Semilinear Elliptic Problems with Pairs of Decaying Positive Solutions

1987 ◽  
Vol 39 (5) ◽  
pp. 1162-1173 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Operator ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Elliptic Problems ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Image Position

Our main objective is to prove the existence of a pair of positive, exponentially decaying, classical solutions of the semilinear elliptic eigenvalue problem1.1in a smooth unbounded domain Ω ⊂ RN, N ≧ 2, where λ is a positive parameter and L is a uniformly elliptic operator in Ω defined by

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 ◽  
Image Position

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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