Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball
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| 024 | 7 | 0 | |a 10.1007/s10231-014-0404-8 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10231-014-0404-8 | ||
| 100 | 1 | |a Miyamoto |D Yasuhito |u Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914, Tokyo, Japan |4 aut | |
| 245 | 1 | 0 | |a Classification of bifurcation diagrams for elliptic equations with exponential growth in a ball |h [Elektronische Daten] |c [Yasuhito Miyamoto] |
| 520 | 3 | |a Let $$B\subset \mathbb {R}^N$$ B ⊂ R N , $$N\ge 3$$ N ≥ 3 , be the unit ball. We study the global bifurcation diagram of the solutions of $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u+\lambda f(u)=0 &{}\quad \text {in}\ B,\\ u=0 &{} \quad \text {on}\ \partial B,\\ u>0 &{} \quad \text {in}\ B, \end{array}\right. } \end{aligned}$$ Δ u + λ f ( u ) = 0 in B , u = 0 on ∂ B , u > 0 in B , where $$f(u)=e^u+g(u)$$ f ( u ) = e u + g ( u ) and $$g(u)$$ g ( u ) is a lower order term. The solution set is a curve $$\mathcal {C}$$ C parametrized by the $$L^{\infty }$$ L ∞ -norm of the solution. We show that this problem has the singular solution $$(\lambda ^*,u^*)$$ ( λ ∗ , u ∗ ) and that the curve $$\mathcal {C}$$ C has infinitely many turning points around $$\lambda ^*$$ λ ∗ if $$3\le N\le 9$$ 3 ≤ N ≤ 9 . We show that under a certain condition on $$g$$ g , the curve $$\mathcal {C}$$ C has no turning point if $$N\ge 10$$ N ≥ 10 . We also study the Morse index of $$u^*$$ u ∗ . | |
| 540 | |a Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 | ||
| 690 | 7 | |a Bifurcation diagram |2 nationallicence | |
| 690 | 7 | |a Exponential growth |2 nationallicence | |
| 690 | 7 | |a Intersection number |2 nationallicence | |
| 690 | 7 | |a Elliptic Dirichlet problem |2 nationallicence | |
| 690 | 7 | |a Singular solution |2 nationallicence | |
| 773 | 0 | |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 931-952 |x 0373-3114 |q 194:4<931 |1 2015 |2 194 |o 10231 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10231-014-0404-8 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10231-014-0404-8 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Miyamoto |D Yasuhito |u Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914, Tokyo, Japan |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 931-952 |x 0373-3114 |q 194:4<931 |1 2015 |2 194 |o 10231 | ||