Existence of Hölder Continuous Solutions for a Class of Degenerate Fourth-Order Elliptic Equations

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of a homogeneous Dirichlet problem related to a class of nonlinear fourth-order elliptic equations with strengthened degenerate ellipticity condition.


Introduction
In this paper, we prove the existence of bounded and locally Hölder continuous weak solutions of the homogeneous Dirichlet problem related to a class of nonlinear fourth-order elliptic equations whose model is is an open-bounded set, α = (α 1 , . . . , α N ) is a multiindex with nonnegative integer components and length |α| = α 1 + · · · + α N and D α u(x) = ∂ |α| u(x) ∂x α1 1 ∂x α2 2 . . . ∂x αN N . Here, θ and p α are real numbers, such that 0 ≤ θ < 1 and with 1 < p < N 2 , 2p < q < N, and f ∈ L t (Ω) with t > N q . In the case θ = 0, Eq. (1.1) is the fourth-order prototype of a class of nonlinear higher order elliptic equations introduced by I. V. Skrypnik in [42]. the ellipticity condition does not ensure the boundedness of a solution u ∈ W m,p (Ω), unless mp > N (as a consequence of Sobolev's embedding theorem) or mp = N (see [20]) or N − mp is sufficiently small (see [49]), while in the case where N > mp, examples of equations with unbounded weak solutions are available.
In [42], I. V. Skrypnik has selected a subclass of (1.3) imposing a strengthened ellipticity condition which, in the model case, takes the following form: where C 1 > 0, p ≥ 2, and mp < q < N.
This condition allowed reaching Hölder continuity of any generalized solution u ∈ W 1,q (Ω) ∩ W m,p (Ω) without any further relation on N , m, p.
Here, we consider a degenerate version of Skrypnik's fourth-order operator in the sense that the differential operator though well defined, is not coercive on W 1,q 0 (Ω) ∩ W 2,p 0 (Ω) when u is large. Due to this lack of coercivity, standard existence theorems for solutions of nonlinear equations cannot be applied. We overcome this difficulty by approximating our problem with a sequence of homogeneous nondegenerate Dirichlet problems and we will prove an L ∞ -a priori estimate on the approximating solutions which, in turn, implies an a priori estimate in the energy space. Once this has been accomplished, a compactness result for the approximating solutions allows us to find a bounded weak solution of the problem (1.1) which is, as well, locally Hölder continuous.
It is worthwhile to note that in the case of second-order equations, the existence of solutions of the Dirichlet problem under various assumptions on f , has been studied in the papers [1,2]. We point out that the equation we are dealing with presents two more difficulties: it involves a fourth-order operator which behaves like a system of PDEs, and moreover, it has non smooth coefficients. Many of the well-known techniques which work for one single equation of second order do not hold anymore in the framework of high-order equations and we need to find a suitable method to overcome the issues.
This article is organized as follows. In Sect. 2, we formulate the hypotheses and state the results. In Sect. 3, we prove two a priori estimates will be used in the proof of the main theorem. At last, in Sect. 4, we give the proofs of the existence of bounded solutions as well as their Hölder's continuity.

Preliminaries and Statement of the Results
Let Ω be an open-bounded set in R N with N ≥ 3. We denote by N (2) the number of different multi-indices α, such that |α| = 1, 2.
Then, there exists a weak solution u of the problem (2.5), in the sense of Definition 2.1, such that where M > 0 is a constant depending on θ, N , q, p, ν 1 , ν 2 , |Ω| and ||f || L t (Ω) .
Under the same assumptions, as in the nondegenerate case, it can be readily proved the local Hölder continuity of any weak solution u ∈W 1,q 2,p (Ω)∩ L ∞ (Ω). Namely where C is a positive constant depending on the same parameters of ρ and d = dist(Ω , ∂Ω).
In the framework of second-order elliptic equations with a lower order term having natural growth with respect to Du, the existence of bounded solutions has been studied in [3] assuming f in L t (Ω), with t > N q , and in [6,7,10,11,14,15] assuming f in a suitable Morrey space.

A Priori Estimates
We begin this section recalling an algebraic lemma due to Serrin (see Lemma 2 in [41]).
Lemma 3.1. Let χ be a positive exponent and a i , β i , i = 1, . . . , N, be two sets of N real numbers, such that 0 < a i < +∞ and 0 < β i < χ. Suppose that z is a positive number satisfying the inequality Then Given n ∈ N, let T n (s) be the truncation function defined by Following the technique already used in [1] and [2] in the framework of second-order elliptic equations, let us define the following Dirichlet problems: for almost every x ∈ Ω and for every (η, ξ) ∈ R × R N (2) , by Leray-Lions existence theorem (see [39]), there exists a solution u n ∈W 1,q 2,p (Ω) of problem (3.1). Moreover, every u n is bounded thanks to the boundedness result of [25] (see also [45]). Now, we are going to prove the following. Next lemma deals with the boundedness of u n in the energy space. To prove the previous two lemmas, we have to state some auxiliary propositions. First of all, we need to ensure that the composition of a suitable function ζ(s) with a function u ∈W 1,q 2,p (Ω) belongs toW 1,q 2,p (Ω).
Proof of Lemma 3.2. Given k ≥ 1 and σ > 1 + pq q−2p (note that σ > 2), let us consider the function v = ζ(u n ) with Due to the boundedness of u n , as a consequence of Lemma 3.4, v is an admissible test function in (3.1), and it holds a.e. in Ω, (3.5) with R |α| (u n ) ≡ 0 if |α| = 1 and if |α| = 2. Choosing v = ζ(u n ) in (3.1), we obtain Using the ellipticity condition (2.1), from the above relation, we get From now on, we will denote by c a positive constant not depending on n (namely, it may depend on N , |Ω|, p, q, ν 1 , ν 2 , ||f || t ) and whose value may vary from line to line.
We are going to evaluate the integrals on the right-hand side of (3.8). Due to the growth condition (2.2) and the estimate (3.6), we get Taking into account the inequality 1 ≤ 1 + |T n (u n )| ≤ 1 + |u n | and using Young's inequality with exponents p p−1 , q 2 and pq q−2p , for all τ > 0, we obtain We denote by A n (k) the level set of |u n |, that is Using hypothesis (2.4) and Hölder's inequality, we evaluate terms on the right-hand side of the above inequality, as follows: (3.14) and where t is the conjugate exponent of t. From (3.12)-(3.15) and dropping the integrals involving second derivatives in the left-hand side of (3.12), we deduce Ω |α|=1 , 1 + δ . The use of Hölder's inequality, together with relations Using (3.17) in (3.16) and Sobolev's embedding theorem, we obtain where γ * = Nγ N −γ .
Using v = u n as test function in the integral identity (3.1), applying the ellipticity condition (2.1), the growth condition (2.2), and Young's inequality, we have Now, taking into account (3.22), from the above inequality, we obtain (3.23) and the Lemma follows.

Proofs of the Results
Before proving Theorem 2.2, we have to premise a compactness result for the approximating solutions u n which, together with the a priori estimates proved in the previous section, will allow us to pass to the limit in the approximate problems (3.1). As a consequence of Lemma 3.2 and Lemma 3.3, there exist a subsequence, still denoted by {u n }, and a function u ∈W 1,q 2,p (Ω) ∩ L ∞ (Ω), such that {u n } is bounded in L ∞ (Ω) and u n → u weakly inW 1,q 2,p (Ω) u n → u almost everywhere in Ω.
We need to prove that the sequences D α u n , |α| = 1, 2 are almost everywhere convergent in Ω. To this aim, we exploit the following compactness result whose proof is in [12].
Then, z n is relatively compact in the strong topology ofW 1,q 2,p (Ω). We are now in the position to prove the (relatively) compactness of {u n }. We take u n − u as test function in the weak formulation of Problem (3.1), and we obtain |α|=1,2 Ω A α (x, T n (u n ), D 1 u n , D 2 u n )D α [u n − u] dx = Ω f (u n − u) dx.
From the above equality, it follows: The right-hand side of (4.3) tends to zero as n tends to +∞, since u n converges to u weakly * in L ∞ (Ω), weakly inW 1,q 2,p (Ω) and A α (x, u, D 1 u, D 2 u) belongs to L p α (Ω), |α| = 1, 2 thanks to (2.2). Due to the boundedness of u n L ∞ , T n (u n ) = u n for sufficiently large n and using Lemma 4.1, we conclude, up to a subsequence, that u n → u strongly inW 1,q 2,p (Ω).
From now on, thanks to the boundedness of the solution u and following the outlines of the proof of [8] [ Theorem 1.4] or [42], we can prove that there exists 0 < μ < 1, such that ω(R) ≤ μ ω(2R) + R r .