Decreasing streamlines of solutions and spectral properties of linearized operators for semilinear elliptic equations (Q1826028)

Let \(\Omega\) be the unit ball in \({\mathbb{R}}^ N\), let \(f: {\mathbb{R}}^+\to {\mathbb{R}}^+\) be a \(C^ 1\) function and let u be a positive solution of the semilinear elliptic equation \(-\Delta u=f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). The author studies the spectrum of the linearized operator \(A=A(u)=-\Delta -f'(u).\) In particular, he proves \(\dim Ker A(u)\leq 1.\) An application to the case \(f(u)=\lambda e^ u\) is given.