Uniform L ∞ -Estimates for Quasilinear Elliptic Systems

. The aim of this work is to provide uniform L ∞ -estimates for the solutions of a quite general class of ( p, q )-quasilinear elliptic systems depending on two parameters α and δ . Mathematics Subject Classiﬁcation. 35J92, 35J50, 35B45.


Introduction
Let us consider the following autonomous quasilinear system where Ω is a smooth bounded domain of R N , N ≥ 3, p, q ∈ [2, N), α ≥ 0 and H : I × R 2 → R is a function, where I ⊂ R is an interval and H(δ, •, •) ∈ C 1 (R 2 , R) for any δ ∈ I.Moreover, we assume that ( * ) there are p ∈ (p, p * ), q ∈ (q, q * ) and C 0 > 0 such that Let X be the product space W 1,p 0 (Ω)×W 1,q 0 (Ω) endowed with the norm z = u 1,p + v 1,q where z = (u, v) ∈ X.In what follows we shall denote respectively by • s and • 1,s the usual norms in L s (Ω) and W 1,s  0 (Ω).

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Weak solutions of problem (1.1) correspond to critical points of the Euler functional I α,δ : X → R defined as By ( * ), the functional I α,δ is C 1 on X and, for any z 0 = (u 0 , v 0 ) and z = (u, v) in X, it results Systems involving this kind of quasilinear operators model some phenomena in non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology; see [7,9,11,12].Existence, nonexistence and regularity results for such quasilinear elliptic systems are obtained by various authors, see for instance [1,3,6,8,14].
More recently we proved that any weak solution of the following system, not depending on δ, In this work we want to extend the previous result to the class of systems (1.1) depending also on δ.Moreover here we show carefully that, for any arbitrary z 0 ∈ X and r > 0, the (L ∞ (Ω)) 2 -norm of the weak solutions to (1.1) belonging to B r (z 0 ) depends just on r and z 0 , but is independent on α and δ.
The main result of this work is the following: Moreover, for any fixed (u 0 , v 0 ) ∈ X, r > 0, α ≥ 0 and δ ∈ I, denoting by there exists C > 0, depending on r and (u 0 , v 0 ) but independent of α and δ, such that This uniform L ∞ -estimate will be used in the forthcoming paper [2] in which we derive some crucial existence results about system (1.1), studying the interaction of the spectrum of the quasilinear operators with the nonlinearity H which grows (p, q)-linearly at infinity, in continuity with the Amann-Zehnder type results obtained in [5] for a class of quasilinear elliptic equations.

Proof of Theorem 1.1
We first introduce the following result.
Proof.By contradiction, assume that there are r, ε > 0, for any n ∈ N.
Up to subsequences, u n strongly converges to some ū in L s (Ω).Moreover, denoting by Now, inspired by [4] and [10], we prove the main result.

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Proof of Theorem 1.1.For every γ, t, k > 1 we define Observe that h k,γ and Φ k,t,γ are C 1 -functions with bounded derivative, depending on γ, t and k.
Setting k = k p q in (2.10) and substituting in (2.9) we obtain Using again Lemma 2.1 and choosing a suitable σ ≥ σ 2 , we find C > 0 such that, for any k, γ > 1 where C depends on r and γ but is independent of k.
Analogously, we can prove that there is C > 0, independent of k, such that

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Thus we can use Fatou Lemma and, passing to the limit for k → +∞, we get where C depends on r and γ.
Since γ > 1 is an arbitrary number, we have that ū, v ∈ L t (Ω) for any t > 1.
We want to prove that for any (ū, v) ∈ D r,α,δ (u 0 , v 0 ), ū and v are in where the constant C 2 > 0 still depends on r but is independent of α and δ.

Lemma 2 . 1 .
Let s ∈ (1, N) and denote by s * the conjugate Sobolev exponent of s, namely s