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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2020

Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients

Résumé

Let Ω ⊂ RN, N≥ 2 , be a smooth bounded domain. We consider the boundary value problem [Figure not available: see fulltext.]where cλ and h belong to Lq(Ω) for some q> N/ 2 , μ belongs to R\ { 0 } and we write cλ under the form cλ: = λc+- c- with c+≩ 0 , c-≥ 0 , c+c-≡ 0 and λ∈ R. Here cλ and h are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence (and uniqueness) of solution to (Pλ) when λ≤ 0. Then, assuming that (P) has a solution, we prove existence and multiplicity results for λ> 0. Our proofs rely on a suitable change of variable of type v= F(u) and the combination of variational methods with lower and upper solution techniques.

Dates et versions

hal-03337292 , version 1 (07-09-2021)

Identifiants

Citer

Colette De Coster, Antonio J. Fernández Sánchez. Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients. Calculus of Variations and Partial Differential Equations, 2020, 59 (3), ⟨10.1007/s00526-020-01755-z⟩. ⟨hal-03337292⟩
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