Special Session 77: Recent developments in variational problems and geometric analysis

Concentration phenomena for nonlinear fractional relativistic Schrodinger equations

Vincenzo Ambrosio Universita' Politecnica delle Marche Italy Co-Author(s):

Abstract: In this talk, we consider the following class of fractional relativistic Schr\odinger equations: $\begin{equation*} \left\{ \begin{array}{ll} (-\varepsilon^{2}\Delta+m^{2})^{s}u + V(x) u= f(u)+\gamma u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \ u \in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \ \end{array} \right. \end{equation*}$ where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $\gamma\in \{0, 1\}$, and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent. Here, the pseudo-differential operator $(-\varepsilon^{2}\Delta+m^{2})^{s}$ is simply defined in Fourier variables by the symbol $(\varepsilon^{2}|\xi|^{2}+m^{2})^{s}$, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Utilizing the extension method and penalization techniques, we first show that there exists a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, that concentrate around a local minimum of $V$ as $\varepsilon\rightarrow 0$. Finally, we combine the generalized Nehari manifold method with the Ljusternik-Schnirelman theory to relate the number of positive solutions to the topology of the set where the potential $V$ attains its minimum value.

Some new results on elliptic equations involving Logarithmic Laplacian

Rakesh Arora Indian Institute of Technology (IIT-BHU) India Co-Author(s):    Jacques Giacomoni and Arshi Vaishnavi

Abstract: This talk presents new existence, non-existence and uniqueness results for the following problem \[ L_{\Delta} u = f(x,u) \ \text{in} \ \Omega, \quad u = 0 \ \text{in} \ \mathbb{R}^N \setminus \Omega \] where $L_\Delta$ is the Logarithmic Laplace operator and $f$ satisfies sub-critical, critical and super-critical nonlinearities growth conditions. Such type of problems are connected to the model problems in population dynamics, optimal control, approximation of fractional harmonic maps, and fractional image denoising.

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR QUASILINEAR EQUATIONS WITH WEIGHTS

Roberta Filippucci University of Perugia Italy Co-Author(s):    Laura Baldelli, Valentina Brizi, Yadong Zheng

Abstract: In this talk, we present some recent results on existence and nonexistence of positive radial solutions for a Dirichlet problem both in the case of the $p$-Laplacian operator and of the mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem is delicate and requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity.

(p,q)-fractional problems involving a sandwich type perturbation and a critical Sobolev nonlinearity

Alessio Fiscella Universidade Estadual de Campinas Brazil Co-Author(s):    Mousomi Bhakta and Shilpa Gupta

Abstract: In this talk, we deal with elliptic problems set on a general open domain Omega, driven by a (p,q)-fractional operator, involving a critical Sobolev nonlinearity and a nonlinear perturbation of sandwich type. More precisely, the subcritical term is intrinsically linked to the double (p,q)-growth of the main operator. Under different settings of involved parameters, we prove existence and multiplicity results for our problems. For this, we combine topological tools and variational methods. Our results, contained in https://arxiv.org/abs/2409.13986, generalize in several directions the theorems proved in https://doi.org/10.1007/s00526-020-01867-6 and in https://doi.org/10.1016/j.aml.2020.106646

One-dimensional half-harmonic maps into the circle

Ali Hyder TIFR-CAM Bangalore India Co-Author(s):    L. Martinazzi

Abstract: Given a function $g$ in the homogeneous fractional Sobolev space $\dot H^{1/2, 2}(R,S^1)$ from the real line into the unit circle, by the direct minimization method, one can construct a half-harmonic map $u$ from the real line into the unit circle such that $u=g$ outside $(-1,1)$. In this talk we will show the existence of another half-harmonic maps with the same boundary condition $g$ by minimizing the fractional Dirichlet energy in a different homotopy class. We will also show that in general it is not possible to minimize the energy in every homotopy class.

Ground state solutions for a $(p, q)$--Choquard equation with a general nonlinearity

Teresa Isernia Universita` Politecnica delle Marche Italy Co-Author(s):    Vincenzo Ambrosio

Abstract: In this talk, we will consider the following $(p, q)$-Choquard equation $\begin{equation*} -\Delta_{p}u -\Delta_{q}u + |u|^{p-2}u + |u|^{q-2}u = \left(I_{\alpha}*F(u) \right) f(u) \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*}$ where $2 \leq p$ < $q$ < $N$, $\Delta_{s}$ is the $s$--Laplacian operator with $s\in \{p, q\}$, $I_{\alpha}$ is the Riesz potential of order $\alpha \in \left( (N-2q)^{+}, N \right)$, $F\in C^{1}(\mathbb{R}, \mathbb{R})$ is a general nonlinearity of Berestycki--Lions type, and $F'=f$. By means of variational methods, we analyze the existence of ground state solutions, along with the regularity, symmetry, and decay properties of these solutions. $\Delta_{s}$ is the $s$-Laplacian operator with $s\in \{p, q\}$, $I_{\alpha}$ is the Riesz potential of order $\alpha \in \left( (N-2q)^{+}, N \right)$, $F\in C^{1}(\mathbb{R}, \mathbb{R})$ is a general nonlinearity of Berestycki--Lions type, and $F'=f$. By means of variational methods, we analyze the existence of ground state solutions, along with the regularity, symmetry, and decay properties of these solutions.

Quantitative stability of the Poincar\`{e}-Sobolev inequality on the hyperbolic space

Debabrata Karmakar TIFR Centre for Applicable Mathematics India Co-Author(s):    Mousomi Bhakta, Debdip Ganguly, Debabrata Karmakar, and Saikat Mazumdar

Abstract: The classical Sobolev inequality in $\mathbb{R}^n$ states that the $L^{\frac{2n}{n-2}}$-norm of smooth compactly supported functions can be controlled, up to an optimal constant $S,$ by the $L^2$-norm of their gradient. The explicit value of $S$ is known, and the cases where equality holds have been obtained and classified as {\it Aubin-Talenti bubbles.} The question of quantitative stability and its applications has garnered significant interest in recent times. In this presentation, we will explain the question of the stability of the optimizers and their counterparts within the framework of the hyperbolic space.

Symmetry of Sobolev Extremals in the Hyperbolic space

Sandeep kunnath TIFR Centre for Applicable Mathematics India Co-Author(s):

Abstract: Extremals of the Sobolev inequality in the Hyperbolic space satisfies a p-Laplace type equation. In this talk we investigate the radial symmetry of extremals of Sobolev inequality in the hyperbolic space or more generally the positive finite energy solutions of the corresponding Euler Lagrange equation. We prove using the moving plane method that the Solutions are radially symmetric with respect to a point in the hyperbolic space. The crucial ingredient in the proof is the sharp asymptotic estimates on the solution and its gradient at infinity, which depends crucially on a classification result for eigenfunctions of the p-Laplace equation in the hyperbolic space with a desired pole at infinity. This is a joint work with Ramya Datta.

Higher order semilinear equations on hyperbolic spaces

Jungang Li University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: In this talk we will introduce some recent progress on higher order semilinear equations on hyperbolic spaces. Our results consists of the existence and symmetry of solutions to Br\`ezis-Nirenberg equations and Shr\odinger equations. Due to the complexity of hyperbolic spaces, classical variational methods such as the blow-up analysis and the mini-max theory have to be modified. Moreover, we will highlight the relation of this modification with the study of sharp Sobolev type inequalities on hyperbolic spaces. Part of these results are joint works with G. Lu. Q. Yang and Z. Wang.

Higher dimensional concentration for singularly perturbed coupled elliptic systems.

BHAKTI BHUSAN MANNA IIT HYDERABAD India Co-Author(s):    Alok Kumar Sahoo

Abstract: In this talk, we shall discuss the existence of positive solutions and their concentration profile for the following problem: $\begin{equation*} -\varepsilon^2\Delta u +c(x)u=b(x) |v|^{q-1}v,\,\,\text{ and } -\varepsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u \quad\text{in } \Omega, \end{equation*}$ with Neumann boundary data on $\partial\Omega$. The domain is bounded, and the coefficients are considered smooth, positive and bounded. We shall first discuss the existence of positive solutions using some direct method of calculus of variations. Then, we explain the concentration profile of the solutions as the perturbation parameter converges to zero. Our emphasis will be on the dependence of the concentration profile on the coefficients. We conclude by applying the result for different kinds of higher dimensional concentrations for some coupled elliptic systems.

Compactness of conformal metrics with constant Q-curvature of higher order.

Saikat Mazumdar Indian Institute of Technology Bombay India Co-Author(s):    Bruno Premoselli

Abstract: In this talk, we will consider the question of compactness for the higher-order Yamabe equation. This amounts to studying compactness (in $C^2k$) of nonnegative solutions of a $2k$-th order critical exponent elliptic equation, involving the GJMS operator, on a closed Riemannian manifold of dimension $\ge 2k+1$. Here $k$ is a positive integer. We will assume positivity conditions on the GJMS operator and establish uniform bounds on the (geometric) solutions under appropriate geometric assumptions. This is a joint project with Bruno Premoselli (ULB Brussels).

An abstract multiplicity result with applications to critical growth elliptic problems

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

Abstract: We present an abstract multiplicity theorem that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $\lambda > 0$. In particular, the number of solutions goes to infinity as $\lambda \to \infty$. Moreover, we give an explicit lower bound on $\lambda$ in order to have a given number of solutions.