Abstract
In this paper we prove the existence of two intervals of positive real parameters λ for a Dirichlet boundary value problem involving the p-Laplacian which admit three weak solutions, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. Our main tool is a three critical points theorem due to G. Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim., 28:249–258, 2004].
Highlights
Where Δpu = div(|∇u|p−2∇u) is the p-Laplacian operator, Ω ⊂ RN (N 1) is a non-empty bounded open set with smooth boundary ∂Ω, p > N, λ is a positive parameter and f : Ω × R → R is an L1- Caratheodory function
Δpu + λf (x, u) = a(x)|u|p−2u in Ω, u=0 on ∂Ω, where Ω ⊂ RN (N 2) is non-empty bounded open set with smooth boundary ∂Ω, p > N, λ > 0, f : Ω × R → R is a continuous function and positive weight function a(x) ∈ C(Ω), admits at least three weak solutions whose norms in
Bonanno in [6] established the existence of two intervals of positive real parameters λ for which the functional Φ + λΨ has three critical points, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals
Summary
In [7], Ricceri’s three critical points theorem [15] has been successfully used to obtain existence of at least three weak solutions to the problem (1.1) in W01,p(Ω). In [1], based on Ricceri’s three critical points theorem [15] we obtained the existence of an interval Λ ⊆ [0, +∞[ and a positive real number q such that for each λ ∈ Λ problem
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