Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation - ε ² Δu + V(x)u = P(x)|u|^{{p - 2}} u + k(x)|u|^{{2 - 2}} u & text for, x ∈ ℝ ^N {u(x) 0} & text{as}, {|x| → ∞ where N ≥ 3, 2*N) Px (N-2) is the Sobolev critical exponent, p ∈ (2, 2*), P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that ≤ Em ; for any m ∈ N, it has m pairs of solutions if ≤ E_m; and suppose there exists an orthogonal involution τ ℝ ^N to ℕ ^N such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that E ≤ E, where E and E_m are sufficiently small positive numbers. Moreover, these solutions u to 0 in H¹ ℝ ^N as 0.