Superlinear indefinite elliptic problems and nonlinear Liouville theorems
The authors study the boundary value problem \[ \begin{aligned} \sum_{i,j=1}^ N a_{ij} (x) {{\partial^ 2 u} \over {\partial x_ i \partial x_ j}}+ \sum_{i=1}^ N b_ i(x) {{\partial u} \over {\partial x_ i}}+ a(x) g(u) &= 0\quad \text{in }\Omega,\\ \sum_{j,k=1}^ N \nu_ j a_{jk} {{\partial u} \over {\partial x_ k}}+ \alpha(x)u &= 0 \quad \text{on } \partial\Omega. \end{aligned} \] Here the above differential operator is uniformly elliptic, \(\alpha(x) \geq 0\) on \(\partial\Omega\), but the coefficient \(a(x)\) may change sign. The nonlinearity \(g\) is assumed to be \(C^ 1\) with \(g(0)= g'(0) =0\), \(g(s)>0\) for large \(s>0\), and such that the limit \(s^{-p} g(s)\) exists, as \(s\to\infty\), for some \(p>1\). The main existence result then states that the above boundary value problem has a solution if \(1<p< (N+2)/ (N-1)\). Many interesting additional statements are given, mainly for the model equation \(\Delta u- m(x)u+ a(x) g(u)=0\).
scientific article; zbMATH DE number 733359