A counterexample to the Fredholm alternative for the p-Laplacian

被引:16
作者
Drábek, P
Takác, P
机构
[1] Univ W Bohemia, Dept Math, CZ-30614 Plzen, Czech Republic
[2] Univ Rostock, Fachbereich Math, D-18055 Rostock, Germany
关键词
nonuniqueness and multiplicity of solutions; resonance for the p-Laplacian; nonlinear Fredholm alternative;
D O I
10.1090/S0002-9939-99-05195-3
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional p-Laplacian with 1 < p < infinity is considered: (*) -(\u'\(p-2)u')' - lambda\u\(p-2)u = f(x) for 0 < x < T; u(0) = u(T) = 0; where f = 1 + h with h is an element of L infinity(0, T) small enough. Solvability properties of Problem (*) with respect to the spectral parameter lambda is an element of R are investigated. We focus our attention on some fundamental differences between the cases p not equal 2 and p = 2. For p not equal 2 we give a counterexample to the classical Fredholm alternative (which is valid for the linear case p = 2).
引用
收藏
页码:1079 / 1087
页数:9
相关论文
共 13 条
[1]  
ANANE A, 1987, CR ACAD SCI I-MATH, V305, P725
[2]   The range of the p-Laplacian [J].
Binding, PA ;
Drabek, P ;
Huang, YX .
APPLIED MATHEMATICS LETTERS, 1997, 10 (06) :77-82
[3]   On the Fredholm alternative for the p-Laplacian [J].
Binding, PA ;
Drabek, P ;
Huang, YX .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1997, 125 (12) :3555-3559
[4]  
Deimling K., 1985, NONLINEAR FUNCTIONAL, DOI DOI 10.1007/978-3-662-00547-7
[5]   A HOMOTOPIC DEFORMATION ALONG P OF A LERAY-SCHAUDER DEGREE RESULT AND EXISTENCE FOR (/U'/P-2U')'+F(T,U)=O, U(O)=U(T)=O, P-GREATER-THAN-1 [J].
DELPINO, M ;
ELGUETA, M ;
MANASEVICH, R .
JOURNAL OF DIFFERENTIAL EQUATIONS, 1989, 80 (01) :1-13
[6]   MULTIPLE SOLUTIONS FOR THE P-LAPLACIAN UNDER GLOBAL NONRESONANCE [J].
DELPINO, MA ;
MANASEVICH, RF .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1991, 112 (01) :131-138
[7]   EXISTENCE AND MULTIPLICITY OF SOLUTIONS WITH PRESCRIBED PERIOD FOR A 2ND-ORDER QUASI-LINEAR ODE [J].
DELPINO, MA ;
MANASEVICH, RF ;
MURUA, AE .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1992, 18 (01) :79-92
[8]  
DIAZ JI, 1987, CR ACAD SCI I-MATH, V305, P521
[9]  
EARL A., 1955, THEORY ORDINARY DIFF
[10]  
FLECKINGER J, 1997, P C REACT DIFF EQ TR