Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in \(\mathbb{R}^ n\)
The author studies the questions of uniqueness, nonuniqueness and nonexistence of positive classical solutions of the problem \[ \Delta u+ \lambda g(x) f(u)=0 \quad \text{in } \mathbb{R}^ n \tag{P} \] (with additional condition at infinity \(\lim_{| x| \to\infty} u(x)=0\) if \(n\geq 3\)). The main result of the paper is the following: If \(g\) satisfies (1) \(g(x)<0\) near infinity or (2) \(n\geq 3\) and \(g(x)= O(| x|^{- 2- \sigma})\) at infinity \((\sigma>0)\), and \(f\) satisfies \(f(0)=0\), \(f'(0) >0\), \(f(u)>0\), \(f'' (u)<0\) (for \(u>0\)), then there is at most one solution of (P). It is shown by a counterexample, that these conditions are sharp in some sense. The proof is based on several integral identities for solutions of (P) and on a theorem which states that in the case of multiple solutions, there is at least one pair satisfying \(u>v\) in \(\mathbb{R}^ n\). In addition some nonexistence results are obtained for several values of \(\lambda\), depending on a critical value \(\lambda_{cr}\).
scientific article; zbMATH DE number 716356