Existence and concentration phenomena for a class of indefinite variational problems with critical growth

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In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems -Δu+V(x)u=A(εx)f(u), x ∈ ⁿ, (P)ε where N ≥ 2, ε>0, A:ⁿ→ is a continuous function that satisfies 0<ₓ∈ⁿA(x)≤|ₓ|→₊∞A(x)<ₓ∈ⁿA(x)=A(0),(A) f:→ is a continuous function having critical growth, V:ⁿ→ is a continuous and ⁿ--periodic function with 0∉σ(Δ+V). By using variational methods, we prove the existence of solution for ε small enough. After that, we show that the maximum points of the solutions concentrate around of a maximum point of A.