` di Parma
Dipartimento di Matematica - Universita
TWO DAYS ON NONLINEAR OPERATORS AND APPLICATIONS
2012, October 5th - 6th
Elliptic problems driven by the fractional Laplacian
Raffaella Servadei (Universit`a della Calabria)
Motivated by the interest shown in the literature for non-local operators of elliptic
type, in some recent papers, joint with Enrico Valdinoci and Alessio Fiscella, we have
studied problems modeled by
(−∆)s u − λu = f (x, u) in Ω
(1)
u=0
in Rn \ Ω ,
where s ∈ (0, 1) is fixed, Ω ⊂ Rn , n > 2s, is open, bounded and with Lipschitz
boundary, λ is a real parameter, (−∆)s is the fractional Laplace operator defined,
up to normalization factors, as
Z
u(x + y) + u(x − y) − 2u(x)
s
dy , x ∈ Rn
−(−∆) u(x) =
n+2s
|y|
n
R
and the function f : Ω × R → R satisfies suitable growth and regularity conditions.
For instance, in the subcritical or critical setting f is modeled by
f (x, t) = |t|q−2 t ,
with 2 < q ≤ 2∗ = 2n/(n − 2s)
(here 2∗ is the fractional critical Sobolev exponent), while in the asymptotically linear
case as a model we can take
f (x, t) = arctan t .
In problem (1) the Dirichlet datum is given in Rn \ Ω and not simply on ∂Ω, consistently with the non-local character of the operator (−∆)s . Problem (1) represents
the non-local counterpart of the following nonlinear elliptic equation
−∆u − λu = f (x, u) in Ω
u=0
on ∂Ω ,
which is widely studied in the literature. Aim of this talk will be to present some new
results which extend the validity of some existence theorems known in the classical
case of the Laplacian to the non-local framework.
1
References
[1] A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for nonlocal elliptic operators, preprint, available at http://www.ma.utexas.edu/mp$\
_$arc-bin/mpa?yn=12-61 .
[2] A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by the fractional Laplacian operator, in preparation.
[3] R. Servadei, The Yamabe equation in a non-local setting, preprint, available
at http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=12-40 .
[4] R. Servadei, A critical fractional Laplace equation in the resonant case,
preprint, available at http://www.ma.utexas.edu/mp$\_$arc-bin/mpa?yn=
12-87 .
[5] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic
operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898.
[6] R. Servadei and E. Valdinoci, Variational methods for non-local operators
of elliptic type, to appear in Discrete Contin. Dyn. Systems.
[7] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional
Laplacian, to appear on Trans. AMS.
[8] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, preprint, http://www.ma.utexas.edu/mp$\
_$arc-bin/mpa?yn=12-41.
[9] R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, preprint, available at http://www.ma.utexas.edu/mp$\
_$arc-bin/mpa?yn=12-58.
[10] R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, preprint, available at
http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-82.
Campus - Parco Area delle Scienze, 53/A - 43124 Parma - Italia
Tel: +39 0521 906900 - Fax: +39 0521 906950
2