Ricerc@Sapienza

We study the Dirichlet problem on a bounded convex domain of RN, with zero boundary data, for truncated Laplacians Pk±, which are degenerate elliptic operators, for k< N, defined by the upper and respectively lower partial sum of k eigenvalues of the Hessian matrix. We establish a necessary and sufficient condition (Theorem 1) in terms of the “flatness” of domains for existence of a solution for general inhomogeneous term. This result, in particular, shows that the strict convexity of the domain is sufficient for the solvability of the Dirichlet problem.

We compute the Morse index of 1-spike solutions of the semilinear elliptic problem
()
where is a smooth bounded domain and is sufficiently large.

When Ω is convex, our result, combined with the characterization in [21], a result in [40] and with recent uniform estimates in [37], gives the uniqueness of the solution to (), for p large. This proves, in dimension two and for p large, a longstanding conjecture.