Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We study the semilinear problem with the boundary reaction $$-\Delta u+u=0\text{in}\Omega ,\frac{\partial u}{\partial \nu }=\lambda f\left(u\right)\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a smooth bounded domain, $f:[0,\infty )\to (0,\infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter ${\lambda }^{*}>0$ such that a classical minimal solution exists for $\lambda <{\lambda }^{*}$, and there is no solution for $\lambda >{\lambda }^{*}$. Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $\lambda ={\lambda }^{*}$. In this paper, we continue to study the spectral properties of $u^{*}$ and show...