On the linearization of some singular, nonlinear elliptic problems and applications (Q1861108)

Let us consider the semilinear problem \[ Lu:=- \sum_{1\leq i,j\leq n} a_{ij}(x)\partial_i\partial_j u+ \sum_{1\leq i\leq n} b_i(x)\partial_i u= f(x,u)\text{ in }\Omega,\;u|_{\partial\Omega}= 0,\tag{1} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain and the coefficients \(b_i\) exhibit an appropriate singularity at the boundary. The linearization of (1) around a given positive solution \(u\) leads the authors to consider the linear weighted eigenvalue problem \[ LU- M(x)U= \lambda N(x)U\quad\text{in }\Omega,\quad U|_{\partial\Omega}= 0,\tag{2} \] where \(M(x):= f_u(x,u(x))\). They describe the spectrum of (2) in the case \(N>0\) and analyze the existence and uniqueness of the principal eigenvalue when \(N\) changes sign in \(\Omega\). Then the problem (1) is rewritten in integral form, via a Green operator and the Fréchet differentiability of the resulting problem with respect to \(u\) and with respect to a parameter is considered. Several applications are given which deal with the construction of solutions of (1) as limits of sequences of sub- and/or supersolutions and with the stability of the solutions of (1) as steady states of the associated parabolic problem.