characterization of solutions having finite morse index for some nonlinear pde with supercritical growth

in this paper, we study solutions of the equation −δu = (u⁺)p in ω ⊂ rⁿ, where n+2/n−2≤ p < p_c (n); see (3) for the definition of p_c (n). we first classify all solutions (not necessarily bounded) having finite morse index for ω = rⁿ or ω = rⁿ₊ with dirichlet boundary condition. when ω is a regular bounded domain of rⁿ , for a family of solutions with dirichlet boundary condition (un), we prove that ‖un‖_l^∞(ω) is bounded if and only if the morse index sequence, i(un), of un is bounded. finally, we prove the same result for other nonlinearities f (x, u) which have similar growth respectively at±∞.