A VANISHING THEOREM FOR NONLINEAR ELLIPTI C-EQUATIONS ON COMPACT RIEMANNIAN-MANIFOLDS

Let (M, g) be a compact n-dimensional Riemannian manifold without boun dary, Delta(g) the Laplacian of (M, g) and lambda(1) its first nonzero eigenvalue. We assume that the Ricci tenser of g is nonnegative and l et R be its largest lower bound If q is an element of]1, n + 2/n-2] an d lambda > 0 we give the value of some real number C = C (n, q, R, lam bda(1)) greater than or equal to 0 such that any positive solution of (E) Delta(g) u + u(q) = lambda u on M is constant as soon as lambda le ss than or equal to C.