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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent


Authors: Zongming Guo and Juncheng Wei
Journal: Trans. Amer. Math. Soc. 363 (2011), 4777-4799
MSC (2010): Primary 35B33; Secondary 35B32, 35J61
DOI: https://doi.org/10.1090/S0002-9947-2011-05292-X
Published electronically: March 25, 2011
MathSciNet review: 2806691
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Abstract: We consider the nonlinear eigenvalue problem

$\displaystyle (0.1) \qquad\qquad\qquad \qquad\qquad\left \{\begin{array}{l}-\De... ...; u=0 \;\; \mbox{on $\partial B$}, \end{array} \right.\qquad\qquad\qquad\qquad $

where $ B$ denotes the unit ball in $ \mathbb{R}^N$, $ N \geq 3$, $ \lambda>0$ and $ p>(N+2)/(N-2)$. According to classical bifurcation theory, the point $ (\mu_1,0)$ is a bifurcation point from which emanates an unbounded branch $ \mathscr{C}$ of solutions $ (\lambda, u)$ of (0.1), where $ \mu_1$ is the principal eigenvalue of $ -\Delta$ in $ B$ with Dirichlet boundary data. It is known that there is a unique value $ \lambda=\lambda_* \in (0, \mu_1)$ such that (0.1) has a radial singular solution $ u_* (\vert x\vert)$. Let $ p_c>\frac{N+2}{N-2}$ be the Joseph-Lundgren exponent. We show that the structure of the branch $ \mathscr{C}$ changes for $ p \geq p_c$ and $ (N+2)/(N-2)<p<p_c$. For $ (N+2)/(N-2)<p<p_c$, $ \mathscr{C}$ turns infinitely many times around $ \lambda_*$, which implies that all the singular solutions have infinite Morse index. For $ p \geq p_c$, we show that all solutions (regular or singular) have finite Morse index. For $ N \geq 12$ and $ p>p_c$ large, we show that all solutions (regular or singular) have exactly Morse index one. As a consequence, we prove that any regular solution intersects with the singular solution exactly once and any regular solution exists (and is unique) only when $ \lambda \in (\lambda_{*}, \mu_1)$.


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Additional Information

Zongming Guo
Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453007, People’s Republic of China
Email: gzm@htu.cn

Juncheng Wei
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/S0002-9947-2011-05292-X
Keywords: Branch of positive radial solutions, infinitely many turning points, Morse index, semilinear elliptic problems with super-critical exponent, Bessel function
Received by editor(s): March 31, 2009
Received by editor(s) in revised form: November 5, 2009
Published electronically: March 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society