An existence and approximation theorem for solutions of degenerate nonlinear elliptic equations

The main result establishes that a weak solution of degenerate nonlinear elliptic equations can be approximated by a sequence of solutions for non-degenerate nonlinear elliptic equations.


Introduction.
Let L be a degenerate elliptic operator in divergence form (1.1) Lu(x) = − n i,j=1 where the coefficients a ij are measurable, real-valued functions whose coefficient matrix A(x) = (a ij (x)) is symmetric and satisfies the degenerate ellipticity condition for all ξ ∈ R n and almost everywhere x ∈ Ω ⊂ R n a bounded open set, ω is a weight function, λ and Λ are positive constants.
The main purpose of this paper (see Theorem 1.2) is to establish that a weak solution u ∈ W 1,2 0 (Ω, ω) for the nonlinear Dirichlet problem can be approximated by a sequence of solutions of non-degenerate nonlinear elliptic equations.By a weight, we shall mean a locally integrable function ω on R n such that ω(x) > 0 for a.e.x ∈ R n .Every weight ω gives rise to a measure on the measurable subsets on R n through integration.This measure will be denoted by µ.Thus, µ(E) = E ω(x) dx for measurable sets E ⊂ R n .
In general, the Sobolev spaces W k,p (Ω) without weights occur as spaces of solutions for elliptic and parabolic partial differential equations.For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces (see [1]- [5], [8] and [10]).
A class of weights, which is particularly well understood, is the class of A p -weights (or Muckenhoupt class) that was introduced by B. Muckenhoupt (see [7]).These classes have found many useful applications in harmonic analysis (see [9]).Another reason for studying A p -weights is the fact that powers of the distance to submanifolds of R n often belong to A p (see [6]).There are, in fact, many interesting examples of weights (see [5] for padmissible weights).
The following lemma can be proved in exactly the same way as Lemma 2.1 in [3] (see also, Lemma 3.1 and Lemma 4.13 in [1]).Our lemma provides a general approximation theorem for A p weights (1 ≤ p < ∞) by means of weights which are bounded away from 0 and infinity and whose A pconstants depend only on the A p -constant of ω.Lemma 1.1 is the key point for Theorem 1.2, and the crucial point consists of showing that a weak limit of a sequence of solutions of approximate problems is in fact a solution of the original problem.

Lemma 1.1. Let α, β > 1 be given and let
) and let a ij = a ji be measurable, real-valued functions satisfying for all ξ ∈ R n and a.e.x ∈ Ω.Then there exist weights ω αβ ≥ 0 a.e. and measurable real-valued functions a αβ ij such that the following conditions are met.
(iv) There exists a closed set F αβ such that ω αβ ≡ ω in F αβ and ω αβ ∼ ω1 ∼ ω2 in F αβ with equivalence constants depending on α and β (i.e., there are positive constants c αβ and The following theorem will be proved in Section 3.

Definitions and basic results.
Let ω be a locally integrable nonnegative function in R n and assume that 0 < ω(x) < ∞ almost everywhere.We say that ω belongs to the Muckenhoupt class A p , 1 < p < ∞, or that ω is an A p -weight, if there is a constant C = C(p, ω) such that for all balls B ⊂ R n , where |.| denotes the n-dimensional Lebesgue measure in R n .If 1 < q ≤ p, then A q ⊂ A p (see [4], [5] or [10] for more information about A p -weights).The weight ω satisfies the doubling condition if there exists a positive constant C such that µ(B(x; 2r)) ≤ Cµ(B(x; r)) for every ball As an example of A p -weight, the function ω µ(B) whenever B is a ball in R n and E is a measurable subset of B (see 15.5 strong doubling property in [5]).Therefore, µ(E) = 0 if and only if |E| = 0; so there is no need to specify the measure when using the ubiquitous expression almost everywhere and almost every, both abbreviated a.e.Definition 2.1.Let ω be a weight, and let Ω ⊂ R n be open.For 0 < p < ∞ we define L p (Ω, ω) as the set of measurable functions f on Ω such that 1) is locally integrable and we have for every open set Ω (see Remark 1.2.4 in [10]).It thus makes sense to talk about weak derivatives of functions in L p (Ω, ω).
It is evident that the weight function ω which satisfies 0 < c 1 ≤ ω(x) ≤ c 2 for x ∈ Ω (c 1 and c 2 positive constants), gives nothing new (the space W 1,2 0 (Ω, ω) is then identical with the classical Sobolev space W 1,2 0 (Ω)).Consequently, we shall be interested above in all such weight functions ω which either vanish somewhere in Ω ∪ ∂Ω or increase to infinity (or both).
The dual space of W 1,2 0 (Ω, ω) is the space and where the dot denotes the Euclidean scalar product in R n .(b) Since the matrix A(x) = (a ij (x)) is symmetric, we have

Proof of
To prove Theorem 1.2, we define B : Step and by (H3) Since B(u, .) is linear, for each u ∈ W 1,2 0 (Ω, ω), there is a linear continuous functional on W 1,2 0 (Ω, ω) denoted by Au such that Au, ϕ = B(u, ϕ) for all ϕ ∈ W 1,2 0 (Ω, ω) (where f, x denotes the value of the functional f at the point x).Moreover, by (3.1), we have . Hence, we obtain the operator Consequently, problem (P ) is equivalent to the operator equation Step 2. The operator A is strictly monotone and coercive.In fact, if u 1 , u 2 ∈ W 1,2 0 (Ω, ω) we have, by (1.2) and Remark 2.3, Therefore, the operator A is strongly monotone, and this implies that A is strictly monotone.Moreover, if u ∈ W 1,2 0 (Ω, ω), we have Hence, A is continuous and this implies that A is hemicontinuous.
Step 3. We need to show that ũ is a solution of problem (P ), i.e, for every ϕ ∈ W 1,2 0 (Ω, ω) we have Using the fact that u m is a solution of (P m ), we have for every ϕ ∈ W 1,2 0 (Ω, ω m ).Moreover, over F k (for m ≥ k) we have the following properties: (a) We see that the functional G 1 is linear and continuous.In fact, we have (by Lemma 1.1 (iv)) ω ∼ ω1 in F k (ω ≤ cω 1 ).By (2), we obtain (b) We see that G 2 is a continuous functional.In fact, if u 1 , u 2 ∈ W 1,2 0 (Ω, ω1 ), we obtain by (H1) . Using (a), (b), properties (i), (ii) and (iii), and basing on the fact that u m is the solution of (P m ), we obtain where E c denotes the complement of a set E ⊂ R n .(I) By the Lebesgue Dominated Convergence Theorem and ω2 ∈ A 2 , we obtain (as m → ∞) (II) Since the matrix A m (x) = (a mm ij )(x) is symmetric, we have Then, by Lemma 1.1 (vi) and (3.5), we obtain

Theorem 1 . 2 . Part 1 .Theorem 3 . 1 .
Existence and uniqueness of solution.The basic idea is to reduce the problem (P ) to an operator equation Au = T and apply the theorem below.Let A : X → X * be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach spaces X.Then the following assertions hold: (a) for each T ∈ X * the equation Au = T has a solution u ∈ X; (b) if the operator A is strictly monotone, then equation Au = T is uniquely solvable in X.