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David G. Costa

Publications and source records attributed to David G. Costa.

7 recordsLinked to original sources
arXiv2025

A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces

After a brief review of the definition of the Trudinger-Moser functions in dimension $N=2$ and some basic notions in the theory of ``Reproducing Kernel Hilbert Spaces (RKHS)'', we will show that there is a close connection between those two topics. More precisely, among other things, we start by considering a properly chosen multiple of the classical Trudinger-Moser family of functions in dimension $N=2$, which we denote by $ γ_t (r) := \frac{1}{2π}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $γ_t$ can be seen as a ``bounded'' (linear) evaluation functional $u \longrightarrow u(t)$ for functions $u$ in a suitable Hilbert Space ${\cal H}$. A slightly different definition for a ''Trudinger-Moser'' type function will also be considered for $N\geq 3$.

math.FA↗
arXiv2019

The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth \begin{align*} L(u)&-λh(x)|x|^{-2(1+a)}u=μf(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N, \end{align*} where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $λ, μ$ are positive real parameters and $1 0$. Using the idea {of the constrained minimization on} Nehari manifold we show the existence of at least two positive solutions for suitable choices of $λ$ and $μ$.

math.AP↗
arXiv2017

Existence and concentration of positive solutions for nonlinear Kirchhoff type problems with a general critical nonlinearity

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) Δu+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in C(\mathbb{R}^+,\mathbb{R}^+)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$ and $f(s)$ is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of $V$ as $\varepsilon\to 0$ under certain conditions on $f(s)$, $M$ and $V$. In particular, the monotonicity of $f(s)/s$ and the Ambrosetti-Rabinowitz condition are not required.

math.AP↗
arXiv2014

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$, $a>0$, $α\geq 0$ and $2^*=\frac{2N}{N-2}$. We show that the the exponent $q=\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem $$ \left\{\begin{array}{ll} -Δu+au=u^{2^*-1}-αu^{q-1}&\mbox{in}\ Ω,\\ u>0&\mbox{in}\ Ω,\\ \frac{\partial u}{\partialν}=0&\mbox{on}\ \partialΩ. \end{array}\right. $$ Namely, we prove that when $q=\frac{2(N-1)}{N-2}$ there exists an $α_{0}>0$ such that the problem has a least energy solution if $α<α_{0}$ and has no least energy solution if $α>α_{0}$.

math.AP↗
arXiv2012

Concentration profiles for the Trudinger-Moser functional are shaped like toy pyramids

This paper answers the conjecture of Adimurthi and Struwe that the semilinear Trudinger-Moser functional (as well as functionals with more general critical nonlinearities) satisfies the Palais-Smale condition at all levels except n/2 for integer n. In this paper we construct critical sequences at any level greater than 1/2 corresponding to a large family of distinct concentration profiles, indexed by all closed subsets C of (0,1) that arise in the two-dimensional case instead of the "standard bubble" in higher dimensions. The approach is based on the profile decomposition in the style of Solimini.

math.AP↗