Special Session 72: Nonlinear elliptic PDEs

Positive Solutions to Singular Second Order BVPs on Time Scales

Shalmali Bandyopadhyay The University of Tennessee at Martin USA Co-Author(s):    Curtis Kunkel

Abstract: We study singular second order BVPs with nonlinear boundary conditions on general time scales. We prove existence of a positive solution using sub and super solution methos, fixed point theory and perturbation methods used in approximating regular problems.

Normalized solutions of Sobolev critical Schrodinger equations in bounded domains

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

Abstract: In this talk, we will discuss the existence of normalized solutions for Sobolev critical Schr\odinger equations within general bounded domains. While recent studies have extensively explored normalized solutions in whole spaces, less is known about their existence in bounded domains. By integrating a recent abstract minimax principle, which incorporates Morse index information for constrained functionals, with novel blow-up analyses, we will demonstrate the existence of normalized solutions of the mountain pass type.

The Br\`ezis-Nirenberg problem for mixed local-nonlocal quasilinear operators

Alessio Fiscella Universidade Estadual de Campinas Brazil Co-Author(s):    Jo\~ao Vitor da Silva and Victor A. Blanco Viloria

Abstract: In this talk, we present existence and multiplicity results, in the spirit of the celebrated paper by Br\`ezis and Nirenberg https://doi.org/10.1002/cpa.3160360405, for a perturbed critical problem driven by the sum of local classical p-Laplacian plus the nonlocal fractional p-Laplacian. More precisely, we face our problem in the case of superlinear perturbations. For this, we first retrace the historical path and we make comparisons with the local classical situation and with the nonlocal fractional situation. We conclude the talk presenting some interesting open questions. The results discussed in this talk are freshly published in https://doi.org/10.1016/j.jde.2024.07.028

Monotonicity trick in nonsmooth critical point theory and its application

Norihisa Ikoma Keio University Japan Co-Author(s):    Jaeyoung Byeon, Andrea Malchiodi, Luciano Mari

Abstract: A motonicity trick due to Struwe and Jeanjean is a powerful tool for functionals of class $C^1$ when it is hard to check the Palais-Smale condition. However, some functionals corresponding to equations appearing in physics or geometry are not of class $C^1$. Szulkin extended the mountain pass and symmetric mountain pass theorem due to Ambrosetti and Rabinowitz into nonsmooth functionals. The aim of this talk is to provide an extension and an application of the monotonicity trick for nonsmooth functionals in a setting which is close to Szulkin`s setting. In particular, we consider Born-Infeld type equations and prove the existence of infinitely many solutions. This talk is based on joint work with Jaeyoung Byeon, Andrea Malchiodi and Luciano Mari.

Some recent results on nonlinear PDEs on lattice graphs

Chao Ji East China University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will introduce some recent results on nonlinear PDEs on lattice graphs. Specifically, we will explore the existence and multiplicity of solutions for the logarithmic Schrodinger equation on lattice graphs, and introduce the double phase problems on lattice graphs. We will also provide some differences between problems on graphs and continuous problems.

Applications of the heat equation to elliptic problems

Haoyu Li Universidade Federal de Sao Carlos Brazil Co-Author(s):    Wang Zhi-Qiang, Ishiwata Michinori

Abstract: In this talk, we first review the application of the heat equation to elliptic problems. Then, we report some recent results concerning the coupled Schr\odinger system \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u_j+ u_j=\mu_j u_j^3+\sum_{i\neq j}\beta_{ij}u_j u_i^2\mbox{ in }\mathbb{R}^n ,\nonumber\ u_j\in H_r^1(\mathbb{R}^n)\mbox{ for }j=1,\cdots,N\nonumber \end{array} \right. \end{equation} and the asymptotically linear problem \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u+\lambda u=\frac{u^3}{1+s u^2}\mbox{ in }\Omega\nonumber,\ u\in H_{0}^1(\Omega)\nonumber \end{array} \right. \end{equation} based on the heat equation approach.

Critical planar Schrodinger-Poisson equations: existence, multiplicity and concentration

Yiqing Li Shandong University of science and technology Peoples Rep of China Co-Author(s):    Vicentiu D. Radulescu, Binlin Zhang

Abstract: In this talk, we will consider the study of the 2-D Schr\{o}dinger-Poisson equation with critical exponential growth. By variational methods, we first prove the existence of ground state solutions for this Schr\{o}dinger-Poisson system with the periodic potential. Then we obtain that there exists a positive ground state solution of the Schr\{o}dinger-Poisson system concentrating at a global minimum of potential function in the semi-classical limit under some suitable conditions. Meanwhile, the exponential decay of this ground state solution is detected. Finally, we establish the multiplicity of positive solutions by using the Ljusternik-Schnirelmann theory.

Variational methods for scaled problems with applications to the Schrodinger-Poisson-Slater equation

Kanishka Perera Florida Institute of Technology USA Co-Author(s):    Carlo Mercuri

Abstract: We develop novel variational methods for solving scaled equations that do not have the mountain pass geometry, classical linking geometry based on linear subspaces, or $\mathbb{Z}_2$ symmetry, and therefore cannot be solved using classical variational arguments. Our contributions here include critical group estimates, nonlinear saddle point and linking geometries based on scaling, a scaling-based notion of local linking, and scaling-based multiplicity results for symmetric functionals. We develop these methods in an abstract setting involving scaled operators and scaled eigenvalue problems. Applications to subcritical and critical Schrodinger-Poisson-Slater equations are given.

One-dimensional boundary blow up problem with a nonlocal term

Futoshi Takahashi Osaka Metropolitan University Japan Co-Author(s):    Taketo Inaba

Abstract: In this talk, we study a nonlocal boundary blow up problem on an interval and obtain the precise asymptotic formula for solutions when the bifurcation parameter in the problem is large. This talk is based on a joint work with Taketo Inaba (Fujitsu Ltd.).

Multiplicity of vector solutions and existence of ground state solution to some indefinite elliptic systems

Rushun Tian Capital Normal University Peoples Rep of China Co-Author(s):    Xu, Ruijin; Su, Jiabao

Abstract: In this talk, we first discuss the existence of infinitely many nonnegative vector solutions of a fully symmetric system, where the coupling parameter represents strong competition. Secondly, we prove the existence of ground state solutions to another indefinite system. A few of information on critical energy levels for coupling parameter $\beta$ in certain ranges will also be given.

Structures and evolution of bifurcation diagrams of a p-Laplacian generalized logistic problem with constant yield harvesting

Shin-Hwa Wang National Tsing Hua University, TAIWAN Taiwan Co-Author(s):    Kuo-Chih Hung and Jhih-Jyun Zeng

Abstract: We study evolutionary bifurcation diagrams for a $p$-Laplacian generalized logistic problem where $p > 1$ and $\mu > 0$ is the harvesting parameter. We mainly prove that, for fixed $\mu > 0$, on the $( \lambda, \left \Vert u \right \Vert _{\infty} )$-plane, the bifurcation diagram always consists of a $\subset$-shaped curve and then we study the structures and evolution of bifurcation diagrams for varying $\mu > 0$. We give two interesting applications. It is a joint work with Kuo-Chih Hung and Jhih-Jyun Zeng.

Coupled elliptic equations with mixed couplings

Zhi-Qiang Wang Utah State University USA Co-Author(s):

Abstract: Abstract. We report results on the existence of solutions for a class of coupled nonlinear elliptic equations with mixed couplings. The solutions exhibit a combined synchronization-segregation effects in some parameter regime. Some asymptotic analysis is provided for qualitative property of the solutions for large couplings.

Multi-bump solutions for the critical Choquard equation

Jiankang Xia Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Xu Zhang

Abstract: In this talk, I will present our recent results in constructing multi-bump solutions for the critical Choquard equation. These solutions are obtained by combining the variational gluing method with a penalization technique. In contrast to the local Yamabe equation, we find that for all dimensions $N\geq 3$, there are infinitely many $\ell$-bump ($\ell\geq2$) positive solutions with polynomial decay. This occurs when the potential function displays periodicity in one variable and features a global maximum with a rapid decay rate in the vicinity of that maximum point. This talk is based on the joint work with Professor Xu Zhang from Central South University, China.

Towards Finding Multiple KKT Points: Part 1-Computing an Inequality/Equality Constrained Local Minimum Point

Jianxin Zhou Texas A&M University, College Station, TX, USA USA Co-Author(s):    Suhan Zhong and Jianxin Zhou

Abstract: The well-known Karush-Kuhn-Tucker (KKT) Theorem provides a set of necessary conditions for a local minimum point subject to finitely many inequality/equality constraints. This long term research project is to develop computational theory and methods for finding multiple KKT points in an infinite-dimensional space setting. In Part 1, an equivalent condition is established for a KKT point, from which a numerical method can be devised to compute a KKT point as a constrained local minimum point. This method is mathematically validated. Numerical examples will be presented to illustrate the algorithm. By an implementation strategy, a convergence result is established. It turns out that this equivalent condition opens a door for people to design numerical methods for computing multiple KKT points or solutions to some differential inclusion problems.