Special Session 113: New Achievements in Nonlinear PDEs and Applications

Normalized solutions to the Born-Infeld and related problems

Laura Baldelli University of Granada Spain Co-Author(s):    Jaroslaw Mederski, Alessio Pomponio

Abstract: It is well known that in the classical Maxwell electromagnetic theory the so called infinite energy problem appears: the energy of the electrostatic field generated by a point charge is infinite. In the early years of the last century, Born and Infeld proposed to solve such a problem by introducing a nonlinear modification of Maxwell`s theory. The aim of this talk is therefore to introduce the Born-Infeld equation and to present some recent existence results concerning normalized solutions for a large class of operators by means of variational methods.

Travelling waves for nonlinear Schrodinger equations

Bartosz Bieganowski University of Warsaw Poland Co-Author(s):    Laura Baldelli, Jaroslaw Mederski

Abstract: We look for travelling wave solutions to the nonlinear Schrodinger equation with a subsonic speed, covering several physical models with Sobolev subcritical nonlinear effects. Our approach is based on a variant of Sobolev type inequality involving the momentum and we show existence of its minimizers solving the nonlinear Schrodinger equation.

Positive and nodal solutions for a quasi-linear equation depending on the gradient

Francesca Faraci University of Catania Italy Co-Author(s):    D. Motreanu, D.Puglisi

Abstract: In this talk we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. Our aim is to combine variational techniques with fixed point theory in order to prove the existence of a positive solution.We prove also the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the p-Laplace operator. Based on the papers F. Faraci, D.Motreanu, D. Puglisi, Quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations (2015) and F. Faraci, D. Puglisi, Nodal solutions of p-Laplacian equations depending on the gradient, Proc. Roy. Soc. Edinburgh A (2024).

ON A CLASS OF QUASILINEAR CRITICAL SCHR\H ODINGER EQUATIONS IN $\mathbb R^N$

Roberta Filippucci University of Perugia Italy Co-Author(s):    Laura Baldelli

Abstract: In this talk, we present multiplicity results, obtained via variational tools, for solutions of generalized quasilinear Schr\H odinger potential free equations, also in the singular case, defined in the entire $\mathbb R^N$ and involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation can be seen as models of several physical phenomena in plasma physics.

Poincare-Sobolev equations with the critical exponent and a potential in the hyperbolic space

DEBDIP GANGULY Indian Institute of Technology Delhi India Co-Author(s):    Mousomi Bhakta, Diksha Gupta, A.K.Sahoo

Abstract: In this talk, I will discuss the following Poincare-Sobolev-type equation $\begin{equation*} -\Delta_{\mathbb{H}^N} u - \lambda u = a(x) |u|^{p-1} \, u\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \end{equation*}$ where $\mathbb{B}^N$ denotes the hyperbolic space, $16$ in the critical case, whereas in the subcritical case, we use the min-max procedure in the spirit of Bahri-Li in the hyperbolic space and using a series of new estimates involving interacting hyperbolic bubbles.

On a classification of steady solutions to two-dimensional Euler equations

Changfeng Gui University of Macau Macau Co-Author(s):    Chunjing Xie, Huan Xu

Abstract: In this talk, I shall provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. This talk is based on a joint work with Chunjing Xie and Huan Xu.

Nearly parallel helical vortex filaments in the three dimensional Euler equations

Monica Musso University of Bath England Co-Author(s):    I. Guerra

Abstract: Klein, Majda, and Damodaran have previously developed a formalized asymptotic motion law describing the evolution of nearly parallel vortex filaments within the framework of the three-dimensional Euler equations for incompressible fluids. In this study, we rigorously justify this model for two stationary configurations: the central configuration consisting of rotating regular polygons of $N$ helical-filaments, and the central configurations of $N+1$ vortex filaments, where an $N$-polygonal central configuration surrounds a central straight filament.

Embeddings in the orthotropic Sobloev space on the Heisenberg groups

Abdolrahman Razani Imam Khomeini International University Iran Co-Author(s):

Abstract: Quantum mechanics revolutionized the understanding of the physical world by introducing a new theory that challenged the limitations of classical physics, especially when dealing with phenomena at the submicroscopic level. One of the key contributions of quantum mechanics was Heisenberg`s Uncertainty Principle, which imposed a fundamental restriction on the precision of measuring certain properties such as position and momentum. Additionally, quantum mechanics brought forth the concept of superposition, where particles could exist in multiple states simultaneously until observed, at which point they would collapse into a single state. The Heisenberg groups $\mathbb{H}^n$ serve as a notable example of noncommutative graded Lie groups, providing a clear illustration of the noncommutative nature of the relationships between position and momentum operators in quantum mechanics. This presentation begins by studying the representation of the orthotropic $p$-Laplacian on the Heisenberg groups $\mathbb{H}^n$, shedding light on the abstract nature of these mathematical structures. Then we present some open questions about embedding theorems within this framework.

On Calderon-Zygmund theory for the p-Laplacian

Armin Schikorra University of Pittsburgh USA Co-Author(s):

Abstract: Calderon-Zygmund theory for the Laplace equation is among the most classical results in Harmonic Analysis. It was conjectured by Iwaniec in 1983 that an analogue theory holds for the p-Laplace. I will discuss how to disprove this conjecture.

Least energy sign-changing solution for degenerate Kirchhoff double phase problems

Patrick Winkert University of Technology Berlin Germany Co-Author(s):    \`{A}ngel Crespo-Blanco, Leszek Gasi\`{n}ski

Abstract: In this talk, we present existence and multiplicity results for Kirchhoff Dirichlet equations of double phase type with right-hand sides that grow superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which turns out to be a least energy sign-changing solution. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincar\`{e}-Miranda existence theorem. This is a joint work with \`{A}ngel Crespo-Blanco (Berlin) and Leszek Gasi\`{n}ski (Krakow).