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

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Abstract

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.

Original languageEnglish
Pages (from-to)140-159
Number of pages20
JournalJournal of Functional Analysis
Volume76
Issue number1
DOIs
Publication statusPublished - 1988
Externally publishedYes

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Degenerate Elliptic Equations
Nontrivial Solution
Nonexistence
Weak Solution
Non-negative
Star
Verify
Estimate
Class

ASJC Scopus subject areas

  • Analysis

Cite this

@article{d6852eb747104dcab203775f1e152543,
title = "Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations",
abstract = "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.",
author = "Mitsuharu Otani",
year = "1988",
doi = "10.1016/0022-1236(88)90053-5",
language = "English",
volume = "76",
pages = "140--159",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "1",

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

AU - Otani, Mitsuharu

PY - 1988

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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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