Abstract
The existence of three distinct weak solutions for a perturbed mixed boundary value problem involving the one-dimensional p-Laplacian operator is established under suitable assumptions on the nonlinear term. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces.
Highlights
IntroductionUsing two kinds of three critical points theorems obtained in [4, 8] which we recall (Theorems 2.1 and 2.2), we ensure the existence of at least three weak solutions for the problem (1); see Theorems 3.1 and 3.2
Consider the following perturbed mixed boundary value problem−(ρ(x)|u′|p−2u′)′ + s(x)|u|p−2u = λf (x, u) + μg(x, u) in ]a, b[ u(a) = u′(b) = 0, (1)where p > 1, λ > 0 and μ ≥ 0 are real numbers, a, b ∈ R with a < b, ρ, s ∈ L∞([a, b]) with ρ0 = essinfx∈[a,b]ρ(x) > 0, s0 = essinfx∈[a,b]s(x) ≥ 0 and f, g : [a, b] × R → R are two L1-Caratheodory function.Using two kinds of three critical points theorems obtained in [4, 8] which we recall (Theorems 2.1 and 2.2), we ensure the existence of at least three weak solutions for the problem (1); see Theorems 3.1 and 3.2
Existence and multiplicity of solutions for mixed boundary value problems have been studied by several authors and, for an overview on this subject, we refer the reader to the papers [2, 3, 12, 15, 18]
Summary
Using two kinds of three critical points theorems obtained in [4, 8] which we recall (Theorems 2.1 and 2.2), we ensure the existence of at least three weak solutions for the problem (1); see Theorems 3.1 and 3.2. There is λ∗ > 0 such that for each λ > λ∗ and for every L1-Caratheodory function g : [a, b] × R → R satisfying the asymptotical condition t sup g(x, s)ds lim sup x∈[a,b].
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