Special Session 54: Trends in Nonlinear Analysis

Liouville properties for differential inequalities with $(p,q)$ Laplacian operator

Mousomi Bhakta Indian Institute of Science Education and Research Pune (IISER Pune) India Co-Author(s):    Anup Biswas, Roberta Filippucci

Abstract: We will discuss several Liouville-type theorems for nonnegative solutions associated with a class of nonhomogenenous quasilinear inequalities, namely \begin{equation*}\tag{$P_s$} -\Delta_p u-\Delta_q u\geq u^{s-1} \, \text{ in }\, \Omega, \end{equation*} where $p>q>1$, $s>1$ and $\Omega$ is any exterior domain of $\mathbb{R}^N$ and \begin{equation*}\tag{$P_{sm}$} -\Delta_p u-\Delta_q u \geq u^s |\nabla u|^m \quad \text{ in }\mathbb{R}^N, \end{equation*} where $p>q>1$, $N>q$ and $s, \, m\geq 0$.

Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part

Bartosz Bieganowski University of Warsaw Poland Co-Author(s):

Abstract: We will present an abstract critical point theory that allows to study the multiplicity of critical points of strongly indefinite functionals with sign-changing nonlinear part. We are going to apply it to nonlinear Schr\{o}dinger-type equations that appear e.g. in nonlinear optics. This is a joint work with Federico Bernini and Daniel Strzelecki.

Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations

Nsoki Mavinga Swarthmore College USA Co-Author(s):    Emer Lopera, Diana Sanchez

Abstract: We consider $(p,q)$-Laplacian problems that include nonlinear perturbation terms in both the differential equations and the boundary. We use variational methods and critical point theory to prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically below the infimum of the set of eigenvalues of the $(p,q)$-Laplacian problem with weights and a spectral parameter present in both the differential equation and the boundary. We also establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for $q$-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary.

ON COMPACT EMBEDDINGS INTO L p AND FRACTIONAL SPACES

Olimpio Miyagaki UFSCAR Brazil Co-Author(s):    H.P. BUENO, A.H.S. MEDEIROS, O.H. MIYAGAKI, AND G.A. PEREIRA

Abstract: The study of the fractional Laplacian operator $(-\Delta)^s$ in $\mathbb{R}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in $\mathbb{R}^N\times (0, \infty)=\mathbb{R}^{N+1}_+$, a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: is the embedding (in the trace sense) on the ground space $L^q(\mathbb{R}^{N})$ compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators. Passing to an abstract setting, let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X`$ a continuous and symmetric operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $\Omega\times(0,\infty)$ or $\mathbb{R}^{N+1}_+$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(\Omega)$ is compact, even in the case $\Omega=\mathbb{R}^N$.

Quasilinear elliptic problems with nonlinear perturbations involving critical Sobolev exponents

M N Nkashama University of Alabama at Birmingham USA Co-Author(s):    N. Mavinga

Abstract: We will present some recent results on the existence of weak minimal and maximal solutions between an ordered pair of sub- and super-solutions for nonlinear (Carath\`{e}odory) perturbations of quasilinear elliptic equations (including the $p$-Laplacian). No monotonicity conditions are imposed on the nonlinear perturbations. Unlike previous results in this setting, we allow the growth of the nonlinear perturbations to go all the way to the critical Sobolev exponents in the appropriate Lebesgue spaces. The approach makes careful use of topological degree theory arguments for demicontinuous operators of class $(S_+)$ with some ingredients from pseudomonotone operators, Zorn`s lemma and a Kato inequality with appropriate estimates.

Travelling waves for Maxwell`s equations in nonlinear symmetric media

Jacopo Schino University of Warsaw Poland Co-Author(s):    Jaros{\\l}aw Mederski

Abstract: We consider a semilinear curl-curl problem arising from Maxwell's equations in a cylindrically symmetric medium. Exploiting a variant of the fountain theorem, we obtain an unbounded sequence of travelling-wave solutions that consist of transverse magnetic field modes, which makes them different from those obtained in the seminal work by McLeod, Stuart, and Troy.

The non-local eigenvalue problems for the p-Laplacian

Mieko Tanaka Tokyo University of Science Japan Co-Author(s):

Abstract: \begin{document} I`ll talk about the non-local eigenvalue problems for the $p$-Laplacian, which derived from the Sobolev-Poincar\`e inequality. In particular, I`d like to focus on the least eigenvalue among those having sign-changing eigenfunctions. \end{document}

Existence and multiplicity of solutions to the mean-field games model with mixed interactions

Yuanze Wu School of Mathematics, Yunnan Normal University Peoples Rep of China Co-Author(s):    Xinfu Li, Xiangqing Liu, Juncheng Wei

Abstract: I will report our recent results on the stationary version of the Mean-Field Games (MFG) models. By developing the minimization method on the Pohozaev manifold, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions under more general conditions.