Some remarks on the critical point theory

Chong Li

DOI: http://dx.doi.org/10.12775/TMNA.2007.027

Abstract


In this paper we discuss some problems about critical point
theory. In the first part of the paper we study existence and
multiplicity results of semilinear second order elliptic equation:
$$
\begin{cases}
-\Delta u=f(x,u) &\text{for } x\in \Omega, \\
u=0 &\text{for } x\in \partial \Omega,
\end{cases}
$$
In [Z. Li. Liu and S. J. Li,
< i> Contractibility of level sets of functionals associated with some
elliptic boundary value problems and applications< /i> , NoDEA < b> 10< /b> (2003), 133–170], the authors study the contractibility of level
sets of functionals associated with some elliptic boundary value
problems. In this paper by using Morse theory and minimax method
we give a more precise description of topological construction of
level set of critical value of energy functional for mountain pass
type critical point. It is well known that nondegenerate critical
point is isolated, so if a critical point is not isolated, it must
be a degenerate critical point. In the second part we will give an
example that all the critical points of functional of a class of
oscillating equation with Neumann boundary condition are isolated
and the equation has only constant solutions. Moreover, critical
groups of each critical point of the functional are trivial. The
elliptic sine-Gordon equation originates from the static case of
the hyperbolic sine-Gordon equation modelling the Josephson
junction in superconductivity, which is of contemporary interest
to physicists. The problem is similar to the elliptic sine-Gordon
equation so we believe that it derives from profound physical
backdrop.

Keywords


Mountain pass theorem; order intervals; Morse theory; minimax method

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