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Infinitely many solutions for a class of Dirichlet quasilinear elliptic systems. (English) Zbl 1253.34026

From the introduction: The aim of this paper is to investigate the existence of infinitely many classical solutions for the following Dirichlet quasilinear elliptic system

-(p i -1)|u i ' (x)| p i -2 u i '' (x)=λF u i (x,u 1 ,,u n )h i (x,u i ' ),x(a,b),
u i (a)=u i (b)=0,for1in,

where p i >1 for 1in, λ is a positive parameter, a,b with a<b, h i :[a,b]×[0,+) is a bounded and continuous function with m i :=inf (x,t)[a,b]× h i (x,t)>0 for 1in, F:[a,b]× n is a function such that the mapping (t 1 ,t 2 ,,t n )F(x,t 1 ,t 2 ,,t n ) is in C 1 in n for all x[a,b], F u i is continuous in [a,b]× n for 1in, where F u i denotes the partial derivative of F with respect to u i , and

sup |(t 1 ,,t n )|M |F u i (x,t 1 ,,t n )|L 1 ([a,b])

for all M>0 and all 1in.

MSC:
34B15Nonlinear boundary value problems for ODE
58E50Applications of variational methods in infinite-dimensional spaces
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