ABSTRACT
Lu = [– |u | q – 2 u + u + g(x,u) for x , x=0 on . (1.10) ( Note, may be the empty set.) Here, Lu is given by (1.8), 2¡q¡N , > j where j is the j-th eigenvalue given in (VL – 2) above, and g(x, s) meets the following conditions:
(g-1) the Caratheodory conditions: the map x g(x, s) is measurable for all s R, and the map s g(x, s) is continuous for a.e. x ;
(g-2) given > 0, b ( ) where q= q-1 such that |g(x, s)| |s|q-1 + b (x) s R and for a.e. x ;
(g-3) sg(x, s) 0 for s R and a.e. x ; (g-4) g(x, s) = –g(x, –s) s R and for a.e. x . By a weak solution of the boundary value problem (1.10), we shall mean
: u H1p, ( , ) such that
L(u,v) = [–|u|q-2u + u + g(x,u)]v v H1p, ( , ) (1.11) where L(u,v) is defined in (1.6). The theorem we shall establish concerning (1.10) is the following:
Theorem 1. Assume ( , ) is a Simple V L -region, that > j, that (g1)-(g-4) holds, and that 2¡q¡N . Then the boundary value problem (1.10) possesses at least j distinct pairs of nontrivial weak solutions, i.e., 1,..., j H1p, ( , ) such that
L( i,v) = –|
i|q-2 i + i + g(x, i) v v H1p, ( , ) (1.12) for i = 1,...,j with i 0 and i ± k for i k, i, k = l , . . , j .
