### 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) |
---|---|

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 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**Solutions of perturbed Schrödinger equations with critical nonlinearity.** / Ding, Yanheng; Lin, Fang-Hua.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 30, no. 2, pp. 231-249. https://doi.org/10.1007/s00526-007-0091-z

}

TY - JOUR

T1 - Solutions of perturbed Schrödinger equations with critical nonlinearity

AU - Ding, Yanheng

AU - Lin, Fang-Hua

PY - 2007/10

Y1 - 2007/10

N2 - 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 ≤ Em; 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 Em are sufficiently small positive numbers. Moreover, these solutions u to 0 in H1 ℝ N as 0.

AB - 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 ≤ Em; 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 Em are sufficiently small positive numbers. Moreover, these solutions u to 0 in H1 ℝ N as 0.

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U2 - 10.1007/s00526-007-0091-z

DO - 10.1007/s00526-007-0091-z

M3 - Article

VL - 30

SP - 231

EP - 249

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

ER -