Singular perturbation problems for nonlinear elliptic equations in degenerate settings

Here N ≥ 1, g(s) ∈ C(R,R) is a function with a subcritical growth, V (x) ∈ C(R ,R) is a positive function and 0 < ε 1. Among solutions of (0.1)ε, we are interested in concentrating families (uε) of solutions, which have the following behavior: (i) uε(x) has a local maximum at xε ∈ R and xε converges to some x0 ∈ R as ε → 0. (ii) rescaled function vε(y) = uε(εy + xε) converges as ε → 0 to a solution ω(y) ∈ H(R ) of the limit equation: −∆ω + V (x0)ω = g(ω), ω > 0 in R , ω ∈ H(R ). (0.2)