Prescribed number of solutions for a multiple potential well quasilinear Schrödinger equation

We show that the quasilinear Schrödinger equation minus epsilon squared normal upper Delta u plus upper Q left parenthesis x right parenthesis u minus normal upper Delta left parenthesis u squared right parenthesis u equals h left parenthesis u right parenthesis − ϵ 2 Δ u + Q ( x ) u − Δ ( u 2 ) u = h ( u ) -ε ²Δ u + Q(x)u -Δ (u²)u= h(u) in double struck upper R Superscript upper N R N RN has normal script upper P ℘ solutions related to the normal script upper P ℘ global minima of the potential upper Q Q Q . We work with the Nehari manifold associated with the energy functional. We evaluate the energy of the solutions to distinguish them from each other.