Special Session 2: Recent advances in nonlinear Schrodinger systems

Normalized solutions for a class of gradient-type Schrodinger systems under Neumman boundary condition

Xiaojun Chang Northeast Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we focus on the existence of normalized solutions for a class of gradient-type Schrodinger systems within bounded domains, subject to Neumann boundary conditions. By utilizing a parameterized minimax principle that incorporates Morse index information for constrained functionals, along with a novel blow-up analysis of the gradient-type Schrodinger system under these Neumann boundary conditions, we establish the existence of mountain pass-type normalized solutions.

Time periodic solutions of the wave equations in a ball

Jianyi Chen Qingdao Agricultural University Peoples Rep of China Co-Author(s):    Kui Li, Zhitao Zhang

Abstract: We discuss several results on the existence of the time-periodic solutions for the Dirichlet problem of the nonlinear wave equation on the ball. Our method is based on critical point theory and the spectral properties of the radial wave operator. This is joint work with Kui Li and Zhitao Zhang.

Self-similar Blow-up Solutions of the Nonlinear Schrodinger Equation with Moving Mesh methods

Shaohua Chen Cape Breton University Canada Co-Author(s):

Abstract: This paper deals with the initial-value problem for the radially symmetric nonlinear Schrodinger equation with cubic non linearity in d = 2 and 3 space dimensions. A very simple, robust and efficient moving mesh method is proposed for numerically solving the radially symmetric nonlinear Schrodinger equation. It is employed in two schemes. The first numerical simulation aims to reproduce the stable self-similar blow-up solution for d=3. The computed data is used to compare it with the exact blow-up solution. The two solutions overlap when the amplitude of the solution is less than 10 to the 5th power. Next, a typical initial function is used to simulate the blow-up solution for d=3 and is compared with the corresponding exact blow-up solution. The graphs of the two solutions almost overlap when the amplitude of the solution reaches 10 to the 60th power and the adjacent mesh points near 0 are as small as 10 to the -61th power. The two solution curves and their derivatives are smooth in the whole domain and show slow oscillations in both r and t directions. However, when d = 2, the numerical solution becomes unstable due to high oscillations. A comparison with the corresponding asymptotic solution reveals that the amplitude of the two solutions almost overlap. Furthermore, both mass and energy are well conserved for d = 2 and 3.

Asymptotic behavior for a dissipative nonlinear Schrodinger equation with time-dependent damping

Bamri Chourouk Faculty of science of tunis Tunisia Co-Author(s):    Slim TAYACHI

Abstract: In this talk, we investigate the effect of a time-dependent damping on the asymptotic behavior of solutions for the dissipative nonlinear Schrodinger equation in the subcritical case. In particular, we reveal that the time-dependent damping plays a crucial role in shaping the long-time behavior of the solution, particularly the asymptotic profile, the $L^\infty$ and the $L^2-$decay rates. Our results are unchanged through an $L^1-$perturbation. This talk is a part of my Ph.D. research, supervised by: Slim Tayachi.

A study on the well posedness and stability of a KGS type system with an infinite memory term

michail E Filippakis University of Piraeus, Department of Digital Systems Greece Co-Author(s):    Poulou Maria Eleni

Abstract: \footnote{The publication of this paper has been partly supported by the University of Piraeus Research Center}} \begin{abstract} \noindent This paper is concerned with the existence, uniqueness and uniform decay of the solutions of a Klein-Gordon-Schr\{o}dinger type system with linear memory term. The existence is proved by means of the Faedo-Galerkin method and the asymptotic behavior is obtained by making use of the multiplier technique combined with integral inequalities. \end{abstract}

Self-organizing pheonomena in Schrodinger type systems

Yeyao Hu Central South University Peoples Rep of China Co-Author(s):    Aleks Jevnikar and Weihong Xie

Abstract: Self-organizing patterns are widely observed in various physical and biological systems. In Schrodinger-type reaction-diffusion systems, such as the Gierer-Meinhardt system, assemblies of interior and boundary spikes can be constructed. In contrast, the FitzHugh-Nagumo system reveals bubble-like assemblies. More recently, infinitely many solutions of Schrodinger-Newton systems have been constructed based on the ground state of the two-component systems. This talk will also cover new progress on cluster solutions and other related aspects.

New solutions for the Lane-Emden problem in planar domains

Isabella Ianni Sapienza Universita di Roma Italy Co-Author(s):

Abstract: We consider the Lane-Emden problem $\begin{eqnarray*} && -\Delta u=|u|^{p-1}u \quad \mbox{ in } \Omega, \ &&u=0 \quad \mbox{ on }\partial\Omega, \end{eqnarray*}$ where $\Omega\subset\mathbb R^2$ is a smooth bounded domain. When the exponent $p$ is large, the existence and multiplicity of solutions strongly depend on the geometric properties of the domain, which also deeply affect their qualitative behaviour. Remarkably, a wide variety of solutions, both positive and sign-changing, have been found when $p$ is sufficiently large. In this talk, we focus on this topic and show the existence of new sign-changing solutions that exhibit an unexpected concentration phenomenon as $p\rightarrow +\infty$. These results are obtained in collaboration with L. Battaglia and A. Pistoia.

The existence of $L^2$-normalized solutions in the $L^2$-critical setting

Norihisa Ikoma Keio University Japan Co-Author(s):    Silvia Cingolani, Marco Gallo, Kazunaga Tanaka

Abstract: This talk is concerned with the existence of $L^2$-normalized solutions. In particular, we deal with the $L^2$-critical nonlinearity both at $0$ and $\infty$. A model case is the power nonlinearity and this problem has a solution only for one mass. We consider a perturbation of this power nonlinearity and our problem is delicate since we could not expect the existence of solutions for general masses. This is joint work with Silvia Cingolani, Marco Gallo and Kazunaga Tanaka.

Bubbling solutions of slightly subcritical and critical Lane-Emden systems

Seunghyeok Kim Hanyang University Korea Co-Author(s):

Abstract: The Lane-Emden system is an extremal equation associated with a specific Sobolev embedding, and it is closely related to the Calderon-Zygmund estimates. This system is one of the simplest elliptic Hamiltonian systems, as the nonlinear Schroedinger system is that of elliptic gradient systems. In this talk, we examine recent developments in the theory of the Lane-Emden system. This includes discussions on slightly subcritical Lane-Emden systems on smooth bounded domains and the critical Lane-Emden system on the entire Euclidean space or smooth bounded domains with small spherical holes.

Multiple normalized solutions to a system of nonlinear Schroedinger equations

Jaroslaw Mederski Institute of Mathematics, Polish Academy of Sciences Poland Co-Author(s):    Andrzej Szulkin

Abstract: We present recent results concerning normalized solutions to a system of coupled nonlinear Schr\odinger equations. The problem appears in different areas of mathematical physics, e.g. in the analysis of Bose-Einstein condensation or in nonlinear optics. By means of spectral results, the Cwikel-Lieb-Rozenblum theorem, the Morse index and new Liouville-type results we show the existence of multiple normalized solutions for sufficiently large coupling. The talk is based on joint work with Andrzej Szulkin.

Quasilinear Schrodinger Equations Involving Stein-Weiss Convolution Type exponential Critical Nonlinearity

Sarika Sarika Netaji Subhas University of Technology Dwarka Delhi India India Co-Author(s):    Dr. Reshmi Biswas and Prof. Konijeti Sreenadh

Abstract: In this talk, we will discuss about the existence result of quasilinear Schrodinger equation involving critical Choquard type/Stein-Weiss convolution type exponential nonlinearity in bounded domain as well as in unbounded domain.

Normalized solutions of $L^2$-supercritical NLS equations on metric graphs

Nicola Soave Universit\`a degli Studi di Torino Italy Co-Author(s):    Jack Borthwick, Xiaojun Chang, Louis Jeanjean

Abstract: We present some results regarding existence of non-trivial bound states of prescribed mass for the $L^2$-supercritical nonlinear Schr\odinger equation on metric graphs. In recent years, the NLSE on graphs was studied by many authors in the $L^2$-subcritical or critical case. In parallel, the search for prescribed mass solutions to the $L^2$-supercritical NLSE in the Euclidean space attracted a lot of attention. However, the $L^2$-supercritical NLSE on graphs was essentially untouched. In such case, the mass constraint introduces severe complications in proving the existence of bounded Palais-Smale sequences. Several approaches have been developed to overcome these issues in the Euclidean case, but ultimately most of them seem to rely on the fact that critical points satisfy a natural constraint induced by a Pohozaev-type identity, on which the functional can be shown to be coercive. These methods allow to treat cases where the functional enjoys some nice scaling properties, but are not applicable if scaling is not allowed, such as on metric graphs. In this talk we present some existence results obtained by developing a new method based upon a variational principle which combines the monotonicity trick and a min-max theorem with second order information for constrained functionals, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.

An Overview on Nonlinear Schrodinger systems

Giusi Vaira University of Bari Aldo Moro Italy Co-Author(s):

Abstract: In this talk I will discuss about the existence of positive nonradial solutions for a system of Schrodinger equations in a fully attractive or repulsive regime in presence of an external trapping and radial potential. We also discuss the critical growth case and the case of logistic type interaction.

Extremal value of the L^2-Pohozaev manifold and its applications

Yuanze Wu China University of Mining and Technology Peoples Rep of China Co-Author(s):    Taicheng Liu

Abstract: We introduce the extremal value of the L2-Pohozaev manifold. As applications, we prove the existence and multiplicity of normalized solutions of Schrodinger equations under mild conditions.

Symmetric non-radial solutions for nonlinear Schr\odinger systems with mixed couplings

Jiankang Xia Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Yohei Sato; Zhi-Qiang Wang

Abstract: In this talk, I will present our recent results on bound state solutions for the coupled nonlinear elliptic system featuring both attractive and repulsive couplings. These solutions can be characterized as ground states within the subspace of even symmetric functions, whereas it is known that ground states do not exist in the full space of Sobolev functions. This scenario is referred to as the `repulsive-mixed case` according to the recent classification by Wei and Wu (J. Math. Pures Appl., 141:50-88, 2020), which established the non-existence of ground states for small attractive coupling. We extend this non-existence to all cases. This talk is based on the joint work with Professor Yohei Sato from Saitama University, Japan and Professor Zhi-Qiang Wang from Utah State University, USA.

Existence of solutions for a calss of quasilinear elliptic equation

Zonghu Xiu Qingdao Agricultural University Peoples Rep of China Co-Author(s):

Abstract: \documentclass{amsart} \begin{document} By variational methods, we give the sufficient conditions on the existence of solutions. We conducted a detailed discussion on the influence of potential functions, parameters, and nonlinear terms on the existence of solutions. \end{document}

Some new results on normalized solutions of Schrodinger equations and systems

Zhitao Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: We introduce some new results on normalized solutions, especially for normalized solutions of mass subcritical Schrodinger equations in exterior domains; normalized solutions to p-Laplacian equations with combined nonlinearities; normalized solutions to Schrodinger systems etc.

Integrability, regularity and symmetry of positive solutions for Wolff type integral systems

Zhitao Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Zhitao Zhang

Abstract: We are concerned with the optimal integrability, regularity and symmetry of integrable solutions for the Wolff type integral systems. Firstly, we prove the optimal integrability and boundedness of solutions by constructing a nonlinear contracting operator and applying the regularity lifting lemma. Moreover, we exploit the general regularity lifting theorem to derive the Lipschitz continuity as $\gamma>2$. We also prove that the solutions $u$ and $v$ vanish at infinity. The results are valuable for the corresponding$\gamma$-Laplace and $k$-Hessian systems. Secondly, we use the method of moving planes to prove the symmetry and monotonicity of solutions as $\gamma>2$. Minkowski`s inequality is crucial in our proofs. We believe that our arguments can be used to prove similar results for other Wolff type integral systems when $\gamma>2$. We also introduce our other new advances on this topic (joint with Yan Bai, Zexin Zhang).