Existence of nontrivial solutions for quasilinear elliptic equations at critical growth

Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents

“…Note that |∇u ε | ≤ |v ε | and |u ε | ≤ |v ε |, we get lim ε→0 u ε X = 0. Secondly, similar to the proof for Theorem 4.1 in [11], we conclude that v ε ∈ L ∞ (R N ) and by [9], we have v ε ∈ C 1,α (B R ). Now let x ε denote the maximum point of v ε in B R and let σ := sup{s > 0 : g(t) < V 0 t for every t ∈ [0, s]}.…”