L∞-Norm Estimates of Weak Solutions via Their Morse Indices for the m-Laplacian Problems

This work is devoted to obtain the L p and the L ∞ -estimates of solutions via their Morse indices to the following m -Laplacian problems - Δ m u = f ( x , u ) in Ω u = 0 , on ∂ Ω , ( 1 ) where Ω ⊂ R N is a bounded domain with smooth boundary, N > m > 2 and f ∈ C ( Ω ¯ × R ) which will be specified later. As far as we know, it seems to be the first time that such explicit estimates are obtained for a nonlinear degenerate problems. So, our main results extend and complement previously L ∞ -estimates results in the literature.