### Some remarks on the critical point theory

#### 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.

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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