NODAL SOLUTIONS OF ELLIPTIC-EQUATIONS WITH CRITICAL EXPONENT

Consider the problem -DELTA-u = \u\p-2 u + lambda-u on OMEGA, u = 0 on partial derivative OMEGA; where OMEGA subset-of R(N) is a bounded open set, lambda is-an-element-of (0, lambda-1), (lambda-1 is the first eigenvalue of -DELTA on OMEGA-under Dirichlet conditions) and p = 2 N/(N - 2) is the critical Sobolev exponent. In this Note we shall prove that for N greater-than-or-equal-to 6, this problem admits a nodal solution (i.e. a solution which changes sign in OMEGA). Similarly for the equation -DELTA-u = \u\p-2 u + lambda \u\q-2 u on OMEGA, u = 0 on partial derivative OMEGA; we obtain the existence of a nodal solution when lambda > 0, q is-an-element-of ((N + 2)/(N - 2), 2N/(N-2)) for N = 3, 4, 5; and q is-an-element-of (2, 2N/(N - 2)) for N greater-than-or-equal-to 6.