### Abstract

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 H^{1} ℝ ^{N} as 0.

Original language | English (US) |
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Pages (from-to) | 231-249 |

Number of pages | 19 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2007 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics