Pierpaolo Esposito

Concentrating solutions for the Hénon equation in ℝ2

We consider the boundary value problem Δu+⋎x⋎2αup=0, α>0, in the unit ballB with homogeneous Dirichlet boundary condition andp a large exponent. We find a condition which ensures the existence of a positive solutionup concentrating outside the origin atk symmetric points asp goes to +∞. The same techniques lead also to a more general result on general domains. In particular, we find that concentration at the origin is always possible, provided α⊄IN.

COMPACTNESS OF A NONLINEAR EIGENVALUE PROBLEM WITH A SINGULAR NONLINEARITY

We study the Dirichlet boundary value problem on a bounded domain Ω ⊂ ℝN. For 2 ≤ N ≤ 7, we characterize compactness for solutions sequence in terms of spectral informations. As a by-product, we give an uniqueness result for λ close to 0 and λ* in the class of all solutions with finite Morse index, λ* being the extremal value associated to the nonlinear eigenvalue problem.

Nondegeneracy of entire solutions of a singular Liouvillle equation

We establish nondegeneracy of the explicit family of finite mass solutions of the Liouvillle equation with a singular source of integer multiplicity, in the sense that all bounded elements in the kernel of the linearization correspond to variations along the parameters of the family.

Two-dimensional Euler flows with concentrated vorticities

For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields w e for a fluid in a bounded region Ω, with concentrated vorticities ω e , for e > 0 small. More precisely, given a positive integer α and a sufficiently small complex number a, we find a family of stream functions ψ e which solve the Liouville equation with Dirac mass source, Δψ e + e 2 e ψe = 4παδ pa,e in Ω, ψ e = 0 on ∂Ω, for a suitable point p = p a,e ∈ Ω. The vorticities ω e := -Δψ e concentrate in the sense that α+1 ω e + 4παδ pa.e — 8π α+1/Σ/j=1 δ pa,e + a j ⇀ 0 as e → 0, j=1 where the satellites a 1 , ... , a α+1 denote the complex (α + 1)-roots of a. The point p a,e lies close to a zero point of a vector field explicitly built upon derivatives of order < α + 1 of the regular part of Greens function of the domain.

Linear instability of entire solutions for a class of non-autonomous elliptic equations

We study the effect of the potential | y | α on the stability of entire solutions for elliptic equations on ℝ N , N ≥ 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a by-product of our results, we will improve the known results about the regularity of such solutions.

Blowing-up solutions for the Yamabe equation

Let ðM; gÞ be a smooth, compact Riemannian manifold of dimension N b 3. We consider the almost critical problem

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension \(N \geq 3\) we are interested in the critical equation

Mathematical analysis of partial differential equations modeling electrostatic MEMS

\Delta_g{u} + \left(\frac{N-2}{4(N-1)}S_{g}+\varepsilon\mathrm{h}\right)u\;=\;u^{\frac{N+2}{N-2}}\qquad \mathrm{in}\;M,\quad u>0\quad \mathrm{in}\;M