### Abstract

Let Ω ⊂ R^{N} be a bounded open set with smooth boundary and p = 2N/(N -2) be the critical Sobolev exponent. In this note we extend the results of [10] and [21] concerning nodal solutions (i.e., a solution which changes sign) for the Dirichlet problem: -Δu = |u|^{P-2} u + λu on Ω and u = 0 on ∂Ω, when N ≥ 6 and λ ∈ (0, λ_{1}) with λ_{1} the first eigenvalue of -Δ. in H^{1}_{0}(Ω). Similarly, for the problem -Δu = |u|^{P-2} u + λ|u|^{q-2} u on Ω and u = 0 on ∂Ω we obtain a nodal solution when λ > 0, (N + 2)/(N-2) < q < 2N/(N-2) for N = 3, 4, 5 and 2 < q < 2N/(N-2) for N ≥ 6.

Original language | English (US) |
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Pages (from-to) | 25-42 |

Number of pages | 18 |

Journal | Differential and Integral Equations |

Volume | 5 |

Issue number | 1 |

State | Published - Jan 1 1992 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

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## Cite this

Tarantello, G., & Brezis, H. (1992). Nodal solutions of semilinear elliptic equations with critical exponent.

*Differential and Integral Equations*,*5*(1), 25-42.