A new critical point theorem with an application to a scalar equation without periodicity and symmetry

In this paper we first establish a new critical point theorem which may produce a relationship between the sign-changing critical point, the Morse index, the upper bound of the critical value, the value of the type inf k ≤ dim ⁡ Y < ∞ Y ⊂ E , ⁡ sup Y ⁡ G and the number of nodal domains. An unified approach is introduced for these composite properties. As one application of the abstract theorem, we study the existence of infinitely many sign-changing bound states for the following nonlinear stationary Schrödinger equation: − Δ u + a ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 2 , p > 2 and p < 2 N / ( N − 2 ) : = 2 ⁎ when N > 2 ; the potential a ( x ) ∈ C ( R N , R ) is a nonnegative function verifying suitable decay assumptions, but being not symmetric or periodic. We obtain the information on the precise estimates of the energies, the Morse indices and the number of nodal domains for these sign-changing bound states. We believe that the abstract theorem in this article will have more applications.