On the solvability of a semilinear elliptic equation via an associated eigenvalue problem
We study a boundary value problem for the equation \(- \Delta u= \lambda u- h(x) f(u)\), where \(f\) grows superlinearly and \(h(x)\geq 0\) may vanish on a set with a nonempty interior. We observe that the existence, nonexistence, and multiplicity of solutions is determined by the solvability of an associated linear eigenvalue problem. In particular, under certain assumptions we show that the nonlinear problem possesses a nontrivial solution if and only if the associated eigenvalue problem is solvable.
scientific article; zbMATH DE number 922489