Special Session 26: Nonlinear partial differential equations and their applications

On some sharp Hardy-Littlewood-Sobolev type inequalities

Qianqiao Guo School of Mathematics and Statistics, Northwestern Polytechnical University Peoples Rep of China Co-Author(s):

Abstract: We establish some sharp Hardy-Littlewood-Sobolev type inequalities. For some cases, the extremal functions are classifified via the method of moving spheres, and the best constants are computed. This is joint work with Jingbo Dou, Zhe Pu, Jiankang Xia, Xiqiang Zhang and Meijun Zhu.

Eventual monotonicity of the first nonzero Steklov eigenvalue for perimeter-normalized regular N-gons

Yeyao Hu Central South University Peoples Rep of China Co-Author(s):    Zhuo Cheng, Changfeng Gui, Qinfeng Li, Ruofei Yao

Abstract: In this talk, we show that the first nonzero Steklov eigenvalue of the regular N-gon with fixed perimeter is strictly increasing as N increases, provided that N is sufficiently large; for instance, this holds for N greater than or equal to 20. The talk is based on joint work with Zhuo Cheng (CSU), Changfeng Gui (University of Macau), Qinfeng Li (HNU), and Ruofei Yao (SCUT).

Uniqueness results for the planar Dirichlet Lane-Emden problem

Isabella Ianni Sapienza University Italy Co-Author(s):    De Marchis, Grossi, Luo, Pacella, Yan

Abstract: We present uniqueness results for non-negative solutions of semilinear equations with a power nonlinearity, set in bounded planar domains with Dirichlet boundary conditions. The talk is mainly based on the following joint works: [1] F. De Marchis, M. Grossi, I. Ianni, F. Pacella, Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains, J. Math. Pures Appl. 128, 2019. [2] M. Grossi, I. Ianni, P. Luo, S. Yan, Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two, J. Math. Pures Appl. 128, 2022. [3] I. Ianni, P. Luo, S. Yan, Morse index, topological degree and local uniqueness of multi-spikes solutions to the Lane-Emden problem in dimension two, submitted

On the bifurcation diagram for free boundary problems arising in plasma physics

Aleks Jevnikar University of Udine Italy Co-Author(s):    D. Bartolucci, J. Wei, R. Wu

Abstract: We are concerned with qualitative properties of the bifurcation diagram of a free boundary problem arising in plasma physics, showing in particular uniqueness and monotonicity of its solutions. We then discuss a new approach to study the Rabinowitz continuum of classical Gelfand problems.

Existence of positive solutions for a class of almost critical problems on an annulus

Gabriele Mancini University of Bari Aldo Moro Italy Co-Author(s):    Giuseppe Rago, Giusi Vaira

Abstract: In this talk I will give a brief overview of some results concerning the existence of positive multi-peak solutions with Dirichlet boundary conditions for a class of slightly subcritical or slightly supercritical elliptic problems. I will also present some recent results, obtained in collaboration with Giuseppe Rago and Giusi Vaira, on arbitrary dimensional annuli. By exploiting the explicit form of the Green function and the Robin function for the annulus, we prove that, in the slightly subcritical case, the annulus must become thinner and thinner as the number of peaks increases, whereas, in the slightly supercritical case, the inner hole of the annulus must be very small.

The nonlinear p-curl-curl problem

Jaroslaw Mederski Institute of Mathematics, Polish Academy of Sciences Poland Co-Author(s):    Jaros{\l}aw Mederski and Andrzej Szulkin

Abstract: Nonlinear curl-curl problems have recently emerged in the study of exact electromagnetic wave propagation in nonlinear media modeled by Maxwell`s equations. In particular, the quintic effect gives rise to a critical partial differential equation involving the curl-curl operator. Ground state solutions of this problem are closely related to the optimizers of a new Sobolev-type inequality. In this work, we present recent results on the existence of ground state solutions and discuss certain symmetry properties of the problem. Applications to zero modes of the Dirac equations are also considered. This is joint work with Andrzej Szulkin.

Multiple nodal solutions of Kirchhoff-Choquard equations with logarithmic potential and critical exponential nonlinearity

Olimpio Miyagaki UFSCAR Brazil Co-Author(s):    E. Boer, E. Barboza, O.H. Miyagaki and C. Santana

Abstract: In this work, we study a class of planar Kirchhoff-Choquard equations involving a sign-changing logarithmic kernel and an exponential nonlinearity. Our main goal is to construct solutions with $k$ nodes for any given $k \in N, $ thereby establishing the multiplicity of solutions for this class of problems. To achieve this, we employ a gluing method inspired by the framework developed in [T. Bartsch, M. Willem-ARMA 1993, and, D. Cao, X. Zhu- Acta Math. Sci. 1998]. The analysis presents substantial challenges due to the interaction among the nonlocal Kirchhoff term, the sign-changing nature of the Choquard kernel, and the lack of compactness induced by the exponential nonlinearity, which is critical in two-dimensional settings. Overcoming these difficulties requires a careful and refined analytical approach to successfully implement the gluing construction.