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

We prove existence results of two solutions of the problem $$\{\begin{array}{ll}L(u)+u^{m-1}=\lambda u^{p-1}& \text{ in }\mathrm{\Omega },\\ u>0& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $L(v)=-\mathrm{d}\mathrm{i}\mathrm{v}(M(x)\mathrm{\nabla }v)$ is a linear operator, $p\in (2,2^{\ast }]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviors.