Large positive solutions of semilinear elliptic equations with critical and supercritical growth
The authors study large positive solutions of the problem \[ \begin{cases} -\Delta u=u^p- \varepsilon u^q\quad &\text{in }\Omega\\ u=0\quad &\text{on }\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N (N \geq 3)\) with smooth boundary \(\partial\Omega\) and \(q>p\geq p_N:= {N+2\over N-2}\), \(\varepsilon>0\). They obtain existence and uniqueness of large positive solutions for (1) and study their asymptotic behaviour as \(\varepsilon \to 0\). It is shown that they develop a boundary layer and the boundary derivative estimate of the large solution is also established.
scientific article; zbMATH DE number 1801635
scientific article