29 Gennaio, 2016 11:00 oclock

Sezione di Analisi

On nonlocal critical equations

Giovanni Molica Bisci, Università degli Studi Mediterranea

Aula seminari III piano

A very interesting area of nonlinear analysis lies in the study of elliptic equations involving fractional operators. Recently, a great attention has been focused on these problems, both for the pure mathematical research and in view of concrete real-world ap- plications. Indeed, this type of operators appear in a quite natural way in different contexts, such as the description of several physical phenomena. In particular, nonlocal critical equations are relevant for their relations with problems arising in differential geometry and in physics, where a lack of compactness occurs (see [4, Part III]). Motivated by this wide interest in the current literature, also in connection with the celebrated Brezis-Nirenberg problem (see [1]), in the first part of the talk we will describe the state of the art for nonlocal critical problems involving the fractional Laplacian operator or its generalizations. Successively, some recent existence and multiplicity results will be discussed [2, 3, 5]. In conclusion, certain open problems will be briefly presented.

References:
[1] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equa- tions involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
[2] A. Fiscella, G. Molica Bisci and R. Servadei, Bifurcation and multi- plicity results for critical nonlocal fractional problems, Bull. Sci. Math. 140 (2016), 14–35.
[3] J. Mawhin and G. Molica Bisci, A Brezis-Nirenberg type result for a non- local fractional operator, preprint 2016.
[4] G. Molica Bisci, V. Ra ?dulescu, and R. Servadei, Variational Meth- ods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, No. 142, Cambridge University Press, Cambridge, 2016.
[5] G. Molica Bisci and R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, Adv. Differential Equations 20 (2015), 635–660.