Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207–222

PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS

Fumi-Yuki Maeda and Takayori Ono

Hiroshima Institute of Technology, Miyake, Saeki-ku Hiroshima, 731-5193 Japan Fukuyama University, Gakuencho Fukuyama, 729-0292 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, ∇u(x)) + ℬ(x, u(x)) = 0

on a domain Ω in R^{N} (N ≥ 2), where 𝒜 (x, ξ): Ω×R^{N} → R^{N} satisfies weighted structure conditions of p-th order with a weight w(x) and ℬ(x, t): Ω × 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}(Ω; μ) of bounded continuous functions with finite (p, μ)-Dirichlet integrals on Ω (dμ = w dx) and its subspace 𝒟_{0}^{p}(Ω; μ) consisting of f ∈ 𝒟^{p}(Ω; μ) which are uniformly bounded limits of φ_{n} ∈ C_{0}^{∞}(Ω) such that ∇φ_{n} → ∇f in L^{p}(Ω; μ). Throughout this paper we assume that Ω is (p, μ)-hyperbolic, namely 1 ∉ 𝒟_{0}^{p}(Ω; μ). Given θ ∈ 𝒟^{p}(Ω; μ), the solution u of (E_{𝒜 ,ℬ} ) satifying u − θ ∈ 𝒟_{0}^{p}(Ω; 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 Ω, 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)} .

2000 Mathematics Subject Classification: Primary 31C45; Secondary 31B25.

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_{−M_{2}≤t≤M_{1}} |ℬ_{n}(x, t)−ℬ(x, t)| → 0 in L^{1}(Ω), then u_{(𝒜 ,ℬ_{n},θ)} → u_{(𝒜 ,ℬ,θ)} locally uniformly on Ω, 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 ,ℬ_{n},θ)} to ∇u_{(𝒜 ,ℬ,θ)} in L^{p}(Ω; 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: Ω×R^{N} → R^{N} and ℬ: Ω×R → R satisfy the following conditions for 1 < p < ∞ and a weight w which is padmissible in the sense of [HKM]:

(A.1) x ↦ 𝒜 (x, ξ) is measurable on Ω for every ξ ∈ R^{N} and ξ ↦ 𝒜 (x, ξ) is

continuous for a.e. x ∈ Ω; (A.2) 𝒜 (x, ξ) · ξ ≥ α_{1}w(x)|ξ|^{p} for all ξ ∈ R^{N} and a.e. x ∈ Ω with a constant

α_{1} > 0; (A.3) |𝒜 (x, ξ)| ≤ α_{2}w(x)|ξ|^{p−1} for all ξ ∈ R^{N} and a.e. x ∈ Ω with a constant

α_{2} > 0; (A.4) (𝒜 (x, ξ_{1}) − 𝒜 (x, ξ_{2})) · (ξ_{1} − ξ_{2}) > 0 whenever ξ_{1}, ξ_{2} ∈ R^{N}, ξ_{1} ≠ ξ_{2} , for a.e.

x ∈ Ω; (B.1) x ↦ ℬ(x, t) is measurable on Ω for every t ∈ R and t ↦ ℬ(x, t) is contin-

uous for a.e. x ∈ Ω; (B.2) For any open set D ⋐ Ω, there is a constant α_{3}(D) ≥ 0 such that |ℬ(x, t)| ≤

α_{3}(D)w(x)(|t|^{p−1} + 1) for all t ∈ R and a.e. x ∈ D; (B.3) t ↦ ℬ(x, t) is nondecreasing on R for a.e. x ∈ Ω.

For the nonnegative measure μ : dμ(x) = w(x) dx and an open set D, we consider the weighted Sobolev spaces H^{1,p}(D; μ), H_{0}^{1,p}(D; μ) and H_{loc}^{1,p}(D; μ) (see [HKM] for details).

Let D be an open subset of Ω. Then u ∈ H_{loc}^{1,p}(D; μ) is said to be a (weak) solution of (E_{𝒜 ,ℬ} ) in D if

∫_{D} 𝒜 (x, ∇u) · ∇φ dx + ∫_{D} ℬ(x, u)φ dx = 0

for all φ ∈ C_{0}^{∞}(D). u ∈ H_{loc}^{1,p}(D; μ) is said to be a supersolution (respectively subsolution) of (E_{𝒜 ,ℬ} ) in D if

∫_{D} 𝒜 (x, ∇u) · ∇φ dx + ∫_{D} ℬ(x, u)φ dx ≥ 0 (respectively ≤ 0)

for all nonnegative φ ∈ C_{0}^{∞}(D) .

A continuous solution of (E_{𝒜 ,ℬ} ) in an open set D ⊂ Ω is called (𝒜 , ℬ)- harmonic in D.

We say that an open set D in Ω is (𝒜 , ℬ)-regular, if D ⋐ Ω and for any θ ∈ H_{loc}^{1,p}(Ω; μ) which is continuous at each point of ∂D, there exists a unique h ∈ C(\overline{D}) ∩ 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 ⋐ Ω 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 ∈ C(\overline{G}) 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}(Ω; μ) = {f ∈ H_{loc}^{1,p}(Ω; μ) | |∇f| ∈ L^{p}(Ω; μ), f is bounded continuous},
𝒟_{0}^{p}(Ω; μ) = {f ∈ 𝒟^{p}(Ω; μ) | there exist φ_{n} ∈ C_{0}^{∞}(Ω) such that φ_{n} → f a.e.,
{φ_{n}} is uniformly bounded, ∇φ_{n} → ∇f in L^{p}(Ω; μ)}.

Note that H_{0}^{1,p}(Ω; μ)∩𝒟^{p}(Ω; μ) ⊂ 𝒟_{0}^{p}(Ω; μ) and the inclusion becomes equality if Ω is bounded.

Lemma 1.1. Suppose ℬ satisfies

(B.5)  ∫_{Ω} |ℬ(x, t)| dx < ∞ for any t ∈ R.

For u ∈ 𝒟^{p}(Ω; μ), if u is a solution (respectively supersolution, subsolution) of (E_{𝒜 ,ℬ} ), then

∫_{Ω} 𝒜 (x, ∇u) · ∇φ dx + ∫_{Ω} ℬ(x, u)φ dx = 0 (respectively ≥ 0, ≤ 0)

for all φ ∈ 𝒟_{0}^{p}(Ω; μ) (respectively for all nonnegative φ ∈ 𝒟_{0}^{p}(Ω; μ)).

Proof. Choose φ_{n} ∈ C_{0}^{∞}(Ω) such that φ_{n} → φ a.e., {φ_{n}} is uniformly bounded and ∇φ_{n} → ∇φ in L^{p}(Ω; μ) (respectively and further φ_{n} ≥ 0). Since u is a solution (respectively supersolution, subsolution) in Ω, we have

(1.1) ∫_{Ω} 𝒜 (x, ∇u) · ∇φ_{n} dx + ∫_{Ω} ℬ(x, u)φ_{n} dx = 0 (respectively ≥ 0, ≤ 0).

Then by (A.3)

∫_{Ω} 𝒜 (x, ∇u) · ∇φ_{n} dx → ∫_{Ω} 𝒜 (x, ∇u) · ∇φ dx

(cf. the proof of [HKM; Lemma 3.11]). By (B.3) and (B.5), ∫_{Ω} |ℬ(x, u)| dx < ∞. Hence, by Lebesgue’s convergence theorem

∫_{Ω} ℬ(x, u)φ_{n} dx → ∫_{Ω} ℬ(x, u)φ dx.

Therefore, letting n → ∞ in (1.1), we have

∫_{Ω} 𝒜 (x, ∇u) · ∇φ dx + ∫_{Ω} ℬ(x, 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 Ω is (p, μ)-hyperbolic and ℬ satisfies (B.5). For u, v ∈ 𝒟^{p}(Ω; μ), if u is a supersolution of (E_{𝒜 ,ℬ} ), v is a subsolution of (E_{𝒜 ,ℬ} ) and min(u − v, 0) ∈ 𝒟_{0}^{p}(Ω; μ), then u ≥ v on Ω.

Proof. Set η = min(u − v, 0). Since η ∈ 𝒟_{0}^{p}(Ω; μ) and η ≤ 0, by Lemma 1.1 we have

∫_{Ω} 𝒜 (x, ∇u) · ∇η dx + ∫_{Ω} ℬ(x, u)η dx ≤ 0

and

∫_{Ω} 𝒜 (x, ∇v) · ∇η dx + ∫_{Ω} ℬ(x, v)η dx ≥ 0.

By (A.4) and (B.3),

∫_{Ω}(𝒜 (x, ∇v) − 𝒜 (x, ∇u)) · ∇η dx
= − ∫_{{u<v}}(𝒜 (x, ∇v) − 𝒜 (x, ∇u)) · (∇v − ∇u) dx
≤ 0

and

∫_{Ω}(ℬ(x, v) − ℬ(x, u))η dx = − ∫_{{u<v}}(ℬ(x, v) − ℬ(x, u))(v − u) dx ≤ 0.

Thus

0 ≤ (∫_{Ω} 𝒜 (x, ∇v) · ∇η dx + ∫_{Ω} ℬ(x, v)η dx)
− (∫_{Ω} 𝒜 (x, ∇u) · ∇η dx + ∫_{Ω} ℬ(x, u)η dx)
≤ 0,

and hence

∫_{{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 Ω 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}(Ω; μ), let θ^{∗} = max(min(θ, K_{2}), K_{1}). If u − θ ∈ 𝒟_{0}^{p}(Ω; μ) and K_{1} ≤ u ≤ K_{2} in Ω, then u − θ^{∗} ∈ 𝒟_{0}^{p}(Ω; μ).

Proof. Choose φ_{n} ∈ C_{0}^{∞}(Ω) such that φ_{n} → u − θ a.e., {φ_{n}} is uniformly bounded and ∇φ_{n} → ∇(u − θ) in L^{p}(Ω; μ). Setting

φ_{n}^{∗} = max(min(φ_{n}, u − K_{1}), u − K_{2}),

we have φ_{n}^{∗} → max(min(u − θ, u − K_{1}), u − K_{2}) = u − θ^{∗} a.e. in Ω. Also, by [HKM; Lemma 1.2.2], ∇φ_{n}^{∗} → ∇(u − θ^{∗}) in L^{p}(Ω; μ). Since u − K_{1} ≥ 0 and u − K_{2} ≤ 0, suppφ_{n}^{∗} is compact in Ω 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 Ω and a bounded subsolution of (E_{𝒜 ,ℬ} ) in Ω.

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 Ω is (p, μ)-hyperbolic and suppose that conditions (C_{1}) and (B.5) are satisfied. If θ ∈ 𝒟^{p}(Ω; μ), then there exists a unique (𝒜 , ℬ)-harmonic function u_{(𝒜 ,ℬ,θ)} on Ω such that u_{(𝒜 ,ℬ,θ)} − θ ∈ 𝒟_{0}^{p}(Ω; μ).

212 Fumi-Yuki Maeda and Takayori Ono

2. A class of L^{1} -functions on Ω Hereafter, we always assume that Ω is (p, μ)-hyperbolic, namely, 1 ∉ 𝒟_{0}^{p}(Ω; μ).

Note that any bounded domain is (p, μ)-hyperbolic.

We consider the following function spaces:

ℱ (𝒜) = {f ∈ L^{1}(Ω) | f/w is locally bounded in Ω and
− div 𝒜 (x, ∇u) = f has a solution in 𝒟_{0}^{p}(Ω; 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_{0}^{p}(Ω; μ) will be denoted by U^{f}. Obviously, 0 ∈ ℱ (𝒜) and U^{0} = 0. If f ∈ L^{1}(Ω), f/w is locally bounded on Ω 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 ,−f_{1}} ), U^{f_{1}} is a solution of (E_{𝒜 ,−f_{1}} ) and min(U^{f_{2}} − U^{f_{1}}, 0) ∈ 𝒟_{0}^{p}(Ω; 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 Ω is bounded, any measurable function f satisfying 0 ≤ f(x) ≤ βw(x) (respectively −βw(x) ≤ f(x) ≤ 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_{1}(x, ξ) = \{ \begin{matrix}𝒜 (x, ξ), & x ∈ Ω, \\ w(x)|ξ|^{p−2}ξ, & x ∈ R^{N} \ Ω, \end{matrix}

and take an open ball B ⊃ Ω. By Proposition A, there exists u ∈ C(\overline{B}) ∩ 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 Ω. It follows from Theorem A that U^{βw} exists and hence βw ∈ ℱ^{+}(𝒜). Similarly, we see that −βw ∈ ℱ^{−}(𝒜).

Example 2.1. Let Ω = B(0, R) = {|x| < R} for 0 < R < ∞, w(x) = |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 f(x) = |x|^{δ}g(|x|). Then f ∈ ℱ^{+}(𝒜) and

U^{f}(x) = ∫_{|x|}^{R}(\frac{1}{r^{N+δ−1}} ∫_{0}^{r} g(t)t^{N+δ−1} dt)^{1/(p−1)} dr.

If we take g(t) ≡ 1, i.e., f(x) = |x|^{δ}, then

U^{f}(x) = (N + δ)^{−1/(p−1)}p′^{−1}(R^{p′} − |x|^{p′}),

where 1/p + 1/p′ = 1.

We can take unbounded f, e.g., f(x) = |x|^{δ}(R − |x|)^{−α} with 0 < α < 1.

Example 2.2. Let Ω = R^{N}, w(x) = |x|^{δ} with p < N + δ and 𝒜 (x, ξ) = |x|^{δ}|ξ|^{p−2}ξ. Note that Ω 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 ∫_{0}^{∞} g(t)t^{N+δ−1} dt < ∞. Then f(x) = |x|^{δ}g(|x|) belongs to ℱ^{+}(𝒜) and

U^{f}(x) = ∫_{|x|}^{∞}(\frac{1}{r^{N+δ−1}} ∫_{0}^{r} g(t)t^{N+δ−1} dt)^{1/(p−1)} dr.

Proof of Examples 2.1 and 2.2. Obviously, f ≥ 0 and f/|x|^{δ} is locally bounded on Ω. Also, it is easily verified that f ∈ L^{1}(Ω). Set

G(r) = ∫_{0}^{r} g(t)t^{N+δ−1} dt and u(x) = ∫_{|x|}^{R}(\frac{1}{r^{N+δ−1}} G(r))^{1/(p−1)} dr,

where we set R = ∞ for Example 2.2. Then G(r) is bounded for 0 < r < R and G(r) ≤ c_{ϱ}r^{N+δ} for 0 < r < ϱ, ϱ < R. From these it follows that u(x) ∈ C^{1}(Ω) and

∇u(x) = −\frac{x}{|x|}(\frac{1}{|x|^{N+δ−1}} G(|x|))^{1/(p−1)}.

In case R < ∞, since |∇u(x)| is bounded, ∫_{Ω} |∇u(x)|^{p}|x|^{δ} dx < ∞. In case R = ∞, since

∫_{|x|≥1} |∇u(x)|^{p}|x|^{δ} dx ≤ c ∫_{|x|≥1} |x|^{−(pN+δ−p)/(p−1)} dx

and p < N + δ, we have ∫_{Ω} |∇u(x)|^{p}|x|^{δ} dx < ∞. Moreover, if |x| → R, then u(x) → 0. Thus, setting φ_{n}(x) = max(u(x)−1/n, 0), we have φ_{n} ∈ 𝒟^{p}(Ω; |x|^{δ} dx) and suppφ_{n} is compact. Since {φ_{n}} is uniformly bounded, φ_{n} → u in Ω and ∇φ_{n} → ∇u in L^{p}(Ω; |x|^{δ} dx), it follows that u ∈ 𝒟_{0}^{p}(Ω; |x|^{δ} dx). Finally, we have

−div (|x|^{δ}|∇u(x)|^{p−2}∇u(x)) = div (x|x|^{−N}G(|x|))

= −N|x|^{−N}G(|x|) + |x|^{δ}g(|x|) + N|x|^{−N}G(|x|)

= |x|^{δ}g(|x|) = f(x),

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 Ω.

Example 3.1. Let ζ(t) be a nondecreasing continuous function on R such that |ζ(t)| ≤ c|t|^{p−1} for |t| ≥ 1 and ζ(t_{0}) = 0 for some t_{0} ∈ R. We set ℬ(x, t) = b(x)ζ(t) with b ∈ L^{1}(Ω) such that b ≥ 0 and b/w is locally bounded on Ω. Then ℬ satisfies (B.4) with T_{1} = t0^{+} , T_{2} = t0^{−} 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}(Ω; μ), we define

sup_{∂Ω} θ = inf{k | (θ−k)^{+} ∈ 𝒟_{0}^{p}(Ω; μ)} and inf_{∂Ω} θ = sup{k | (θ−k)^{−} ∈ 𝒟_{0}^{p}(Ω; μ)}.

Theorem 3.1. Suppose ℬ satisfies (B.4). Then for any θ ∈ 𝒟^{p}(Ω; μ), there exists a unique (𝒜 , ℬ)-harmonic function u_{(𝒜 ,ℬ,θ)} on Ω such that u_{(𝒜 ,ℬ,θ)} −θ ∈ 𝒟_{0}^{p}(Ω; μ). Further it satisfies

min(−T_{2}, inf_{∂Ω} θ) + U^{f_{2}}(x) ≤ u_{(𝒜 ,ℬ,θ)} (x) ≤ max(T_{1}, sup_{∂Ω} θ) + U^{f_{1}}(x)

on Ω.

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 = max(T_{1}, sup_{∂Ω} θ)+U^{f_{1}} +ε. Since ℬ(x, v) ≥ ℬ(x, T_{1}) ≥ −f_{1} by (B.4), v is a supersolution of (E_{𝒜 ,ℬ} ). Also, since v ≥ sup_{∂Ω} θ+ε, we see that max(θ−v, 0) ∈ 𝒟_{0}^{p}(Ω; μ), and hence that min(v − u_{(𝒜 ,ℬ,θ)} , 0) ∈ 𝒟_{0}^{p}(Ω; μ). Hence by Lemma 1.2, v ≥ u_{(𝒜 ,ℬ,θ)} in Ω. Since ε > 0 is arbitrary, max(T_{1}, sup_{∂Ω} θ) + U^{f_{1}} ≥ u_{(𝒜 ,ℬ,θ)} in Ω. The other inequality follows similarly.

Perturbation theory for nonlinear Dirichlet problems 215

In view of Example 3.1, we can state Corollary 3.1. Let ℬ(x, t) = b(x)ζ(t) be as in Example 3.1. Then

min(−t0^{−}, inf_{∂Ω} θ) ≤ u_{(𝒜 ,ℬ,θ)} (x) ≤ max(t0^{+}, sup_{∂Ω} θ)

on Ω.

In view of Propositions 2.4 and 2.2, we also have Corollary 3.2. Suppose Ω is bounded, 𝒜 satisfies (A.5) (in Proposition 2.2)

and |ℬ(x, 0)| ≤ βw(x) a.e. in Ω. Then

min(inf_{∂Ω} θ, 0) − β^{1/(p−1)}U^{w}(x) ≤ u_{(𝒜 ,ℬ,θ)} (x) ≤ max(sup_{∂Ω} θ, 0) + β^{1/(p−1)}U^{w}(x)

on Ω.

Given nonnegative numbers T_{1} , T_{2} , functions f_{1} ∈ ℱ^{+}(𝒜), f_{2} ∈ ℱ^{−}(𝒜) and given θ ∈ 𝒟^{p}(Ω; μ), let

M^{+}(T_{1}, f_{1}, θ) = max(T_{1}, sup_{∂Ω} θ) + sup_{Ω} U^{f_{1}},
M^{−}(T_{2}, f_{2}, θ) = max(T_{2}, − inf_{∂Ω} θ) − inf_{Ω} 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}(Ω; μ),

∫_{Ω}|∇u_{(𝒜 ,ℬ,θ)} (x)|^{p} dμ ≤ (\frac{α_{2}}{α_{1}} )^{p} ∫_{Ω} |∇θ(x)|^{p} dμ + \frac{pM}{α_{1}} ∫_{Ω} |ℬ(x, θ^{∗}(x))| dx,

where

M = M^{+}(T_{1}, f_{1}, θ) + M^{−}(T_{2}, f_{2}, θ)

and

θ^{∗} = max(min(θ, M^{+}(T_{1}, f_{1}, θ)), −M^{−}(T_{1}, f_{1}, θ)).

Proof. Since u_{(𝒜 ,ℬ,θ)} − θ^{∗} ∈ 𝒟_{0}^{p}(Ω; μ) by Lemma 1.3, we have

∫_{Ω} 𝒜 (x, ∇u_{(𝒜 ,ℬ,θ)} ) · (∇u_{(𝒜 ,ℬ,θ)} − ∇θ^{∗}) dx
+ ∫_{Ω} ℬ(x, u_{(𝒜 ,ℬ,θ)} )(u_{(𝒜 ,ℬ,θ)} − θ^{∗}) dx
= 0.

By Theorem 3.1, |u_{(𝒜 ,ℬ,θ)} − θ^{∗}| ≤ M. Hence, using (A.2), (A.3) and (B.3), we have

(3.1)

α_{1} ∫_{Ω} |∇u_{(𝒜 ,ℬ,θ)} |^{p} dμ ≤ α_{2} ∫_{Ω} |∇u_{(𝒜 ,ℬ,θ)} |^{p−1}|∇θ^{∗}| dμ
− ∫_{Ω} ℬ(x, u_{(𝒜 ,ℬ,θ)} )(u_{(𝒜 ,ℬ,θ)} − θ^{∗}) dx
≤ α_{2} ∫_{Ω} |∇u_{(𝒜 ,ℬ,θ)} |^{p−1}|∇θ| dμ
− ∫_{Ω} ℬ(x, θ^{∗})(u_{(𝒜 ,ℬ,θ)} − θ^{∗}) dx

≤ α_{2}(∫_{Ω} |∇u_{(𝒜 ,ℬ,θ)} |^{p} dμ)^{(p−1)/p}(∫_{Ω} |∇θ|^{p} dμ)^{1/p}
+ M ∫_{Ω} |ℬ(x, 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}(Ω; μ) and set u_{j} = u_{(𝒜 ,ℬ_{j},θ)} , j = 1, 2. Then

\begin{aligned}∫_{Ω}(𝒜 (x, ∇u_{1}) − 𝒜 (x, ∇u_{2})) & · (∇u_{1} − ∇u_{2}) dx \\ & ≤ M ∫_{Ω} sup_{−M_{2}≤t≤M_{1}}|ℬ_{1}(x, t) − ℬ_{2}(x, t)| dx, \end{aligned}

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_{0}^{p}(Ω; μ),

\begin{aligned}∫_{Ω}(𝒜 (x, ∇u_{1}) − 𝒜 (x, ∇u_{2})) & · (∇u_{1} − ∇u_{2}) dx \\ & + ∫_{Ω}(ℬ_{1}(x, u_{1}) − ℬ_{2}(x, u_{2}))(u_{1} − u_{2}) dx = 0. \end{aligned}

Hence using (B.3) we have

\begin{aligned}∫_{Ω}(𝒜 (x, ∇u_{1}) − 𝒜 (x, ∇u_{2})) & · (∇u_{1} − ∇u_{2}) dx \\ & ≤ ∫_{Ω}(ℬ_{1}(x, u_{2}) − ℬ_{2}(x, u_{2}))(u_{2} − u_{1}) dx. \end{aligned}

Since −M_{2} ≤ u_{j} ≤ M_{1} , j = 1, 2, by Theorem 3.1, we obtain the desired estimate.

4. 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_{0}^{p}(Ω; μ) such that {∫_{Ω} |∇u_{n}|^{p} dμ} is bounded and u_{n} → u a.e. in Ω. If u is continuous, then u ∈ 𝒟_{0}^{p}(Ω; 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 Ω and x_{0} be any point of Ω. If 0 < R < ∞ is such that \overline{B}(x_{0}, R) ⊂ Ω and if |u| ≤ L in B(x_{0}, R), then there are constants c and 0 < λ < 1 such that

sup_{B(x_{0},ϱ)} u − inf_{B(x_{0},ϱ)} u ≤ c(\frac{ϱ}{R})^{λ},

whenever 0 < ϱ < R. Here c and λ depend only on N, p, α_{1} , α_{2} , α_{3}(B(x_{0}, 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}(Ω; μ). Assume further that there exists a nonnegative function b on Ω such that b/w is locally bounded in Ω and

(4.1)  ℬn^{+}(x, M_{1}) + ℬn^{−}(x, −M_{2}) ≤ b(x) a.e. on Ω

for all n, where M_{1} , M_{2} are as in Theorem 3.3. If

(4.2)  ∫_{Ω} sup_{−M_{2}≤t≤M_{1}}|ℬ_{n}(x, t) − ℬ(x, t)| dx → 0 (n → ∞),

then u_{(𝒜 ,ℬ_{n},θ)} → u_{(𝒜 ,ℬ,θ)} as n → ∞ locally uniformly on Ω.

Proof. If we set

ℬ_{n}^{∗}(x, t) = \{ \begin{matrix}ℬ_{n}(x, M_{1}), & t ≥ M_{1}, \\ ℬ_{n}(x, t), & −M_{2} < t < M_{1}, \\ ℬ_{n}(x, −M_{2}), & t ≤ −M_{2}, \end{matrix}

then u_{(𝒜 ,ℬ_{n},θ)} is a solution of (E_{𝒜 ,ℬ_{n}^{∗}} ), and hence u_{(𝒜 ,ℬ_{n},θ)} = u_{(𝒜 ,ℬ_{n}^{∗},θ)} . Thus by (4.1), we may assume that |ℬ_{n}(x, t)| ≤ b(x) (n = 1, 2, …) for any t ∈ R. Then, for any D ⋐ Ω, we can take α_{3}(D) = sup_{D} b/w in (B.2) for ℬ_{n} for all n.

Let u_{n} = u_{(𝒜 ,ℬ_{n},θ)} and u = u_{(𝒜 ,ℬ,θ)} . By Theorem 3.1, {u_{n}} is uniformly bounded in Ω. Hence, by Lemma 4.2, we see that {u_{n}} is equi-continuous at each point of Ω. Hence it follows from Ascoli–Arzela’s theorem that any subsequence of {u_{n}} has a locally uniformly convergent subsequence.

Let {u_{n_{k}}} be any subsequence of {u_{n}} which converges locally uniformly to u^{∗}. Since {sup_{−M_{2}≤t≤M_{1}} |ℬ_{n}(x, t)|} is bounded in L^{1}(Ω) by (4.2), Theorem 3.2 yields that {∇u_{n}} is bounded in L^{p}(Ω; μ). Thus, by Lemma 4.1 and [HMK; Lemma 1.3.3], we see that u^{∗} − u ∈ 𝒟_{0}^{p}(Ω; μ) and ∇u_{n_{k}} → ∇u^{∗} weakly in L^{p}(Ω; μ). On the other hand, by Theorem 3.3 and [HKM; Lemma 3.73], ∇u_{n} → ∇u weakly in L^{p}(Ω; μ), and hence ∇u^{∗} = ∇u. Since Ω is (p, μ)- hyperbolic, it follows that u^{∗} = u. Therefore, u_{n} → u locally uniformly in Ω.

In view of Example 3.1, we can state

Corollary 4.1. Let ζ(t) 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 Ω

for some function b_{0} such that b_{0}/w is locally bounded in Ω and

(4.4)  ∫_{Ω} |b_{n}(x) − b(x)| dx → 0 (n → ∞).

Then, for ℬ_{n}(x, t) = b_{n}(x)ζ(t), n = 1, 2, …, and ℬ(x, t) = b(x)ζ(t), u_{(𝒜 ,ℬ_{n},θ)} → u_{(𝒜 ,ℬ,θ)} locally uniformly on Ω.

The following example shows that we cannot assert the uniform convergence on Ω in the above theorem and corollary:

Example 4.1. Let Ω = B(0, 1), 1 < p ≤ N, w(x) = 1 and 𝒜 (x, ξ) = |ξ|^{p−2}ξ. Let {a_{n}} be a sequence of points in B(0, 1) such that |a_{n}| → 1 and set b_{n}(x) = 2^{−n}r_{n}^{−N}χ_{B(a_{n},r_{n})}(x) with 0 < r_{n} < 1 − |a_{n}|, n = 1, 2, …. For a nondecreasing continuous function ζ on R such that ζ(0) = 0, ζ(1) > 0 and |ζ(t)| ≤ c|t|^{p−1} for |t| ≥ 1, we set

ℬ_{n}(x, t) = ∑_{k=1}^{n} b_{k}(x)ζ(t) and ℬ(x, t) = ∑_{n=1}^{∞} b_{n}(x)ζ(t).

Then, for θ = 1, ℬ_{n} and ℬ satisfy the conditions in Theorem 4.1. If we choose {r_{n}} suitably, then {u_{(𝒜 ,ℬ_{n},1)} } does not converge uniformly on B(0, 1).

Proof. Let B_{n} = B(a_{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_{0}^{p}(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, by Example 2.1 we have

v_{n}(a_{n}) = (2^{−n}r_{n}^{−N})^{1/(p−1)}\{∫_{0}^{r_{n}}(\frac{1}{r^{N−1}} ∫_{0}^{r} t^{N−1} dt)^{1/(p−1)} dr

+ ∫_{r_{n}}^{1−|a_{n}|}(\frac{1}{r^{N−1}} ∫_{0}^{r_{n}} t^{N−1} dt)^{1/(p−1)} dr\}

= \{ \begin{matrix}(2^{n}N)^{1/(1−N)}\{1 − \frac{1}{N} + log \frac{1 − |a_{n}|}{r_{n}}\} if p = N, \\ (2^{n}N)^{1/(1−p)}\{\frac{N(p − 1)}{p(N − p)}r_{n}^{(p−N)/(p−1)} − \frac{p − 1}{N − p}(1 − |a_{n}|)^{(p−N)/(p−1)}\} \\ if p < N. \end{matrix}

Thus, U^{b_{n}}(a_{n}) ≥ v_{n}(a_{n}) ≥ 1 for sufficiently small r_{n}.

Let u_{n} = u_{(𝒜 ,ℬ_{n},1)} and u = u_{(𝒜 ,ℬ,1)} for simplicity. By considering the extension of ℬ_{n} by 0 outside B(0, 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 B(0, 1). Then u(x) → 1 as |x| → 1. Choose ε > 0 such that ε^{p−1} < ζ(1 − ε). Then there is r_{0} < 1 such that u(x) ≥ 1 − ε for |x| ≥ r_{0} . For large n, B(a_{n}, r_{n}) ∩ B(0, r_{0}) = ∅ and

\begin{aligned}−div (|∇(1 − u)|^{p−2}∇(1 − u)) & = div(|∇u|^{p−2}∇u) \\ & = ∑_{n=1}^{∞} b_{n}(x)ζ(u(x)) ≥ ζ(1 − ε)b_{n}(x). \end{aligned}

Thus, by Lemma 1.2, 1 − u ≥ ζ(1 − ε)^{1/(p−1)}U^{b_{n}}, so that

u(a_{n}) ≤ 1 − ζ(1 − ε)^{1/(p−1)}U^{b_{n}}(a_{n}) < 1 − ε

for large n. This contradicts our assumption that u(x) ≥ 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}(Ω; μ). Assume further that 𝒜 satisfies

(A.4s) (𝒜 (x, ξ_{1}) − 𝒜 (x, ξ_{2})) · (ξ_{1} − ξ_{2}) ≥ α_{4}w(x)(|ξ_{1}| + |ξ_{2}|)^{p−2}|ξ_{1} − ξ_{2}|^{2}

with α_{4} > 0. If (4.2) holds, then ∇u_{(𝒜 ,ℬ_{n},θ)} → ∇u_{(𝒜 ,ℬ,θ)} in L^{p}(Ω; μ); further u_{(𝒜 ,ℬ_{n},θ)} → u_{(𝒜 ,ℬ,θ)} in H^{1,p}(Ω; μ) in case Ω is bounded.

Proof. If p ≥ 2, the first assertion is obvious from Theorem 3.3. In case 1 < p < 2, we have

∫_{Ω} |∇u_{n} − ∇u|^{p} dμ
= ∫_{Ω}{(|∇u_{n}| + |∇u|)^{p−2}|∇u_{n} − ∇u|^{2}}^{p/2}(|∇u_{n}| + |∇u|)^{p(2−p)/2} dμ

≤ (∫_{Ω}(|∇u_{n}| + |∇u|)^{p−2}|∇u_{n} − ∇u|^{2} dμ)^{p/2}

× (∫_{Ω}(|∇u_{n}| + |∇u|)^{p} dμ)^{(2−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.

5. Boundedness and convergence of solutions for Dirichlet problems

Given a compactification Ω^{∗} of Ω and a bounded function ψ on ∂^{∗}Ω = Ω^{∗}\Ω, let

𝒰_{ψ} = {u : (𝒜 , ℬ)-superharmonic in Ω and
lim inf_{x→ξ} u(x) ≥ ψ(ξ) for all ξ ∈ ∂^{∗}Ω}

and

ℒ_{ψ} = {v : (𝒜 , ℬ)-subharmonic in Ω and
lim sup_{x→ξ} v(x) ≤ ψ(ξ) for all ξ ∈ ∂^{∗}Ω}.

If both 𝒰_{ψ} and ℒ_{ψ} are nonempty, then

\overline{H} (ψ; Ω^{∗}) =\overline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) := inf 𝒰_{ψ}

and

\underline{H}(ψ; Ω^{∗}) = \underline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) := sup ℒ_{ψ}

are (𝒜 , ℬ)-harmonic in Ω and \underline{H}(ψ; Ω^{∗}) ≤\overline{H} (ψ; Ω^{∗}) ([MO1; Theorem 3.1]). We say that ψ is (𝒜 , ℬ)-resolutive if both 𝒰_{ψ} and ℒ_{ψ} are nonempty and \underline{H}(ψ; Ω^{∗}) = \overline{H} (ψ; Ω^{∗}). In this case we write H(ψ; Ω^{∗}) = H^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) for \underline{H}(ψ; Ω^{∗}) = \overline{H} (ψ; Ω^{∗}). Ω^{∗} is said to be an (𝒜 , ℬ)-resolutive compactification if every ψ ∈ C(∂^{∗}Ω) is (𝒜 , ℬ)-resolutive.

In the proof of [MO1; Proposition 3.1] we have shown Lemma 5.1. If ψ_{1} and ψ_{2} are (𝒜 , ℬ)-resolutive functions on ∂^{∗}Ω, then

|H(ψ_{1}; Ω^{∗}) − H(ψ_{2}; Ω^{∗})| ≤ sup_{∂^{∗}Ω} |ψ_{1} − ψ_{2}|.

Theorem 5.1. Let Ω^{∗} be a compactification of Ω. If ℬ satisfies (B.4) and if ψ is a bounded function on ∂^{∗}Ω, then both 𝒰_{ψ} and ℒ_{ψ} are nonempty and

min(−T_{2}, inf_{∂^{∗}Ω} ψ) + U^{f_{2}} ≤ \underline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) ≤\overline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗})

≤ max(T_{1}, sup_{∂^{∗}Ω} ψ) + U^{f_{1}}.

Proof. By (B.4) we see that

max(T_{1}, sup_{∂^{∗}Ω} ψ) + U^{f_{1}} ∈ 𝒰_{ψ}

and

min(−T_{2}, inf_{∂^{∗}Ω} 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 ℬ(x, t) = b(x)ζ(t) be as in Example 3.1. If ψ is a bounded function on ∂^{∗}Ω, then both 𝒰_{ψ} and ℒ_{ψ} are nonempty and

min(−t0^{−}, inf_{∂^{∗}Ω} ψ) ≤ \underline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) ≤\overline{H}^{(𝒜 ,ℬ)} (ψ; Ω^{∗}) ≤ max(t0^{+}, sup_{∂^{∗}Ω} ψ).

Corollary 5.2. Suppose Ω is bounded and let Ω^{∗} be a compactification of Ω. If 𝒜 satisfies (A.5) and if |ℬ(x, 0)| ≤ βw(x) for a.e. x ∈ Ω, then

|H^{(𝒜 ,ℬ)} (ψ; Ω^{∗})| ≤ sup_{∂^{∗}Ω} |ψ| + β^{1/(p−1)}U^{w}

for any bounded (𝒜 , ℬ)-resolutive function ψ on ∂^{∗}Ω.

We recall ([MO1; Theorem 3.2]) that under conditions (C_{1}) and (B.5), the Q-compactification Ω_{Q}^{∗} of Ω (see [CC]) is an (𝒜 , ℬ)-resolutive compactification if Q ⊂ 𝒟^{p}(Ω; μ).

Theorem 5.2. Let Q ⊂ 𝒟^{p}(Ω; μ), Ω_{Q}^{∗} be the Q-compactification of Ω, Γ = Ω_{Q}^{∗} \ Ω and let ψ ∈ C(Γ). 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_{Γ} ψ) + sup_{Ω} U^{f_{1}} and M_{2} = max(T_{2}, − min_{Γ} ψ) − inf_{Ω} U^{f_{2}}.

Assume further that there exists a nonnegative function b(x) on Ω such that b(x)/w(x) is locally bounded in Ω and ℬ_{n} satisfy (4.1) for all n. If (4.2) holds, then H^{(𝒜 ,ℬ_{n})} (ψ; Ω_{Q}^{∗}) → H^{(𝒜 ,ℬ)} (ψ; Ω_{Q}^{∗}) locally uniformly on Ω.

Proof. Since the set of continuous extensions of functions in Q is dense in C(Γ) with respect to the uniform convergence, given ε > 0 there exists θ ∈ Q such that sup_{ξ∈Γ} |θ^{∗}(ξ) − ψ(ξ)| ≤ 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 ,ℬ_{n})} (ψ; Ω_{Q}^{∗}), H_{n}(θ) = H^{(𝒜 ,ℬ_{n})} (θ^{∗}; Ω_{Q}^{∗}), H(ψ) = H^{(𝒜 ,ℬ)} (ψ; Ω_{Q}^{∗}) and H(θ^{∗}) = H^{(𝒜 ,ℬ)} (θ^{∗}; Ω_{Q}^{∗}). By Lemma 5.1, we see that |H(θ^{∗}) − H(ψ)| ≤ 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 ,ℬ_{n},θ)} and H(θ^{∗}) = u_{(𝒜 ,ℬ,θ)} . Also, since ℬn^{+}(x, M^{+}(T_{1}, f_{1}, θ)) ≤

ℬn^{+}(x, M_{1}) and ℬn^{−}(x, −M^{−}(T_{2}, f_{2}, θ)) ≤ ℬn^{−}(x, −M_{2}), by Theorem 4.1, for any

open G ⋐ Ω, there is n_{0} such that

sup_{G}|u_{(𝒜 ,ℬ_{n},θ)} − u_{(𝒜 ,ℬ,θ)} | ≤ ε

for n ≥ n_{0} . Thus, for n ≥ n_{0}

sup_{G} |H_{n}(ψ) − H(ψ)|

≤ sup_{G}{|H_{n}(ψ) − H_{n}(θ^{∗})| + |H_{n}(θ^{∗}) − H(θ^{∗})| + |H(θ^{∗}) − H(ψ)|}

≤ 3ε.

Hence the theorem follows.

Like Corollary 4.1, we obtain

Corollary 5.3. Let Ω_{Q}^{∗} and Γ be as in Theorem 5.2 and let ℬ_{n}(x, t) = b_{n}(x)ζ(t) and ℬ(x, t) = b(x)ζ(t) with nonnegative measurable functions b_{n}, b on Ω and ζ as in Example 3.1. If (4.3) and (4.4) are satisfied then H^{(𝒜 ,ℬ_{n})} (ψ; Ω_{Q}^{∗}) → H^{(𝒜 ,ℬ)} (ψ; Ω_{Q}^{∗}) locally uniformly on Ω for any ψ ∈ C(Γ).

References

[CC] Constantinescu, C., and A. Cornea: Ideale Ränder Riemannscher Flächen. - Springer-Verlag, 1963.

[HKM] Heinonen, J., T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Clarendon Press, 1993.

[MO1] Maeda, F.-Y., and T. Ono: Resolutivity of ideal boundary for nonlinear Dirichlet problems. - J. Math. Soc. Japan 52, 2000, 561–581.

[MO2] Maeda, F.-Y., and T. Ono: Properties of harmonic boundary in nonlinear potential theory. - Hiroshima Math. J. 30, 2000, 513–523.

[O] Ono, T.: On solutions of quasi-linear partial differential equations −div 𝒜 (x, ∇u) + ℬ(x, u) = 0. - RIMS Kokyuroku 1016, 1997, 146–165.

Received 2 July 2002