### Abstract

In connection with the maximizing problem for the functional R(u) = ∥u∥_{Lq
} ∥▽u∥_{Lp
} in W_{0}
^{1,p}(Ω)β{0}, we consider the equation -div(|▽u|^{p - 2} ▽u(x)) = |u|^{q - 2} u(x), x ε{lunate} Ω, 1 < p, q < ∞, p ≠ q, (E) u(x) = 0, x ε{lunate} ∂Ω. It is shown that for the case q < p^{*} (p^{*} = ∞ if p ≧ N, and p^{*} = Np (N - p) if p < N), (E) has always a nonnegative nontrivial solution belonging to W_{0}
^{1,p}(Ω) ∩ L^{∞}(Ω), and for the case p < N and q > p^{*} (resp. q = p^{*}), (E) has no nontrivial (resp. nonnegative nontrivial) solution belonging to the class P = {u ε{lunate} W_{0}
^{1,p}(Ω) ∩ L^{q}(Ω); x_{i}|u|^{q - 2}u ε{lunate} L^{ p (p - 1)}(Ω), i = 1, 2, ..., N} ⊂ W_{0}
^{1,p}(Ω) ∩ ^{∞}(Ω), provided that Ω is star shaped. The crucial point of the proof of our result is to obtain an L^{∞}-estimate of weak solutions and to verify a certain "Pohozaev-type inequality" for weak solutions belonging to P.

Original language | English |
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Pages (from-to) | 140-159 |

Number of pages | 20 |

Journal | Journal of Functional Analysis |

Volume | 76 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1988 |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis