Dynamical boundary problem for Dirichlet-to-Neumann operator with critical Sobolev exponent and Hardy potential

2021 ◽  
Vol 62 ◽  
pp. 103346
Author(s):  
Yanhua Deng ◽  
Zhong Tan ◽  
Minghong Xie
Keyword(s):  
Boundary Problem ◽  
Critical Sobolev Exponent ◽  
Hardy Potential ◽  
Neumann Operator ◽  
Sobolev Exponent ◽  
Dirichlet To Neumann Operator

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential

2010 ◽  
Vol 62 (1) ◽  
pp. 19-33
Author(s):  
Mohammed Bouchekif ◽  
Yasmina Nas
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Critical Sobolev Exponent ◽  
Existence Results ◽  
Hardy Potential ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of ℝN. By variational methods we study the existence results.

scholarly journals Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

2021 ◽  
Author(s):  
Tim Binz
Keyword(s):  
Boundary Conditions ◽  
Elliptic Operator ◽  
Compact Manifold ◽  
Analytic Semigroup ◽  
Continuous Functions ◽  
Analytic Semigroups ◽  
Abstract Theory ◽  
Neumann Operator ◽  
Wentzell Boundary Conditions ◽  
Dirichlet To Neumann Operator

AbstractWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

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