Special Session 117: Advances on nonlinear elliptic PDEs

Nonlinear scalar field $(p_{1}, p_{2})$-Laplacian equations in $\mathbb{R}^{N}$: existence and multiplicity

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

Abstract: In this talk, we focus on the following class of $(p_{1}, p_{2})$-Laplacian problems: $$\begin{equation*} \left\{ \begin{array}{ll} -\Delta_{p_{1}}u-\Delta_{p_{2}}u= g(u) \mbox{ in } \mathbb{R}^{N}, \ u\in W^{1, p_{1}}(\mathbb{R}^{N})\cap W^{1, p_{2}}(\mathbb{R}^{N}), \end{array} \right. \end{equation*}$$ where $N\geq 2$, $1$<$p_{1}$<$p_{2}\leq N$, $\Delta_{p_{i}}$ is the $p_{i}$-Laplacian operator for $i=1, 2$, and $g:\mathbb{R}\to \mathbb{R}$ is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems.

Existence results for quasilinear Choquard equations in $\mathbb{R}^N$

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

Abstract: In this talk I will present some existence results for quasilinear Choquard equations driven by the $p$-Laplacian operator including a $C^1$ nonlinearity $G$, in $\mathbb{R}^N$. Assuming \textit{Berestycki--Lions} type conditions on $G$, we prove the existence of ground state solutions $u\in W^{1, p}(\mathbb{R}^N)$ by means of variational methods. Moreover, we establish some qualitative properties of the solutions when $G$ is even and non--decreasing. The talk is based on a joint work with Vincenzo Ambrosio and Teresa Isernia.

Fractional Schrodinger equations with mixed nonlinearities

Mousomi Bhakta Indian Institute of Science Education and Research Pune (IISER Pune) India Co-Author(s):    Paramananda Das, Debdip Ganguly

Abstract: In this talk I will discuss fractional Schr\"odinger equations with a vanishing parameter, namely $$ (-\Delta)^s u+u=|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N, \quad u \in H^s(\mathbb{R}^N), $$ where $ s \in (0,1)$, $N > 2s$, $2 < q < p\leq 2^*_s=\frac{2N}{N-2s} $ are fixed parameters and $\lambda > 0$ is a vanishing parameter. We investigate the asymptotic behavior of positive ground state solutions for $\lambda$ small, when $p$ is subcritical, or critical Sobolev exponent $2^*_s$. For $p < 2_s^*$, the ground state solution asymptotically coincides with unique positive ground state solution of $(-\Delta)^s u+u=u^p$, whereas for $p=2_s^*$ the asymptotic behavior of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for $\lambda>0$ small, we will discuss the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions.

Multiplicity of solutions to stronglu indefinite problems with sign-changing nonlinearities

Bartosz Bieganowski University of Warsaw Poland Co-Author(s):    Federico Bernini, Daniel Strzelecki

Abstract: We will present an abstract critical point theory that allows to study the multiplicity of critical points of strongly indefinite functionals with sing-changing nonlinear part. We are going to apply it to nonlinear Schrodinger-type equations that appear e.g. in nonlinear optics.

Limiting cases in Choquard type equations and Schroedinger-Poisson systems

Daniele Cassani University of Insubria & RISM Italy Co-Author(s):

Abstract: Quantitative and qualitative informations on nonlinear Schr\odinger equations strongly coupled with Poisson`s equation can be derived from nonlocal Choquard type equations. Limiting cases appear when the underlying function space setting is not well defined for the equation, as a consequence of the limiting Sobolev emedding which provides logarithmic kernels competing with exponential nonlinearities. We present two possible approaches to overcome this difficulty. The first one by establishing a suitable weighted Trudinger-Moser type inequality which eventually yields a proper functional setting. Alternatively, one can exploit a uniform approximation of the $\log$-kernel and then pass to the limit in the approximating equations. Both methods reveal new aspects which throw some light on the problem.

Special wave forms for a generalized semilinear wave equation

Julia Henninger KIT Karlsruhe Germany Co-Author(s):    Wolfgang Reichel, Sebastian Ohrem

Abstract: We study the generalized semilinear wave equation $\begin{align*} V(x) u_{tt} - d(t) M(x, \partial_{x} ) u - V(x) |u|^{p-1} u=0 \quad \text{ for } \quad (x,t) \in \mathbb{R}^N \times \mathbb{R} \end{align*}$ where $M$ is elliptic and $d$ is a positive potential. Our goal is to construct solutions which are localized in space and/or time by means of variational methods. We present our approach with its main difficulties and discuss suitable examples for $M$ and $d$. This is joint work with Sebastian Ohrem and Wolfgang Reichel.

Least energy solutions for nonlinear fractional Choquard-Kirchhoff equations in $\mathbb{R}^{N}$

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

Abstract: In this talk, we will consider the following fractional Choquard--Kirchhoff equation $$\begin{equation*} \left(a+b\iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dxdy \right) (-\Delta)^{s} u + u = \left(I_{\alpha}*F(u) \right) F`(u) \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*}$$ where $N\geq 2$, $a, b>0$ are constants, $(-\Delta)^{s}$ is the fractional Laplacian operator of order $s\in (0,1)$, $I_{\alpha}$ is the Riesz potential of order $\alpha \in \left( (N-4s)^{+}, N \right)$, $F\in C^{1}(\mathbb{R}, \mathbb{R})$ is a general nonlinearity of Berestycki--Lions type. Applying suitable variational methods, we analyze the existence of ground state solutions, along with the regularity, symmetry, and decay properties of these solutions.

Some recent results on normalized solutions for $(2,q)$-Laplacian equations

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 on normalized solutions for $(2,q)$-Laplacian equations. Specifically, we will explore the existence and multiplicity of normalized solutions to a class of $(2, q)$-Laplacian equations in the strongly sublinear regime, and provide existence and multiplicity of normalized solutions for $(2,q)$-Laplacian equations with the mixed nonlinearities.

On some doubly critical elliptic systems

Rafael Lopez-Soriano Universidad de Granada Spain Co-Author(s):

Abstract: We will consider a type of cooperative nonlinear elliptic system in $\mathbb{R}^N$. The interest of this problem is based on the presence of Sobolev or Sobolev-Hardy critical power nonlinearities and a nonlinear coupling, possibly critical, as well as singular potentials of Hardy type. Using variational methods we will focus on the existence of bound and ground states of the underlying energy functional. Finally, for a certain range of parameters, we derive some qualitative properties and a classification criterion.

Quasilinear Schr\odinger Equation: a bifurcational approach

Miguel Martinez-Teruel University of Granada Spain Co-Author(s):    David Arcoya and Jose Carmona

Abstract: This talk deals with existence and multiplicity results of positive solutions for the quasilinear Schr\odinger equation \begin{align*} \left\{\begin{array}{c} \displaystyle-\Delta u-\lambda m(x) u\Delta(u^2)=f(\mu,x,u)\text{ in }\Omega, \ u=0\text{ on }\partial\Omega. \end{array}\right. \end{align*} where $\Omega$ is a bounded open domain in $\mathbb{R}^N$ with smooth boundary and $m$ is bounded positive continuous function. Under suitable assumptions on $f$ and asymptotically linear behaviour, we can use bifurcation theory in order to give an analysis on the set of positive solutions.

Travelling waves for Maxwell`s equations in nonlinear and symmetric media

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

Abstract: We look for travelling wave fields satisfying Maxwell equations in a nonlinear and cylindrically symmetric medium. We obtain a sequence of solutions with diverging energy. The solutions represent the so-called TM-modes.

The problem of prescribing non-constant curvatures in a disk

Francisco Javier Reyes Sanchez Universidad de Granada Spain Co-Author(s):    Rafael Lopez Soriano and David Ruiz

Abstract: In this talk we outlines a general existence result regarding the prescription of curvatures on a disk under conformal changes of the metric. This problem is equivalent to solving a nonlinear partial differential equation with nonlinear Neumann boundary conditions. We addresses the case of negative Gaussian curvature. As a preliminary step, we focus on a symmetric setting. Additionally, we present a non-existence result, demonstrating that the assumptions for existence are necessary. This is a work in collaboration with Rafael Lopez Soriano and David Ruiz from the University of Granada.