Asymptotic Expansion for the Solutions of Control Constrained Semilinear Elliptic Problems with Interior Penalties

2011 ◽  
Vol 49 (6) ◽  
pp. 2494-2517 ◽  
Author(s):  
J. Frédéric Bonnans ◽  
Francisco J. Silva
Keyword(s):  
Asymptotic Expansion ◽  
Elliptic Problems ◽  
Semilinear Elliptic ◽  
Interior Penalties ◽  
Semilinear Elliptic Problems

Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

2021 ◽  
pp. 1-18
Author(s):  
Lucio Boccardo ◽  
Liliane Maia ◽  
Benedetta Pellacci
Keyword(s):  
Linear Operator ◽  
Asymptotic Behaviour ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Existence Results ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
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We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \] where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

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