LUCA BATTAGLIA - PERSONAL WEB PAGE

HOME
CURRICULUM
RESEARCH
STUDENTS
TEACHING
NOTES
LINKS

(per la versione italiana cliccare qui)

Research interests:

My research activity mainly deals with nonlinear elliptic partial differential equations.

In particular, I am interested in:

- Liouville-type equations with exponential nonlinearities in dimension two;
- Moser-Trudinger type inequalities;
- Prescribed curvature problems on surfaces with boundary;
- Problems with nonlinear Neumann boundary conditions;
- Choquard-type equations with non-local nonlinearities.

Scientific publications:

Full list with descriptions: English, Italian.

(27) *New solutions for the Lane-Emden problem in planar domains* (with Isabella Ianni and Angela Pistoia), submitted (arXiv).

(26) *On the critical points of Steklov eigenfunctions* (with Angela Pistoia and Luigi Provenzano), submitted (arXiv).

(25) *A mean field problem approach for the double curvature prescription problem* (with Rafael López-Soriano), __Commun. Contemp. Math.__, accepted (arXiv).

(24) *Prescribing nearly constant curvatures on balls* (with Sergio Cruz-Blázquez, Angela Pistoia), __Proc. Roy. Soc. Edinburgh Sect. A__, accepted (arXiv).

(23) *On the shape of solutions to elliptic equations in possibly non convex domains* (with Fabio De Regibus, Massimo Grossi), __Discrete Contin. Dyn. Syst. Ser. S__ 17 (2024), no. 4, 1588-1598 (arXiv).

(22) *Non uniqueness for the nonlocal Liouville equation in R and applications* (with Matteo Cozzi, Antonio J. Fernández, Angela Pistoia), __SIAM J. Math. Anal.__ 55 (2023), no. 5, 4816-4842 (arXiv).

(21) *Prescribing Gaussian curvature on surfaces with conical singularity and geodesic boundary* (with Aleks Jevnikar, Zhi-An Wang and Wen Yang), __Ann. Mat. Pura Appl.__ (4) 202 (2023), no. 3, 1173-1185 (arXiv).

(20) *A blow-up phenomenon for a non-local Liouville-type equation* (with María Medina and Angela Pistoia), __J. Anal. Math.__ 149 (2023), no. 1, 343-367 (arXiv).

(19) *Large conformal metrics with prescribed Gaussian and geodesic curvatures* (with María Medina and Angela Pistoia), __Calc. Var. Partial Differential Equations__ 60 (2021), no. 1, 39 (arXiv).

(18) *Asymptotic behavior of minimal solutions to -Delta u=lambda f(u) as lambda goes to -infinity* (with Francesca Gladiali and Massimo Grossi), __Discrete Contin. Dyn. Syst.__ 41 (2021), no.2, 681-700 (arXiv).

(17) *A double mean field equation related to a curvature prescription problem* (with Rafael López-Soriano), __J. Diff. Equations__ 269 (2020), no. 4, 2705-2740 (arXiv).

(16) *Non-uniqueness of blowing-up solutions to the Gelfand problem* (with Massimo Grossi and Angela Pistoia), __Calc. Var. Partial Differential Equations__ 58 (2019), no. 5, Paper No. 163, 28 pp. (arXiv).

(15) *Uniform bounds for solutions to elliptic problems on simply connected planar domains*, __Proc. Amer. Math. Soc.__ 147 (2019), no. 10, 4289-4299 (arXiv).

(14) *A general existence result for stationary solutions to the Keller-Segel system*, __Discrete Contin. Dyn. Syst.__ 39 (2019), no. 2, 905-926 (arXiv).

(13) *A unified approach of blow-up phenomena for two-dimensional singular Liouville systems* (with Angela Pistoia), __Rev. Mat. Iberoam.__ 34 (2018), no. 4, 1867-1910 (arXiv).

(12) *Groundstates of the Choquard equations with a sign-changing self-interation potential* (with Jean Van Schaftingen), __Z. Angew. Math. Phys.__ 69 (2018), no. 3, 69:86 (arXiv).

(11) *Nonradial entire solutions for Liouville systems* (with Francesca Gladiali and Massimo Grossi), __J. Diff. Equations__ 263 (2017), no. 8, 5151-5174 (arXiv).

(10) *Existence of groundstates for a class of nonlinear Choquard equations in the plane* (with Jean Van Schaftingen), __Adv. Nonlinear Stud.__ 17 (2017), no. 3, 581-594 (arXiv).

(9) *B _{2} and G_{2} Toda systems on compact surfaces: a variational approach*,

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Ph.D. thesis:

In my Ph.D. thesis "Variational systems of singular Liouville systems" I studied singular Liouville systems on compact surfaces, that is system of second order elliptic PDEs with exponential nonlinearities, from a variational point of view. I first gave sufficient and necessary conditions for the existence of global minima solutions; then, I showed some existence results for min-max solutions; finally, I gave some non-existence results (see publications (2) - (7) above).

Here is the full text .pdf and a brief review .pdf (in English).

Master's degree thesis:

In my Master's degree thesis "Sobolev inequality in the limiting case and exponential integrability" I studied the limiting case p=N of Sobolev embedding, showing the asymptotic behavior for large p of the best Sobolev constant and of the functions which realize it, and stressing the relation with exponential integrability. I then considered the classical Moser-Trudinger inequality for bounded Euclidean domains and its possible extensions to unbounded domains and to conformal metrics on the unit ball, finding original results (see above). I finally studied the problem of extremals for Moser-Trudinger inequality, reported the known results for bounded domains and gave some extensions (see publication (1) above).

Here is a brief review (in Italian) .pdf.