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.

Semilinear elliptic equations, positive solutions, asymptotic behavior

We consider the semilinear Lane–Emden problem [Equation not available: see fulltext.]where B is the unit ball of RN, N? 2 , centered at the origin and 1 < p< pS, with pS= + ? if N= 2 and pS=N+2N-2 if N? 3. Our main result is to prove that in dimension N= 2 the Morse index of the least energy sign-changing radial solution up of (Ep) is exactly 12 if p is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in RN in any dimension N? 2. © 2016, Springer-Verlag Berlin Heidelberg.