Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

Colette De Coster (1) , Antonio J. Fernández Sánchez

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

We consider the boundary value problem (Pλ) −∆pu = λc(x)|u| p−2u + µ(x)|∇u| p + h(x) , u ∈ W1,p 0 (Ω) ∩ L∞(Ω) , where Ω ⊂ RN , N ≥ 2, is a bounded domain with smooth boundary. We assume c, h ∈ Lq (Ω) for some q > max{N/p, 1} with c 0 and µ ∈ L∞(Ω). We prove existence and uniqueness results in the coercive case λ ≤ 0 and existence and multiplicity results in the non-coercive case λ > 0. Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case.

Domaines: Mathématiques [math]

Colette De Coster, Antonio J. Fernández Sánchez. Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient. Calculus of Variations and Partial Differential Equations, 2018, 57 (3), 42 p. ⟨10.1007/s00526-018-1346-6⟩. ⟨hal-03137670⟩