Curves of sign-changing solutions for semilinear equations
Positive and sign-changing solution curves for the Dirichlet problem: \[ \Delta u+\lambda f(u)=0\quad\text{ for}\,\, | x| <R\,,\quad u=0 \quad\text{ for}\,\, | x| =R, \leqno{(1)} \] \(x\in \mathbb{R}^N, \lambda>0\), are studied. Under the condition \(f(u)/u\to\infty\) as \(u\to\infty\), an asymptotic shape of positive solutions to (1) is found for large \(\lambda\). This result is then used to prove non-existence of positive solutions for large \(\lambda\) if \(f\) has no stable roots, extending significantly results of \textit{K.J. Brown}, \textit{A. Castro} and \textit{R. Shivaji} [Differ. Integral Equ. 2, No. 4, 541--545 (1989, Zbl 0736.35039)] (positive solutions to (1) are always radially symmetric). Similar asymptotic results are also found for radially symmetric sign-changing solution. Using the non-existence results of radial solutions the author study symmetry-breaking off a branch of positive solutions to (1). The main result in this direction is that the radial symmetry is lost forward at some \(\lambda=\overline\lambda\) and that a family of symmetry breaking-solutions bifurcates at \(\lambda=\overline\lambda\). Some refined results for ODE are provided as well.
scientific article; zbMATH DE number 1836616
scientific article