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

الأصل · EN

We consider the boundary value problem -Δₚ u = λc(x) |u|ᵖ⁻²u + μ(x) | u|ᵖ + h(x), u ∈ W¹,ᵖ₀(Ω) ∩ L∞(Ω), where Ω⊂ Rⁿ, N ≥ 2, is a bounded domain with smooth boundary. We assume c, h ∈ Lq(Ω) for some q > {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.