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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mountain impasse theorem and spectrum of semilinear elliptic problems


Author: Kyril Tintarev
Journal: Trans. Amer. Math. Soc. 336 (1993), 621-629
MSC: Primary 35J60; Secondary 35B45, 35J20, 58E05
MathSciNet review: 1097172
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Abstract: This paper studies a minimax problem for functionals in Hilbert space in the form of $ G(u) = \frac{1} {2}\rho \vert\vert u\vert{\vert^2} - g(u)$, where $ g(u)$ is Fréchet differentiable with weakly continuous derivative. If $ G$ has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence $ {u_j} \in H$ such that

$\displaystyle g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,\vert\vert{u_j}\vert\vert \to \infty ,$

and $ G({u_j})$ approximates the minimax value from above. If

$\displaystyle \gamma (t) = \mathop {\sup }\limits_{\vert\vert u\vert{\vert^2} = t} \;g(u)$

and

$\displaystyle {J_0} = \left( {2\mathop {\inf }\limits_{{t_2} > {t_1} > 0} \frac... ...{t_1} > 0} \frac{{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right),$

then $ g\prime (u) = \rho u$ has a nonzero solution $ u$ for a dense subset of $ \rho \in {J_0}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1097172-6
PII: S 0002-9947(1993)1097172-6
Article copyright: © Copyright 1993 American Mathematical Society