614 research outputs found
On mountain pass theorem and its application to periodic solutions of some nonlinear discrete systems
We obtain a new quantitative deformation lemma, and then gain a new mountain
pass theorem. More precisely, the new mountain pass theorem is independent of
the functional value on the boundary of the mountain, which improves the well
known results (\cite{AR,PS1,PS2,Qi,Wil}). Moreover, by our new mountain pass
theorem, new existence of nontrivial periodic solutions for some nonlinear
second-order discrete systems is obtained, which greatly improves the result in
\cite{Z04}.Comment: 11 page
Blowup solutions of Grushin's operator
In this note, we consider the blowup phenomenon of Grushin's operator. By
using the knowledge of probability, we first get expression of heat kernel of
Grushin's operator. Then by using the properties of heat kernel and suitable
auxiliary function, we get that the solutions will blow up in finite time.Comment:
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