### 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 |
---|---|

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 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

**Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations.** / Otani, Mitsuharu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations

AU - Otani, Mitsuharu

PY - 1988

Y1 - 1988

N2 - In connection with the maximizing problem for the functional R(u) = ∥u∥Lq ∥▽u∥Lp in W0 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 W0 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} W0 1,p(Ω) ∩ Lq(Ω); xi|u|q - 2u ε{lunate} L p (p - 1)(Ω), i = 1, 2, ..., N} ⊂ W0 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.

AB - In connection with the maximizing problem for the functional R(u) = ∥u∥Lq ∥▽u∥Lp in W0 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 W0 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} W0 1,p(Ω) ∩ Lq(Ω); xi|u|q - 2u ε{lunate} L p (p - 1)(Ω), i = 1, 2, ..., N} ⊂ W0 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.

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U2 - 10.1016/0022-1236(88)90053-5

DO - 10.1016/0022-1236(88)90053-5

M3 - Article

VL - 76

SP - 140

EP - 159

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -