PERTURBED NEUMANN PROBLEMS WITH MANY SOLUTIONS

Given f, g : [0, infinity) -> two continuous nonlinearities with f (0) = g (0) = 0 and f having a suitable oscillatory behavior at zero or at infinity, we prove by a direct method that for every k is an element of N there exists epsilon(k) > 0 such that the problem {-Delta(p)u + alpha(x) u(p-1) = f(u) + epsilon g(u) in Omega, partial derivative u/partial derivative n = 0 on partial derivative Omega, has at least k distinct nonnegative weak solutions in W-1,W-p(Omega) whenever vertical bar epsilon vertical bar <= epsilon(k). We also give various W-1,W-p- and L-infinity-estimates of the solutions. No growth assumption on g is needed, and alpha is an element of L-infinity(Omega) may be sign-changing or even negative depending on the rate of the oscillation of f.