Semiclassical states for a class of nonlinear elliptic field equations

In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation

−ħ²Δv+V(x)v−ħ^pΔ_pv+W^′(v)=0,

where v : $\mathbb{R} $ ^N→ $\mathbb{R} $ ^N+1, N≥3, p>N, ħ>0, the potential V is positive and W is an appropriate singular function. Assuming that V has a mountain pass geometry, determined by the existence of two strict local minima, then for small values of the parameter ħ we are able to find a solution which concentrates at the related mountain pass point of V in the semiclassical limit (i.e., as ħ→0⁺). The proof of our results uses a variational approach and the solutions are captured as critical points for the associated energy functional.