Existence and multiplicity for elliptic problems with quadratic growth in the gradient

Louis Jeanjean (1) , Boyan Sirakov (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Departamento de Matematica

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.

Domaines: Equations aux dérivées partielles [math.AP]

Louis Jeanjean, Boyan Sirakov. Existence and multiplicity for elliptic problems with quadratic growth in the gradient. Communications in Partial Differential Equations, 2013, 38 (2), pp.244-264. ⟨hal-00519454v2⟩