scholarly journals On the p-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Principal Eigenvalue ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Sobolev Exponent

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the p-biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

scholarly journals Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

2020 ◽  
Vol 6 (2) ◽  
pp. 198-209
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Dirichlet Problem ◽  
Variational Approach ◽  
Type Inequality ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Hardy Type Inequality ◽  
Principal Curve ◽  
Hardy Type ◽  
Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

On the p-biharmonic operator with critical Sobolev exponent

2014 ◽  
Vol 41 (2) ◽  
pp. 229-237
Author(s):  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Critical Sobolev Exponent ◽  
Biharmonic Operator ◽  
Sobolev Exponent

scholarly journals The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

2021 ◽  
Vol 93 (2) ◽  
Author(s):  
Robert Denk ◽  
Markus Kunze ◽  
David Ploß
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Regularity Of Solutions ◽  
Lipschitz Domains ◽  
Lipschitz Boundary ◽  
Full Characterization ◽  
Wentzell Boundary Conditions ◽  
Adjoint Operators

AbstractWe investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d with Lipschitz boundary $$\Gamma $$ Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$ L 2 ( Ω ) × L 2 ( Γ ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent

1986 ◽  
Vol 103 (3-4) ◽  
pp. 275-285 ◽  
Author(s):  
A. Capozzi ◽  
G. Palmieri
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Multiplicity Results ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisIn this paper we study the following boundary value problemwhere Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

scholarly journals Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Elliot Tonkes
Keyword(s):  
Elliptic Equations ◽  
Principal Eigenvalue ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic Equations ◽  
Asymptotic Nature ◽  
Semilinear Elliptic ◽  
Palais Smale Condition ◽  
Sobolev Exponent

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent.

Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions

1991 ◽  
Vol 117 (3-4) ◽  
pp. 225-250 ◽  
Author(s):  
C. Budd ◽  
M. C. Knaap ◽  
L. A. Peletier
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
Nonlinear Eigenvalue Problem ◽  
Neumann Boundary ◽  
Critical Sobolev Exponent ◽  
Neumann Boundary Conditions ◽  
Supremum Norm ◽  
Asymptotic Estimates ◽  
Sobolev Exponent

SynopsisAsymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu= λ(up−uq) in the unit ballBin ℝN(N> 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Herepis the critical Sobolev exponent (N+ 2)/(N− 2) and 0 <q<p− 1 = 4/(N− 2).

High-energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponents

1995 ◽  
Vol 125 (5) ◽  
pp. 1003-1029 ◽  
Author(s):  
Zhi-Qiang Wang
Keyword(s):  
Boundary Conditions ◽  
Critical Exponents ◽  
Neumann Boundary ◽  
Critical Sobolev Exponent ◽  
High Energy ◽  
Nonlinear Neumann Problem ◽  
Existence Of Positive Solutions ◽  
Sobolev Exponent ◽  
Two Peaks ◽  
Invariant Domains

We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem −Δu + λu = up in antipodal invariant domains including ball domains with Neumann boundary conditions. Here p is the critical Sobolev exponent (N + 2)/(N − 2). The shape of the solutions and the location of the peaks are also studied.

Critical semilinear biharmonic equations in RN

1992 ◽  
Vol 121 (1-2) ◽  
pp. 139-148 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson ◽  
Yang Jianfu
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Biharmonic Equations ◽  
Best Constant ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisAn existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

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