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Boundary eigencurve problems involving the biharmonic operator

100%

Chakrone O., Tsouli N., Rahmani M., Darhouche O.

Applicationes Mathematicae

2014

tom 41

nr 2-3

267-275

The aim of this paper is to study the spectrum of the fourth order eigenvalue boundary value problem ⎧Δ²u = αu + βΔu in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω. where (α,β) ∈ ℝ². We prove the existence of a first nontrivial curve of this spectrum and we give its variational characterization. Moreover we prove some properties of this curve, e.g., continuity, convexity, and asymptotic behavior. As an application, we study the non-resonance of solutions below the first principal eigencurve of the biharmonic problem ⎧Δ²u = f(u,x) + βΔu + h in Ω, ⎨ ⎩Δu = u = 0 ∂Ω, where f: Ω × ℝ → ℝ is a Carathéodory function and h is a given function in L²(Ω).

Three solutions for a nonlinear Neumann boundary value problem

100%

Tsouli N., Chakrone O., Darhouche O., Rahmani M.

Applicationes Mathematicae

2014

tom 41

nr 2-3

257-266

The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form $-Δ_{p(x)}u + a(x)|u^{|p(x)-2}u = μg(x,u)$ in Ω, $|∇u|^{p(x)-2} ∂u/∂ν = λf(x,u)$ on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

Ograniczanie wyników

2 Applicationes Mathematicae

2 Chakrone O.

2 Darhouche O.

2 Rahmani M.

2 Tsouli N.

2 2014

Wyniki wyszukiwania

Boundary eigencurve problems involving the biharmonic operator

Three solutions for a nonlinear Neumann boundary value problem