An abstract critical point theorem and applications.

*(Chinese)*Zbl 0633.58010In the paper of J. Q. Liu and Li Shujie [Kexue Tongbao 17, 1025-1027 (1984)], a local link was defined, and was applied to study the existence of nontrivial critical points. In this paper, the following theorem is proved to extend the previous result as well as the well known theorem due to V. Benci and P. Rabinowitz:

Suppose that \(X=X_ 1\oplus X_ 2\) is a Banach space, with dim \(X_ 2<\infty\), and that \(f\in C(X,R^ 1)\) satisfies the Palais Smale condition. If furthermore we assume that (1) \(\exists \rho >0\), \(\beta >\alpha\), such that \(f|_{\partial B_{\rho}\cap X_ 1}\geq \beta\). (2) \(\exists e\in \partial B_ R\cap X_ 1\), \(R>\rho\) such that \(f|_{T_ R}\leq \alpha\), where \(T_ R=\{x=te=x_ 2|\) \(0\leq t\leq 1\), \(x_ 2\in X_ 2\), \(\| x\| =R\}\). (3) \(\exists\) an odd continuous map \(\phi\) : \(B_ R\cap X_ 2\to f_{\beta}\), the level set, such that \(\phi |_{\partial B_ R\cap X_ 2}=id\). Then f possesses at least a critical value \(c\geq \beta.\)

This theorem is applied to study the BVP of an elliptic equation with subcritical Sobolev exponent. Infinitely many solutions are obtained, the assumptions are similar to those given by Bahri, Berestycki, and by P. Rabinowitz [Trans. Am. Math. Soc. 272, 753-769 (1982; Zbl 0589.35004)].

Suppose that \(X=X_ 1\oplus X_ 2\) is a Banach space, with dim \(X_ 2<\infty\), and that \(f\in C(X,R^ 1)\) satisfies the Palais Smale condition. If furthermore we assume that (1) \(\exists \rho >0\), \(\beta >\alpha\), such that \(f|_{\partial B_{\rho}\cap X_ 1}\geq \beta\). (2) \(\exists e\in \partial B_ R\cap X_ 1\), \(R>\rho\) such that \(f|_{T_ R}\leq \alpha\), where \(T_ R=\{x=te=x_ 2|\) \(0\leq t\leq 1\), \(x_ 2\in X_ 2\), \(\| x\| =R\}\). (3) \(\exists\) an odd continuous map \(\phi\) : \(B_ R\cap X_ 2\to f_{\beta}\), the level set, such that \(\phi |_{\partial B_ R\cap X_ 2}=id\). Then f possesses at least a critical value \(c\geq \beta.\)

This theorem is applied to study the BVP of an elliptic equation with subcritical Sobolev exponent. Infinitely many solutions are obtained, the assumptions are similar to those given by Bahri, Berestycki, and by P. Rabinowitz [Trans. Am. Math. Soc. 272, 753-769 (1982; Zbl 0589.35004)].

Reviewer: K.Chang

##### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

47J05 | Equations involving nonlinear operators (general) |

35J60 | Nonlinear elliptic equations |