Ricerc@Sapienza

Let Omega be a bounded smooth domain in R-N which contains a ball of radius R centered at the origin, N >= 3. Under suitable symmetry assumptions, for each delta is an element of (0, R), we establish the existence of a sequence (u(m,delta)) of nodal solutions to the critical problem-Delta u = vertical bar u vertical bar(2*-2)u in Omega(delta) := {x is an element of Omega : vertical bar x vertical bar > delta} on partial derivative Omega(delta),where 2* := 2N/N-2 is the critical Sobolev exponent.

We study a critical Neumann problem in an unbounded cone Σ_ω:={tx:x∈ω and t>0}, where ω is an open connected subset of the unit sphere S^N−1 in R^N with smooth boundary, N≥3 and 2∗:=2N/N−2. We assume that some local convexity condition at the boundary of the cone is satisfied. If ω is symmetric with respect to the north pole of S^N−1, we establish the existence of a nonradial sign-changing solution.