Ricerc@Sapienza

Morse index of solutions of semilinear elliptic equations : definition, computation and applications

We consider an elliptic problem of the type where Ω is a bounded Lipschitz domain in ℝ N with a cylindrical symmetry, ν stands for the outer normal and. Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [6, 9, 16, 18].

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.

We consider the Lane-Emden problem in the unit ball B of ℝ^2 centered at the origin with Dirichlet boundary conditions and exponent ∈(1,+∞) of the power nonlinearity. We prove the existence of sign-changing solutions having 2 nodal domains, whose nodal line does not touch ∂ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties.

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.