On the oscillations of the solution curve for a class of semilinear equations

We consider positive solutions of the Dirichlet problem View the MathML source Turn MathJax on where B is unit ball in Rn, λ is a positive parameter. Let λ1 denote the principal eigenvalue of the Laplacian on B with zero boundary conditions. We show that for 1⩽n⩽5 the problem has infinitely many positive solutions at λ=λ1, while for n⩾6 the problem has at most finitely many solutions at any λ.