Multiple solutions for an elliptic problem with quadratic growth in the gradient

Louis Jeanjean (1) , Boyan Sirakov (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 MODAL'X - Modélisation aléatoire de Paris X

We consider a divergence-form elliptic problem with quadratic growth in the gradient. Assuming that the zero order term has a given sign we show the existence of at least two solutions. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule. Our proofs are mainly variational but also sub- and super-solutions arguments are used.

Domaines: Equations aux dérivées partielles [math.AP]

Louis Jeanjean, Boyan Sirakov. Multiple solutions for an elliptic problem with quadratic growth in the gradient. 2010. ⟨hal-00519454v1⟩