Eigenproblem for p-Laplacian and nonlinear elliptic equation with nonlinear boundary conditions

2015 ◽  
Vol 31 (4) ◽  
pp. 659-674
Author(s):  
Berrajaa Mohammed ◽  
Chakrone Omar ◽  
Diyer Fatiha ◽  
Diyer Okacha
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Elliptic

OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

2008 ◽  
Vol 18 (11) ◽  
pp. 1869-1893 ◽  
Author(s):  
FUENSANTA ANDREU ◽  
NOUREDDINE IGBIDA ◽  
JOSÉ M. MAZÓN ◽  
JULIÁN TOLEDO
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Maximal Monotone ◽  
Nonlinear Boundary Conditions ◽  
Obstacle Problems ◽  
Laplacian Operator ◽  
Usual Sense ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic

In this paper we study the questions of existence and uniqueness of solutions for equations of type - div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solution has to be introduced.

scholarly journals INFINITE RESONANT SOLUTIONS AND TURNING POINTS IN A PROBLEM WITH UNBOUNDED BIFURCATION

2010 ◽  
Vol 20 (09) ◽  
pp. 2885-2896 ◽  
Author(s):  
J. M. ARRIETA ◽  
R. PARDO ◽  
A. RODRÍGUEZ-BERNAL
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Infinite Number ◽  
Nonlinear Boundary ◽  
Turning Points ◽  
Nonlinear Boundary Conditions ◽  
Steklov Eigenvalue

We consider an elliptic equation -Δu + u = 0 with nonlinear boundary conditions ∂u/∂n = λu + g(λ, x, u), where (g(λ, x, s))/s → 0, as |s| → ∞. In [Arrieta et al., 2007, 2009] the authors proved the existence of unbounded branches of solutions near a Steklov eigenvalue of odd multiplicity and, among other things, provided tools to decide whether the branch is subcritical or supercritical. In this work, we give conditions on the nonlinearity, guaranteeing the existence of a bifurcating branch which is neither subcritical nor supercritical, having an infinite number of turning points and an infinite number of resonant solutions.

Multiplicity and Existence Results for a Nonlinear Elliptic Equation With Sobolev Exponent

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Zakaria Bouchech ◽  
Hichem Chtioui
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)u

scholarly journals Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions

2012 ◽  
Vol 142 (1) ◽  
pp. 137-153 ◽  
Author(s):  
N. Mavinga
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
A Priori ◽  
Nonlinear Problems ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Generalized Spectrum ◽  
Singular Weights

We are concerned with the solvability of nonlinear second-order elliptic partial differential equations with nonlinear boundary conditions. We study the generalized Steklov–Robin eigenproblem (with possibly singular weights) in which the spectral parameter is both in the differential equation and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates.

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