Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 645-658

EXISTENCE OF WEAK SOLUTIONS FOR ELLIPTIC SYSTEMS WITH p,q-GROWTH

Giovanni Cupini, Francesco Leonetti and Elvira Mascolo

Università di Bologna, Dipartimento di Matematica
Piazza di Porta S. Donato 5, 40126 Bologna, Italy; giovanni.cupini 'at' unibo.it

Università di L'Aquila, Dipartimento di Ingegneria e Scienze dell'Informazione, Matematica
67100 L'Aquila, Italy; leonetti 'at' univaq.it

Università di Firenze, Dipartimento di Matematica e Informatica "U. Dini"
Viale Morgagni 67/A, 50134 Firenze, Italy; mascolo 'at' math.unifi.it

Abstract. We consider a non-linear system of m equations in divergence form and a boundary condition:

∑ⁿ_i=1 ∂/∂x_i(A^α_i(x,Du(x))) = 0, 1 ≤ α ≤ m, in Ω
u = \tilde u on ∂Ω.

The functions A^α_i(x,z) are Hölder continuous with respect to x and

|z|^p - c₁ ≤ ∑^m_α=1 ∑ⁿ_i=1A^α_i(x,z)z_i^α ≤ c₂ (1 + |z|)^q, 2 ≤ p ≤ q.

We prove the existence of a weak solution u in (\tilde u + W₀^1,p(Ω;R^m)) ∩ W_loc^1,q(Ω;R^m), provided p and q are close enough and under suitable summability assumptions on the boundary datum \tilde u.

2010 Mathematics Subject Classification: Primary 35J25; Secondary 35J47, 49N60.

Key words: Existence, regularity, weak, solution, elliptic, system, growth.

Reference to this article: G. Cupini, F. Leonetti and E. Mascolo: Existence of weak solutions for elliptic systems with p,q-growth. Ann. Acad. Sci. Fenn. Math. 40 (2015), 645-658.

Full document as PDF file

doi:10.5186/aasfm.2015.4035