Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

Marie-Françoise Bidaut-Véron (1) , Quoc-Hung Nguyen (2, 3) , Laurent Véron (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 SNS - Scuola Normale Superiore di Pisa
3 Centro di Ricerca Matematica Ennio De Giorgi

We study the equation −div(A(x, u)) = g(x, u, u) + µ where µ is a measure and either g(x, u, u) ∼ |u| q 1 u||u| q 2 or g(x, u, u) ∼ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.

Mots clés: key-words: Quasilinear equations; Wolff and Riesz potentials; renormalized solutions; Bessel and Riesz capacities 2010 Mathematics Subject Classication: 31C15; 35J62; 35J92; 35R06; 45G15

Domaines: Equations aux dérivées partielles [math.AP]

Marie-Françoise Bidaut-Véron, Quoc-Hung Nguyen, Laurent Véron. Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data. 2018. ⟨hal-01775096v1⟩