| Popis: |
We deal with the existence of solutions for the quasilinear problem ( P λ ) { − Δ p u = λ u q − 1 + u p ∗ − 1 in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary, N ⩾ p 2 , 1 p ⩽ q p ∗ , p ∗ = N p / ( N − p ) , λ > 0 is a parameter. Using Morse techniques in a Banach setting, we prove that there exists λ ∗ > 0 such that, for any λ ∈ ( 0 , λ ∗ ) , ( P λ ) has at least P 1 ( Ω ) solutions, possibly counted with their multiplicities, where P t ( Ω ) is the Poincare polynomial of Ω. Moreover for p ⩾ 2 we prove that, for each λ ∈ ( 0 , λ ∗ ) , there exists a sequence of quasilinear problems, approximating ( P λ ) , each of them having at least P 1 ( Ω ) distinct positive solutions. |
