scholarly journals A fourth order elliptic equation with nonlinear boundary conditions

2002 ◽  
Vol 49 (8) ◽  
pp. 1037-1047 ◽  
Author(s):  
Julián Fernández Bonder ◽  
Julio D. Rossi
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Order Elliptic Equation ◽  
Fourth Order Elliptic Equation

scholarly journals Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

2019 ◽  
Vol 22 (06) ◽  
pp. 1950057
Author(s):  
Zongming Guo ◽  
Fangshu Wan ◽  
Liping Wang
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Order Elliptic Equation ◽  
Liouville Type ◽  
Related Equation ◽  
Fourth Order Elliptic Equation ◽  
Liouville Type Results

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth-order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.

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.

scholarly journals PRESCRIBING A FOURTH ORDER CONFORMAL INVARIANT ON THE STANDARD SPHERE — PART I: A PERTURBATION RESULT

2002 ◽  
Vol 04 (03) ◽  
pp. 375-408 ◽  
Author(s):  
ZINDINE DJADLI ◽  
ANDREA MALCHIODI ◽  
MOHAMEDEN OULD AHMEDOU
Keyword(s):  
Elliptic Equation ◽  
Fourth Order ◽  
Critical Sobolev Exponent ◽  
Order Elliptic Equation ◽  
Conformal Invariant ◽  
Standard Sphere ◽  
Finite Dimensional ◽  
Fourth Order Elliptic Equation ◽  
Zero Degree ◽  
Sobolev Exponent

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree

scholarly journals A nonlinear boundary problem involving thep-bilaplacian operator

2005 ◽  
Vol 2005 (10) ◽  
pp. 1525-1537 ◽  
Author(s):  
Abdelouahed El Khalil ◽  
Siham Kellati ◽  
Abdelfattah Touzani
Keyword(s):  
Boundary Conditions ◽  
Boundary Problem ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Nondecreasing Sequence ◽  
Nonlinear Boundary Problem

We show some new Sobolev's trace embedding that we apply to prove that the fourth-order nonlinear boundary conditionsΔp2u+|u|p−2u=0inΩand−(∂/∂n)(|Δu|p−2Δu)=λρ|u|p−2uon∂Ωpossess at least one nondecreasing sequence of positive eigenvalues.

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