Abstract
We investigate the degenerate bi-harmonic equation
Delta(2)(m)u = f (x, u) in Omega, u = Delta u = 0 on partial derivative Omega,
with m >= 2, and also the degenerate tri-harmonic equation:
-Delta(3)(m)u = f (x, u) in Omega, u = partial derivative u/partial derivative v = partial derivative(2)u/partial derivative v(2) = 0 on partial derivative Omega,
where Omega subset of R-N is a bounded domain with smooth boundary N > 4 or N > 6 respectively, and f is an element of C-1(Omega x R) satisfies suitable m-superlinear and subcritical growth conditions. Our main purpose is to establish L-P and L-infinity explicit bounds for weak solutions via the Morse index. Our results extend previous explicit estimates obtained in [1]-[4].