Bifurcation for a semilinear elliptic equation on \({\mathbb{R}}^ N\) with radially symmetric coefficients (Q913083)

The author considers the semilinear eigenvalue problem of the type \[ - \Delta u-q(x)| u|^{\alpha}u=\lambda u\quad on\quad {\mathbb{R}}^ N, \] where q is a radially symmetric function, \(N\geq 2\) and \(\sigma\) is a positive constant. In particular, he studies the existence of solutions (u,\(\lambda)\) with \(\lambda <0\) and \(u\neq 0\) and the behavior of such solutions as \(\lambda\) approaches 0. Under suitable hypotheses on q and \(\sigma\), he proves that 0 is a bifurcation point in \(H^ 1\), \(H^ 2\) and in \(L^ p\) for \(p\in [2,+\infty]\).