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

We consider positive solutions of the Dirichlet problemΔu+λ(u+sinu)=0,x∈B,u=0forx∈∂B, where B   is unit ball in RnRn, λ   is a positive parameter. Let λ1λ1 denote the principal eigenvalue of the Laplacian on B   with zero boundary conditions. We show that for 1⩽n⩽51⩽n⩽5 the problem has infinitely many positive solutions at λ=λ1λ=λ1, while for n⩾6n⩾6 the problem has at most finitely many solutions at any λ.