Solutions of the mixed boundary problem and their Morse indices

In this paper, we study the solutions of the following mixed boundary problem {−Δu=f(x,u)inΩ,u=0onΣ1,∂u∂ν=0onΣ2. Under some assumptions on the nonlinearity, we will show that the boundedness of solutions is equivalent to the finiteness of their Morse indices. We extend the result of A. Bahri, P.L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Commun. Pure Appl. Anal. 45 (1992) 1205–1215, from the Dirichlet boundary condition to the mixed boundary condition.