Existence solutions for a weighted equation of p-biharmonic type in the unit ball of RN\mathbb{R}^{N} with critical exponential growth

We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\overline{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of $\mathbb{R}^{N}$. It is proved that there is a nontrivial weak solution to this problem by the mountain Pass Theorem. We avoid the loss of compactness by proving a concentration compactness result and by a suitable asymptotic condition.Comment: arXiv admin note: substantial text overlap with arXiv:2201.10433, arXiv:2201.09858. substantial text overlap with arXiv:2311.1678

A Logarithmic Weighted Adams-type inequality in the whole of RN\mathbb{R}^{N} with an application

In this paper, we will establish a logarithmic weighted Adams inequality in a logarithmic weighted second order Sobolev space in the whole set of $\mathbb{R}^{N}$. Using this result, we delve into the analysis of a weighted fourth-order equation in $\mathbb{R}^{N}$. We assume that the non-linearity of the equation exhibits either critical or subcritical exponential growth, consistent with the Adams-type inequalities previously established. By applying the Mountain Pass Theorem, we demonstrate the existence of a weak solution to this problem. The primary challenge lies in the lack of compactness in the energy caused by the critical exponential growth of the non-linear term $f$.Comment: arXiv admin note: text overlap with arXiv:2201.1043

Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Blowup for Biharmonic NLS

We consider the Cauchy problem for the biharmonic (i.\,e.~fourth-order) NLS with focusing nonlinearity given by $i \partial_t u = \Delta^2 u - \mu \Delta u -|u|^{2 \sigma} u$ for $(t,x) \in [0,T) \times \mathbb{R}^d$, where $0 < \sigma <\infty$ for $d \leq 4$ and $0 < \sigma \leq 4/(d-4)$ for $d \geq 5$; and $\mu \in \mathbb{R}$ is some parameter to include a possible lower-order dispersion. In the mass-supercritical case $\sigma > 4/d$, we prove a general result on finite-time blowup for radial data in $H^2(\mathbb{R}^d)$ in any dimension $d \geq 2$. Moreover, we derive a universal upper bound for the blowup rate for suitable $4/d < \sigma < 4/(d-4)$. In the mass-critical case $\sigma=4/d$, we prove a general blowup result in finite or infinite time for radial data in $H^2(\mathbb{R}^d)$. As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems. In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.Comment: Revised version. Corrected some minor typos, added some remarks and included reference [12

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Hamiltonian elliptic systems: a guide to variational frameworks

Consider a Hamiltonian system of type $$-\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega$$ where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this paper. With respect to the original version, this one contains additional references, and some misprints were correcte

Topics in elliptic problems: from semilinear equations to shape optimization

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.Comment: Review article focused on the author's own work(expanded version of his Habilitation lecture).Draws heavily from: arXiv:2305.02870,arXiv:2211.04839,arXiv:2209.02113, arXiv:2109.14753,arXiv:2106.03661,arXiv:2106.00579,arXiv:1908.11090,arXiv:1807.03082, arXiv:1706.08391, arXiv:1701.05005, arXiv:1508.01783,arXiv:1412.4336,arXiv:1409.5693,arXiv:1405.5549,arXiv:1403.6313,arXiv:1307.3981,arXiv:1201.520

Existence of sign-changing solutions for a logarithmic weighted Kirchhoff problem in the whole of RN\mathbb{R}^{N} with exponential growth non-linearity

In this work, we establish the existence of solutions that change sign at low energy for a non-local weighted Kirchhoff problem in the set $\mathbb{R}^{N}, N>2$. The non-linearity of the equation is assumed to have exponential growth in view of the logarithmic weighted Trudinger-Moser inequalities. To obtain the existence result, we apply the constrained minimisation in the Nehari set, the quantitative deformation lemma and results from degree theory.Comment: arXiv admin note: substantial text overlap with arXiv:2304.1161