An existence theorem for positive solutions of semilinear elliptic equations
Let \(f\in C^ 1\), \(f(p_ 0)=0\) for some \(p_ 0>0\), and \(F(u)=\int^{u}_{0}f(s)ds\). The authors prove the equivalence of the following three statements: (1) There exist positive solutions of \(\Delta u+f(u)=0\) in \(B_ R\), the n-ball of radius R, and \(u=0\) on \(\partial B_ R\) for which u(0) is arbitrarily close to \(p_ 0\), \(u(0)<p_ 0\). (2) There exist positive solutions of \(u''+f(u)=0\) in \(B_ 1\), \(u=0\) on \(\partial B_ 1\) for which u(0) is arbitrarily close to \(p_ 0\), \(u(0)<p_ 0\). (3) \(F(p)<F(p_ 0)\) for \(0\leq p<p_ 0\). The argument is based on the radial symmetry deduced by \textit{B. Gidas, W. M. Ni}, and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)] and is related to results of \textit{P. Hess} [Commun. Partial Differ. Equations 6, 951-961 (1981; Zbl 0468.35073)].
scientific article; zbMATH DE number 4013144