New Quantitative Deformation Lemma and New Mountain Pass Theorem

Abstract

In this paper, we obtain a new quantitative deformation Lemma so that we can obtain more critical points, especially for supinf critical value $c_1$, $x=\varphi^{-1}(c_1)$ is a new critical point. For $infmax$ critical value $c_2$, we can obtain two new critical points $x = 0$ (valley point) and $x = e$(peak point) ,comparing with Willem's variant of the mountain pass theorem of Ambrosetti-Rabinowitz,in which $\varphi(e)\leq\varphi(0)<c_2$, but in our new mountain pass theorem, $ \varphi(e)=c_2$.