Existence results for a class of nonlinear degenerate (p,q)-biharmonic operators

(H1) x 7→Aj(x, η, ξ) is measurable on Ω for all (η, ξ)∈R×Rn, (η, ξ) 7→Aj(x, η, ξ) is continuous on R×Rn for almost all x∈Ω. (H2) there exist a constant θ1 > 0 such that [A(x, η, ξ)−A(x, η′, ξ ′)].(ξ − ξ ′)≥ θ1 |ξ − ξ ′| , whenever ξ, ξ ′∈Rn, ξ 6=ξ ′, where A(x, η, ξ) = (A1(x, η, ξ), ...,An(x, η, ξ)) (where a dot denote here the Euclidian scalar product in Rn). (H3) A(x, η, ξ).ξ≥ λ1|ξ|, where λ1 is a positive constant. (H4) |A(x, η, ξ)| ≤K1(x) + h1(x)|η| ′ + h2(x)|ξ| ′ , where K1, h1 and h2 are positive functions, with h1, h2∈L(Ω), and K1∈L ′ (Ω, ω) (with 1/p + 1/p ′ = 1).

By a weight, we shall mean a locally integrable function ω on R n such that 0 < ω(x) < ∞ 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. In the particular case where p = q = 2 and ω≡ 1, we have the equation where ∆ 2 u is the biharmonic operator. If p = q, ω≡ 1 and A(x, η, ξ) = |ξ| p−2 ξ, we have the equation Biharmonic equations appear in the study of mathematical model in several real-life processes as, among others, radar imaging (see [1]) or incompressible flows (see [2]). 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 [3][4][5][6]). In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is disturbed in the sense that some degeneration or singularity appears. There are several very concrete problems from practice which lead to such differential equations, e.g. from glaceology, non-Newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, petroleum extraction and reaction-diffusion problems (see some examples of applications of degenerate elliptic equations in [7,8]).
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 [9]). These classes have found many useful applications in harmonic analysis (see [10]). Another reason for studying A p -weights is the fact that powers of distance to submanifolds of R n often belong to A p (see [11]). There are, in fact, many interesting examples of weights (see [12] for p-admissible weights).
In the non-degenerate case (i.e. with ω(x) ≡ 1), for all f ∈ L p (Ω), the Poisson equation associated with the Dirichlet problem [13]), and the nonlinear Dirichlet problem [14]), where ∆ p u = ÷(|∇u| p−2 ∇u) is the p-Laplacian operator. In the degenerate case, the weighted p-Biharmonic operator has been studied by many authors (see [15] and the references therein), and the degenerated p-Laplacian was studied in [6].

Definitions and basic results
Let ω be a locally integrable nonnegative function in R n and assume that 0 < ω < ∞ 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 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 [10,12,16] 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)) ≤ C µ(B(x; r)), for every ball B = B(x; r) ⊂ R n , where µ(B) = B ω(x) dx. If ω∈A p , then µ is doubling (see Corollary 15.7 in [12]).
As an example of A p -weight, the function ω(x) = |x| α , x∈R n , is in A p if and only if −n < α < n(p − 1) (see Corollary 4.4, Chapter IX in [10]).
whenever B is a ball in R n and E is a measurable subset of B (see 15.5 strong doubling property in [12]). Therefore, if µ(E) = 0 then |E| = 0. The measure µ and the Lebesgue measure |.| are mutually absolutely continuous, i.e., they have the same zero sets (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 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 [17]). It thus makes sense to talk about weak derivatives of functions in L p (Ω, ω).

Definition 2.
Let Ω ⊂ R n be a bounded open set, 1 < p < ∞, k be a nonnegative integer and ω ∈ A p . We shall denote by W k,p (Ω, ω), the weighted Sobolev spaces, the set of all functions u ∈ L p (Ω, ω) with weak derivatives If ω ∈ A p , then W k,p (Ω, ω) is the closure of C ∞ (Ω) with respect to the norm (1) (see Corollary 2.1.6 in [17]). We also define the space W k,p 0 (Ω, ω) as the closure of C ∞ 0 (Ω) with respect to the norm (1). We have that the spaces W k,p (Ω, ω) and W k,p 0 (Ω, ω) are Banach spaces. The space W 1,p 0 (Ω, ω) is the closure of C ∞ 0 (Ω) with respect to the norm (1). Equipped with this norm, W 1,p 0 (Ω, ω) is a reflexive Banach space (see [18] for more information about the spaces W 1,p (Ω, ω)). The dual It is evident that a weight function ω which satisfies 0 < c 1 ≤ ω(x) ≤ c 2 for x ∈ Ω (where c 1 and c 2 are constants), give nothing new (the space W 1,p 0 (Ω, ω) is then identical with the classical Sobolev space W 1,p 0 (Ω)). Consequently, we shall be interested above all in such weight functions ω which either vanish somewhere in Ω or increase to infinity (or both).
In this paper we use the following results.
Proof. The proof of this theorem follows the lines of Theorem 2.8.1 in [19].

Theorem 3. (The weighted Sobolev inequality) Let
Ω be an open bounded set in R n and ω∈A p (1 < p < ∞). There exist constants C Ω and δ positive such that for all u ∈ W 1,p 0 (Ω, ω) and all k satisfying 1 ≤ k ≤ n/(n − 1) + δ, Proof. Its suffices to prove the inequality for functions u ∈ C ∞ 0 (Ω) (see Theorem 1.3 in [20]). To extend the estimates (2) to arbitrary u ∈ W 1,p 0 (Ω, ω), we let {u m } be a sequence of C ∞ 0 (Ω) functions tending to u in W 1,p 0 (Ω, ω). Applying the estimates (2) to differences u m 1 − u m 2 , we see that {u m } will be a Cauchy sequence in L kp (Ω, ω). Consequently the limit function u will lie in the desired spaces and satisfy (2).
for all x, y ∈ R n ; (b) There exist two positive constants β p , γ p such that for every x, y ∈ R n Proof. See [14], Proposition 17.2 and Proposition 17.3.
for all ϕ ∈ X.

Proof of Theorem 1
The basic idea is to reduce the Problem (P) to an operator equation Au = T and apply the theorem below.
Theorem 4. Let A : X→X * be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach space 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 A u = T is uniquely solvable in X.
To prove Theorem 1, we define B, B 1 , B 2 , B 3 : X × X → R and T : X → R by Then u ∈ X is a (weak) solution to problem (P) if, for all ϕ ∈ X, Step 1. For j = 1, ..., n we define the operator F j : X →L p (Ω, ω) as We now show that the operator F j is bounded and continuous.
(i) Using (H4), we obtain where the constant C p depends only on p. We have, by Theorem 3 (with k = 1), Therefore, in (3) we obtain (ii) Let u m → u in X as m → ∞. We need to show that F j u m →F j u in L p (Ω, ω). We will apply the Lebesgue Dominated Theorem. If u m → u in X, then |∇u m |→ |∇u| in L p (Ω, ω). Using Theorem 2, there exist a subsequence {u m k } and a function Φ 1 in L p (Ω, ω) such that D j u m k (x) → D j u(x), a.e. in Ω, |∇u m k (x)|≤Φ 1 (x), a.e. in Ω.
By Theorem 3 (with k = 1), Next, applying (H4) we obtain By condition (H1), we have as m k → +∞. Therefore, by the Lebesgue Dominated Convergence Theorem, we obtain We conclude from the Convergence Principle in Banach spaces (see Proposition 10.13 in [22]) that F j u m → F j u in L p (Ω, ω).
Step 6. We need to show that the operator A is continuous. Let u m → u in X as m → ∞. We have F j u m − F j u L p (Ω,ω) + G 1 u m − G 1 u L p (Ω,ω) + G 2 u m − G 2 u L p (Ω,ω) ϕ X .
Conflicts of Interest: "The author declares no conflict of interest."