On the uniqueness of solution for a class of semilinear elliptic problems
The Dirichlet problem \(-\Delta u=f(u)+\rho h(x)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) (\(\Omega\subset\mathbb R^ N \) bounded, \(\partial \Omega\) smooth) is considered, where \(f\in C^1(\mathbb R)\), \(h\in L_ 2(\Omega)\) and \(f_{\pm}=\lim_{s\to \pm \infty} f'(s)\) exist. The equivalent problem (2) \(-\Delta v=\rho^{-1}f(\rho v)+h\) in \(\Omega\), \(v=0\) on \(\partial \Omega\) is linked to the limiting problem (3) \(-\Delta v=f_+v\) \(--f_-v\) \(-+h\) in \(\Omega\), \(v=0\) on \(\partial \Omega\) by Theorem 1: If \(h(x)\neq 0\), (E) \(\| v\| \leq M\) for all solutions of (2) with \(\rho \geq \rho_ 0\) and (ND) any problem \(-\Delta w=(f_+\chi_ A+f_-\chi_ B)w\) in \(\Omega\), \(w=0\) on \(\partial \Omega\) \((A\cup B=\Omega\), \(A\) and \(B\) have positive measure) has only the zero solution, then uniqueness for problem (3) implies uniqueness for problem (2). Several sufficient conditions for (E), (ND) and the unique solvability of (3) are given in terms of the eigenvalues \(\lambda_ 1<\lambda_ 2<\lambda_ 3<...\) are the corresponding eigenspaces \(N_ i\). Theorem 2 states that the assumptions \(f_+,f_-\leq \lambda_ 1\) or \(\lambda_ k\leq f_+\), \(f_-\leq \lambda_{k+1}\) (but not \(f_+=f_-=\lambda_ i)\), \(h\in L^p(\Omega)\) for some \(p>N\) plus a non-orthogonality assumption in case \(f_-\) or \(f_+\) coincides with an eigenvalue imply that (3) has a unique solution and (ND) holds.
scientific article; zbMATH DE number 4034214