p 1) U w x) on. Given nonnegative numbers T 1, T 2, functions f 1 F + A ), f 2 F A ) and given θ D p ; µ), let M + T 1, f 1, θ) = max T 1, sup θ + sup U f 1, M T 2, f 2, θ) = max T 2, inf θ inf U f 2. Theorem 3.1 asserts that M T 2, f 2, θ) u A,B,θ) M + T 1, f 1, θ). Theorem 3.2. Suppose B satisfies B.4). Then for θ D p ; µ), u A,B,θ) x) p p α2 dµ θx) p dµ + pm B x, θ x) ) dx, α 1 α 1 where and M = M + T 1, f 1, θ) + M T 2, f 2, θ) θ = max min θ, M + T 1, f 1, θ) ), M T 1, f 1, θ) ). Proof. Since u A,B,θ) θ D p 0 ; µ) by Lemma 1.3, we have A x, u A,B,θ) ) u A,B,θ) θ ) dx + Bx, u A,B,θ) )u A,B,θ) θ ) dx = 0.

10 216 Fumi-Yuki Maeda and Takayori Ono By Theorem 3.1, u A,B,θ) θ M. Hence, using A.2), A.3) and B.3), we have α 1 u A,B,θ) p dµ α 2 u A,B,θ) p 1 θ dµ Bx, u A,B,θ) )u A,B,θ) θ ) dx α 2 u A,B,θ) p 1 θ dµ 3.1) Bx, θ )u A,B,θ) θ ) dx α 2 + M ) p 1)/p 1/p u A,B,θ) p dµ θ p dµ Bx, θ ) dx, where in the last inequality we have used Hölder s inequality. An application of Young s inequality yields that X AX p 1)/p + C implies X A p + pc for X 0, A 0 and C 0. Hence, from 3.1) we obtain the desired estimate. Theorem 3.3. Suppose B 1 and B 2 satisfy B.4) with the same T 1, T 2, f 1 and f 2. Let θ D p ; µ) and set u j = u A,Bj,θ), j = 1, 2. Then A x, u1 ) A x, u 2 ) ) u 1 u 2 ) dx M sup M 2 t M 1 B 1 x, t) B 2 x, t) dx, where M 1 = M + T 1, f 1, θ), M 2 = M T 2, f 2, θ) and M = M 1 + M 2. Proof. Since u 1 u 2 D p 0 ; µ), A x, u1 ) A x, u 2 ) ) u 1 u 2 ) dx + Hence using B.3) we have A x, u1 ) A x, u 2 ) ) u 1 u 2 ) dx B1 x, u 1 ) B 2 x, u 2 ) ) u 1 u 2 ) dx = 0. B1 x, u 2 ) B 2 x, u 2 ) ) u 2 u 1 ) dx. Since M 2 u j M 1, j = 1, 2, by Theorem 3.1, we obtain the desired estimate.

11 Perturbation theory for nonlinear Dirichlet problems Convergence theorems The following lemma can be shown in the same manner as [MO1; Lemma 5.1]. Lemma 4.1. Let {u n } be a uniformly bounded sequence of functions in D p 0 ; µ) such that { u n p dµ } is bounded and u n u a.e. in. If u is continuous, then u D p 0 ; µ). The next lemma will be used to show Theorem 4.1. Lemma 4.2 [O; Theorem 4.7]). Let u be an A, B)-harmonic function in and x 0 be any point of. If 0 < R < is such that Bx 0, R) and if u L in Bx 0, R), then there are constants c and 0 < λ < 1 such that λ ϱ sup u inf u c, Bx 0,ϱ) Bx 0,ϱ) R whenever 0 < ϱ < R. Here c and λ depend only on N, p, α 1, α 2, α 3 Bx0, R) ), µ, R and L. Theorem 4.1. Suppose B n, n = 1, 2,..., and B all satisfy B.4) with the same T 1, T 2, f 1 F + A ), f 2 F A ). Let θ D p ; µ). Assume further that there exists a nonnegative function b on such that b/w is locally bounded in and 4.1) B + nx, M 1 ) + B n x, M 2 ) bx) a.e. on for all n, where M 1, M 2 are as in Theorem 3.3. If 4.2) sup B n x, t) Bx, t) dx 0 M 2 t M 1 n ), then u A,Bn,θ) u A,B,θ) as n locally uniformly on. Proof. If we set B n x, M 1 ), t M 1, Bnx, t) = B n x, t), M 2 < t < M 1, B n x, M 2 ), t M 2, then u A,Bn,θ) is a solution of E A,B n ), and hence u A,Bn,θ) = u A,B n,θ). Thus by 4.1), we may assume that B n x, t) bx) n = 1, 2,...) for any t R. Then, for any D, we can take α 3 D) = sup D b/w in B.2) for B n for all n. Let u n = u A,Bn,θ) and u = u A,B,θ). By Theorem 3.1, {u n } is uniformly bounded in. Hence, by Lemma 4.2, we see that {u n } is equi-continuous at each

12 218 Fumi-Yuki Maeda and Takayori Ono point of. Hence it follows from Ascoli Arzela s theorem that any subsequence of {u n } has a locally uniformly convergent subsequence. Let {u nk } be any subsequence of {u n } which converges locally uniformly to u. Since { sup M2 t M 1 B n x, t) } is bounded in L 1 ) by 4.2), Theorem 3.2 yields that { u n } is bounded in L p ; µ). Thus, by Lemma 4.1 and [HMK; Lemma 1.3.3], we see that u u D p 0 ; µ) and u n k u weakly in L p ; µ). On the other hand, by Theorem 3.3 and [HKM; Lemma 3.73], u n u weakly in L p ; µ), and hence u = u. Since is p, µ)- hyperbolic, it follows that u = u. Therefore, u n u locally uniformly in. In view of Example 3.1, we can state Corollary 4.1. Let ζt) be as in Example 3.1 and b n, n = 1, 2,..., and b be nonnegative measurable functions such that 4.3) b n x) b 0 x) a.e. on for some function b 0 such that b 0 /w is locally bounded in and 4.4) b n x) bx) dx 0 n ). Then, for B n x, t) = b n x)ζt), n = 1, 2,..., and Bx, t) = bx)ζt), u A,Bn,θ) u A,B,θ) locally uniformly on. The following example shows that we cannot assert the uniform convergence on in the above theorem and corollary: Example 4.1. Let = B0, 1), 1 < p N, wx) = 1 and A x, ξ) = ξ p 2 ξ. Let {a n } be a sequence of points in B0, 1) such that a n 1 and set b n x) = 2 n rn N χ Ban,r n )x) with 0 < r n < 1 a n, n = 1, 2,.... For a nondecreasing continuous function ζ on R such that ζ0) = 0, ζ1) > 0 and ζt) c t p 1 for t 1, we set B n x, t) = n b k x)ζt) and Bx, t) = k=1 b n x)ζt). Then, for θ = 1, B n and B satisfy the conditions in Theorem 4.1. If we choose {r n } suitably, then {u A,Bn,1)} does not converge uniformly on B0, 1). Proof. Let B n = Ba n, 1 a n ) and v n be the solution of the equation div A x, u) = b n on B n which belongs to D p 0 B n; dx). By Lemma 1.2, we see that 0 v n U b n in B n. Noting that A x, ξ) = ξ p 2 ξ is translation invariant, n=1

13 by Example 2.1 we have Perturbation theory for nonlinear Dirichlet problems 219 { rn v n a n ) = 2 n rn N ) 1/p 1) + = 1 an r n 1 r N 1 0 rn 0 1 r 1/p 1) r N 1 t dt) N 1 dr 0 } t N 1 dt) 1/p 1) dr 2 n N) 1/1 N) { 1 1 N + log 1 a n r n { Np 1) 2 n N) 1/1 p) pn p) rp N)/p 1) n p 1 } if p = N, } N p 1 a n ) p N)/p 1) if p < N. Thus, U b n a n ) v n a n ) 1 for sufficiently small r n. Let u n = u A,Bn,1) and u = u A,B,1) for simplicity. By considering the extension of B n by 0 outside B0, 1) and using Proposition A, we see that u n x) 1 as x 1. Now suppose that {u n } converges to u uniformly on B0, 1). Then ux) 1 as x 1. Choose ε > 0 such that ε p 1 < ζ1 ε). Then there is r 0 < 1 such that ux) 1 ε for x r 0. For large n, Ba n, r n ) B0, r 0 ) = and div 1 u) p 2 1 u)) = div u p 2 u) = b n x)ζ ux) ) ζ1 ε)b n x). n=1 Thus, by Lemma 1.2, 1 u ζ1 ε) 1/p 1) U b n, so that ua n ) 1 ζ1 ε) 1/p 1) U b n a n ) < 1 ε for large n. This contradicts our assumption that ux) 1 ε for x r 0. Theorem 4.2. Suppose B n, n = 1, 2,..., and B all satisfy B.4) with the same T 1, T 2, f 1 F + A ), f 2 F A ). Let θ D p ; µ). Assume further that A satisfies A.4s) A x, ξ1 ) A x, ξ 2 ) ) ξ 1 ξ 2 ) α 4 wx) ξ 1 + ξ 2 ) p 2 ξ 1 ξ 2 2 with α 4 > 0. If 4.2) holds, then u A,Bn,θ) u A,B,θ) in L p ; µ); further u A,Bn,θ) u A,B,θ) in H 1,p ; µ) in case is bounded.

14 220 Fumi-Yuki Maeda and Takayori Ono Proof. If p 2, the first assertion is obvious from Theorem 3.3. In case 1 < p < 2, we have u n u p dµ { = un + u ) p 2 u n u 2} p/2 un + u ) p2 p)/2 dµ u n + u ) p 2 u n u 2 dµ 2 p)/2 u n + u ) dµ) p. ) p/2 Hence the first assertion in this case also follows from Theorems 3.2 and 3.3. The second assertion follows from the first assertion and the Poincaré inequality. let 5. Boundedness and convergence of solutions for Dirichlet problems and Given a compactification of and a bounded function ψ on = \, { U ψ = u : A, B)-superharmonic in and } lim inf ux) ψξ) for all ξ x ξ L ψ = { v : A, B)-subharmonic in and If both U ψ and L ψ are nonempty, then lim sup vx) ψξ) for all ξ x ξ }. and H ψ; ) =H A,B) ψ; ) := inf U ψ Hψ; ) = H A,B) ψ; ) := sup L ψ are A, B)-harmonic in and Hψ; ) H ψ; ) [MO1; Theorem 3.1]). We say that ψ is A, B )-resolutive if both U ψ and L ψ are nonempty and Hψ; ) = H ψ; ). In this case we write Hψ; ) = H A,B) ψ; ) for Hψ; ) = H ψ; ). is said to be an A, B )-resolutive compactification if every ψ C ) is A, B )-resolutive. In the proof of [MO1; Proposition 3.1] we have shown

15 Perturbation theory for nonlinear Dirichlet problems 221 Lemma 5.1. If ψ 1 and ψ 2 are A, B)-resolutive functions on, then Hψ 1 ; ) Hψ 2 ; ) sup ψ 1 ψ 2. Theorem 5.1. Let be a compactification of. If B satisfies B.4) and if ψ is a bounded function on, then both U ψ and L ψ are nonempty and min T 2, inf + U f 2 H A,B) ψ; ) H A,B) ψ; ) max T 1, sup ψ + U f 1. Proof. By B.4) we see that max T 1, sup ) ψ + U f 1 U ψ and min T 2, inf ψ + U f 2 L ψ. Thus the theorem follows. Like Corollaries 3.1 and 3.2, we have the following two corollaries. Corollary 5.1. Let Bx, t) = bx)ζt) be as in Example 3.1. If ψ is a bounded function on, then both U ψ and L ψ are nonempty and min t 0, inf ψ H A,B) ψ; ) H A,B) ψ; ) max t + 0, sup ψ. Corollary 5.2. Suppose is bounded and let be a compactification of. If A satisfies A.5) and if Bx, 0) βwx) for a.e. x, then H A,B) ψ; ) sup ψ + β 1/p 1) U w for any bounded A, B)-resolutive function ψ on. We recall [MO1; Theorem 3.2]) that under conditions C 1 ) and B.5), the Q-compactification Q of see [CC]) is an A, B)-resolutive compactification if Q D p ; µ). Theorem 5.2. Let Q D p ; µ), Q be the Q-compactification of, Γ = Q \ and let ψ CΓ). Suppose B n, n = 1, 2,..., and B all satisfy B.4) with the same T 1, T 2, f 1 F + A ), f 2 F A ). Set M 1 = max T 1, max ψ + sup U f 1 and M 2 = max T 2, min ψ inf U f 2. Γ Γ Assume further that there exists a nonnegative function bx) on such that bx)/wx) is locally bounded in and B n satisfy 4.1) for all n. If 4.2) holds, then H A,Bn) ψ; Q ) HA,B) ψ; Q ) locally uniformly on.

16 222 Fumi-Yuki Maeda and Takayori Ono Proof. Since the set of continuous extensions of functions in Q is dense in CΓ) with respect to the uniform convergence, given ε > 0 there exists θ Q such that sup ξ Γ θ ξ) ψξ) ε and inf ψ θ sup ψ on Γ, where θ is the continuous extension of θ to Γ. Note that ψ and θ are A, B)-resolutive as well as A, B n )-resolutive for all n [MO1; Theorem 3.2]). For simplicity, let H n ψ) = H A,Bn) ψ; Q ), H nθ) = H A,Bn) θ ; Q ), Hψ) = HA,B) ψ; Q ) and Hθ ) = H A,B) θ ; Q ). By Lemma 5.1, we see that Hθ ) Hψ) ε and H n θ ) H n ψ) ε for all n. On the other hand, by [MO2; Proposition 2.2], H n θ ) = u A,Bn,θ) and Hθ ) = u A,B,θ). Also, since B n + x, M + T 1, f 1, θ) ) B nx, + M 1 ) and Bn x, M T 2, f 2, θ) ) Bn x, M 2 ), by Theorem 4.1, for any open G, there is n 0 such that sup u A,Bn,θ) u A,B,θ) ε G for n n 0. Thus, for n n 0 sup G H n ψ) Hψ) { sup Hn ψ) H n θ ) + H n θ ) Hθ ) + Hθ ) Hψ) } G 3ε. Hence the theorem follows. Like Corollary 4.1, we obtain Corollary 5.3. Let Q and Γ be as in Theorem 5.2 and let B nx, t) = b n x)ζt) and Bx, t) = bx)ζt) with nonnegative measurable functions b n, b on and ζ as in Example 3.1. If 4.3) and 4.4) are satisfied then H A,Bn) ψ; Q ) H A,B) ψ; Q ) locally uniformly on for any ψ CΓ). References [CC] Constantinescu, C., and A. Cornea: Ideale Ränder Riemannscher Flächen. - Springer- Verlag, [HKM] Heinonen, J., T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Clarendon Press, [MO1] Maeda, F.-Y., and T. Ono: Resolutivity of ideal boundary for nonlinear Dirichlet problems. - J. Math. Soc. Japan 52, 2000, [MO2] Maeda, F.-Y., and T. Ono: Properties of harmonic boundary in nonlinear potential theory. - Hiroshima Math. J. 30, 2000, [O] Ono, T.: On solutions of quasi-linear partial differential equations div A x, u) + Bx, u) = 0. - RIMS Kokyuroku 1016, 1997, Received 2 July 2002