ON AN ELLIPTIC EQUATION WITH CONCAVE AND CONVEX NONLINEARITIES

被引:265
作者
BARTSCH, T [1 ]
WILLEM, M [1 ]
机构
[1] UNIV CATHOLIQUE LOUVAIN,INST MATEMAT PURA & APLIQUEE,B-1348 LOUVAIN,BELGIUM
关键词
D O I
10.2307/2161107
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study the semilinear elliptic equation -Delta u=lambda\u\(q-2)u+mu\u\(p-2)u in an open bounded domain Omega subset of R(N) with Dirichlet boundary conditions; here 1 < q < 2 < p < 2*. Using variational methods we show that for lambda > 0 and mu is an element of R arbitrary there exists a sequence (upsilon(k)) of solutions with negative energy converging to 0 as k --> infinity. Moreover, for mu > 0 and lambda arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brezis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.
引用
收藏
页码:3555 / 3561
页数:7
相关论文
共 10 条
[1]   COMBINED EFFECTS OF CONCAVE AND CONVEX NONLINEARITIES IN SOME ELLIPTIC PROBLEMS [J].
AMBROSETTI, A ;
BREZIS, H ;
CERAMI, G .
JOURNAL OF FUNCTIONAL ANALYSIS, 1994, 122 (02) :519-543
[2]  
Ambrosetti A., 1973, Journal of Functional Analysis, V14, P349, DOI 10.1016/0022-1236(73)90051-7
[3]   MULTIPLICITY OF SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT OR WITH A NONSYMMETRIC TERM [J].
AZORERO, JG ;
ALONSO, IP .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1991, 323 (02) :877-895
[4]  
BARTSCH T, 1994, J REINE ANGEW MATH, V451, P149
[5]   INFINITELY MANY SOLUTIONS OF A SYMMETRICAL DIRICHLET PROBLEM [J].
BARTSCH, T .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1993, 20 (10) :1205-1216
[6]  
BARTSCH T, 1993, LECTURE NOTES MATH, V1560
[7]   ON CRITICAL-POINT THEORY FOR INDEFINITE FUNCTIONALS IN THE PRESENCE OF SYMMETRIES [J].
BENCI, V .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1982, 274 (02) :533-572
[8]  
Rabinowitz P.H, 1986, CBMS REG C SER MATH, V65
[9]  
WILLEM M, 1983, TRABALHO MAT, V199
[10]  
WILLEM M, 1992, RECHERCHES MATH, V19