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The Mountain Pass Theorem is an important result in mathematics concerning funct

April 11, 2024

The Mountain Pass Theorem is an important result in mathematics concerning functional analysis and variational problems. This theorem is typically applied in nonlinear functional analysis and optimization problems. Originally proved by Ambrosetti and Rabinowitz in the 1970s.
The Mountain Pass Theorem is often used to address existence problems, demonstrating the existence of solutions to certain nonlinear equations or variational problems. The core idea of the theorem involves constructing a path, often referred to as a “mountain pass,” connecting the starting point and the endpoint, and locating a critical point, also known as a “mountain peak,” on this path, thereby proving the existence of solutions to the original problem.
Specifically, the Mountain Pass Theorem can be stated as follows:
Let

X be a Banach space,

:



J:X→R a continuously differentiable functional satisfying the following conditions:

(
0
)
=
0
J(0)=0;
There exists a constant

>
0
c>0 and a function

:

+


γ:R+→R such that for all



=

>
0
∥u∥=r>0,

(

)


J(u)≥c and


(

)
(

)
<


(

)
J′(u)(u)<−γ(r).
This theorem asserts the existence of a critical point of

J at a nonzero point in

X, commonly referred to as the mountain pass point.

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