Operator estimates for elliptic problem with rapidly alternating Steklov boundary condition

2020 ◽  
Vol 376 ◽  
pp. 112802
Author(s):  
Aleksandra G. Chechkina ◽  
Ciro D’Apice ◽  
Umberto De Maio
Keyword(s):  
Boundary Condition ◽  
Elliptic Problem

scholarly journals Remarks on a nonlinear nonlocal operator in Orlicz spaces

2019 ◽  
Vol 9 (1) ◽  
pp. 305-326 ◽  
Author(s):  
Ernesto Correa ◽  
Arturo de Pablo
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Sobolev Inequality ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Laplacian Operator ◽  
Nonlocal Operator ◽  
Generalized Eigenvalue

Abstract We study integral operators $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$in a bounded domain $\Omega ,$and boundary condition u ≡ 0 on ${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem $f\left( u \right)=\lambda \psi \left( u \right).$

Semilinear Robin problems resonant at both zero and infinity

2018 ◽  
Vol 30 (1) ◽  
pp. 237-251
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Robin Boundary Condition ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Reaction Term ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Robin Boundary

Abstract We consider a semilinear elliptic problem, driven by the Laplacian with Robin boundary condition. We consider a reaction term which is resonant at {\pm\infty} and at 0. Using variational methods and critical groups, we show that under resonance conditions at {\pm\infty} and at zero the problem has at least two nontrivial smooth solutions.

scholarly journals Symmetric Solutions of a Nonlinear Elliptic Problem with Neumann Boundary Condition

2012 ◽  
Vol 03 (11) ◽  
pp. 1686-1688
Author(s):  
Ana Magnolia Marin Ramirez ◽  
Ruben Dario Ortiz Ortiz ◽  
Joel Arturo Rodriguez Ceballos
Keyword(s):  
Boundary Condition ◽  
Elliptic Problem ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic
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