Nonexistence results for solutions of semilinear elliptic equations
Consider the semilinear elliptic equation (1) \(\Delta u= f(| x|)u+ g(x) u^ q\), \(x\in \mathbb{R}_ 0^ N\) for \(N\geq 3\), \(q>1\), where \(\mathbb{R}_ 0^ N= \mathbb{R}^ N\setminus \{0\}\), \(f\in L^ 1_{\text{loc}} (\mathbb{R}_ 0^ +)\), \(g\in L^ \infty_{\text{loc}} (\mathbb{R}_ 0^ N)\), \(g\geq 0\). The main theorems are sufficient conditions on \(f\), \(g\) for equation (1) to have no nontrivial nonnegative solution \(u\in C(\mathbb{R}_ 0^ N)\). An analogous result is given for the case \(f\equiv 0\). The methods include convexity arguments applied to an ordinary differential equation associated with (1). The theorems extend earlier ones of \textit{F.-H. Lin} [Proc. Am. Math. Soc. 95, 219-226 (1985; Zbl 0584.35031)] and \textit{W.-M. Ni} [Indiana Univ. Math. J. 31, 493-529 (1982; Zbl 0496.35036)].
scientific article; zbMATH DE number 624965
scientific article