LUCA BATTAGLIA - PERSONAL WEB PAGE

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Research interests:

My research activity mainly deals with elliptic partial differential equations.

In particular, I am interested in equation and systems with exponential non-linearities, through variational, perturbative and asymptotic analysis methods.

I have also studied extensions of Moser-Trudinger-type inequalities, including the existence of extremal functions.

I have also worked on some non-local problems on the whole Euclidean plane, such as the non-linear Choquard equation.

Scientific publications:

Full list with descriptions: English, Italian.

(21) "A blow-up phenomenon for a non-local Liouville-type equation" (with María Medina and Angela Pistoia), submitted (arXiv).

(20) "Prescribing Gaussian curvature on surfaces with conical singularity and geodesic boundary" (with Aleks Jevnikar, Zhi-An Wang and Wen Yang), submitted (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", J. Math. Phys. 58 (2017), no. 1, 011506, 25 pp. (arXiv).

(8) "Ground states solutions for a nonlinear Choquard equation", Rend. Sem. Mat. Univ. Politec. Torino, Vol. 74, 2 (2016), 53-60 (arXiv).

(7) "Existence and non-existence results for the SU(3) singular Toda system on compact surfaces" (with Andrea Malchiodi), J. Funct. Anal. 270 (2016), no. 10, 3750-3807 (arXiv).

(6) "Moser-Trudinger inequalities for singular Liouville systems", Math. Z. 282 (2016), no. 3-4, 1169-1190 (arXiv).

(5) "A general existence result for the Toda system on compact surfaces" (with Aleks Jevnikar, Andrea Malchiodi and David Ruiz), Adv. Math. 285 (2015), 937-979 (arXiv).

(4) "A note on compactness properties of the singular Toda system" (with Gabriele Mancini), Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(3):299-307, 2015 (arXiv).

(3) "Existence and multiplicity result for the singular Toda system", J. Math. Anal. Appl. 424 (2015), no. 1, 49-85 (arXiv).

(2) "A Moser-Trudinger inequality for the singular Toda system" (with Andrea Malchiodi), Bull. Inst. Math. Acad. Sin. 9 (2014), no. 1, 1-23 (arXiv).

(1) "Remarks on the Moser-Trudinger Inequality" (with Gabriele Mancini), Adv. Nonlinear Anal. 2 (2013), no. 4, 389-425 (arXiv).

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.