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Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 5 / October 2022
- Published online by Cambridge University Press:
- 28 September 2021, pp. 1233-1250
- Print publication:
- October 2022
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- Article
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We prove existence results of two solutions of the problem
where
\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]
$L(v)=-\textrm {div}(M(x)\nabla v)$ 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.
