The index of a critical point for nonlinear elliptic operators with strong coefficient growth

抄録

This paper is devoted to the computation of the index of a critical point for nonlinear operators with strong coefficient growth. These operators are associated with boundary value problems of the type \begin{center} \displaystyle ∑_{|α|=1}\mathscr{D}<SUP>α</SUP>{ρ<SUP>2</SUP>(u)\mathscr{D}<SUP>α</SUP>u+a<SUB>α</SUB>(x, \mathscr{D}<SUP>1</SUP>u)}=λ a<SUB>0</SUB>(x, u, \mathscr{D}<SUP>1</SUP>u), \ x∈Ω, u(x)=0, \ x∈∂Ω, \end{center} where Ω=\bm{R}<SUP>n</SUP> is open, bounded and such that ∂Ω∈ \bm{C}<SUP>2</SUP>, while ρ:\bm{R}→ \bm{R}<SUB>+</SUB> can have exponential growth. An index formula is given for such densely defined operators acting from the Sobolev space W<SUB>0</SUB><SUP>1, m</SUP>(Ω) into its dual space. We consider different sets of assumptions for m>2 (the case of a real Banach space) and m=2 (the case of a real Hilbert space). The computation of the index is important for various problems concerning nonlinear equations: solvability, estimates for the number of solutions, branching of solutions, etc. The results of this paper are based upon recent results of the authors involving the computation of the index of a critical point for densely defined abstract operators of type (S<SUB>+</SUB>). The latter are based in turn upon a new degree theory for densely defined (S<SUB>+</SUB>)-mappings, which has also been developed by the authors in a recent paper. Applications of the index formula to the relevant bifurcation problems are also included.

収録刊行物

• Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 52(1), 109-137, 2000-01

The Mathematical Society of Japan

各種コード

• NII論文ID(NAID)
10004480038
• NII書誌ID(NCID)
AA0070177X
• 本文言語コード
ENG
• 資料種別
ART
• ISSN
00255645
• NDL 記事登録ID
4974656
• NDL 雑誌分類
ZM31(科学技術--数学)
• NDL 請求記号
Z53-A209
• データ提供元
CJP書誌  NDL  J-STAGE

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