A priori estimates and existence for elliptic equations with gradient dependent terms

• Nathalie Grenon ^[1] ; François Murat ^[2] ; Alessio Porretta ^[3]
  1. [1] Centre Universitaire de Bourges, France
  2. [2] Universit´e Pierre et Marie Curie, France
  3. [3] Universit`a di Roma Tor Vergata, Italia

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• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 1, 2014, págs. 137-205
• Idioma: inglés
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  • We consider, in a bounded domain  ⇢ RN , a class of nonlinear elliptic equations in divergence form as ( ↵0 u − div(a(x, u, Du)) = H(x, u, Du) in , u = 0 on @ where ↵0 # 0, the second order part is a coercive, pseudomonotone operator of Leray-Lions type in the Sobolev space W1,p 0 (), p > 1, and the function H grows at most like |Du|q + f (x), with p−1 < q < p. Assuming f (x) to belong to an (optimal) Lebesgue class Lm, with m < Np , we prove a priori estimates and existence of solutions, discussing several ranges of the exponents m, q and p which include cases of singular data (L1 data or measures). The obtention of a priori estimates is not straightforward because of the “superlinear”character of the first order terms. To this purpose we use a new approach, generalizing the method introduced in our note [29]. We complete the results known in the previous literature where either q  p − 1 or m # Np.