THE FREDHOLM ALTERNATIVE AT THE FIRST EIGENVALUE FOR THE ONE-DIMENSIONAL P-LAPLACIAN

In this work we study the range of the operator u bar right arrow (/u' /(p - 2)u')' + lambda(1) /u/(p - 2)u, u(0) = u(T) = 0, p > 1. We prove that all functions h is an element of C-1 [0, T] satisfying integral( 0)(T) h(t) sin(p) pi(p)t/T dt = 0 lie in the range, but that if p not equivalent to 2 and h = 0, the solution set is bounded. Here sinp pi(p )t/T a first eigenfunction associated to lambda(1). (C) Academie des S ciences/Elsevier, Paris.