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Pierpaolo Esposito

Publications and source records attributed to Pierpaolo Esposito.

At least 19 recordsLinked to original sources
arXiv2026

On the classification of solutions to a class of $N$-Liouville equations in $\mathbb{R}^N$

Given $N\geq 2$ and $α>-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left\{ \begin{aligned} -& Δ_N u = |x|^{Nα} e^u \quad \text{ in } \mathbb{R}^N && , \\ & \int_{\mathbb{R}^N} |x|^{Nα} e^u \, dx < + \infty\,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $α=0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $α>-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\geq 3$ by providing the classification for any $-1<α\leq 0$. In particular, the explicit radial solutions are the unique ones for $-1<α\leq0$ but become degenerate for special values $α_k>0$, a hint that non-radial solutions might arise for $α>0$ as it happens when $N=2$.

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arXiv2026

Entire solutions to a strongly competitive nonlinear Schrödinger system

We build infinitely-many non-radial positive solutions to the Schrödinger system \begin{equation*} \left\{\begin{aligned} &-Δu_1+u_1=u_1^{{\mathfrak p} }-Λu_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-Δu_2+u_2=u_2^{{\mathfrak p} }-Λu_1^{b_1}u_2^{b_2} \ \hbox{in}\ \mathbb R^N\\ \end{aligned}\right. \end{equation*} with sub-critical $\mathfrak p$-growth as $Λ\to +\infty$. The profile of each component is the sum of several copies of the positive solution to $-ΔU+U=U^{{\mathfrak p} }$ in $\mathbb R^N$, centered at suitable {\em peaks} whose mutual distances diverge as $Λ$ increases. More precisely, given two concentric regular polygons with $k$ sides and very large radii, the peaks of the first component are arranged along the edges of the {\em outer} polygon, alternated with those of the second component, and along the $k$ rays joining the vertices of the two polygons. To the best of our knowledge, this provides the first example of non-radial positive solutions for strongly competitive Schrödinger systems in the whole space.

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arXiv2023

The Green function for p-Laplace operators

On a bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 2$, we consider existence, uniqueness and "regularity" issues for the Green function $G_λ$ of the quasi-linear operator $u \to -Δ_p u-λ|u|^{p-2}u$ with $1 0$ is the first eigenvalue of $-Δ_p$.

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arXiv2023

The quasi-linear Brezis-Nirenberg problem in low dimensions

We discuss existence results for a quasi-linear elliptic equation of critical Sobolev growth [H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437--477; M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879--902] in the low-dimensional case, where the problem has a global character which is encoded in sign properties of the ``regular" part for the corresponding Green's function as in [O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 2, 125--142; P. Esposito, On some conjectures proposed by Haïm Brezis, Nonlinear Anal. 54 (2004), no. 5, 751--759].

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arXiv2021

Harnack inequalities and quantization properties for the n-Liouville equation

We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtained by the first author in P. Esposito, A classification result for the quasi-linear Liouville equation. Ann. Inst. H. Poincaré Anal. Non Linèaire 35 (2018), no. 3, 781--801. A fundamental tool is provided here by some Harnack inequality of "sup+inf" type, a question of independent interest that we prove in the quasilinear context through a new and simple blow-up approach.

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arXiv2021

Isolated singularities for the n-Liouville equation

In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.

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arXiv2019

On the mean field equation with variable intensities on pierced domains

We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$\left\{ \begin{array}{ll} -Δu=λ_1\dfrac{V_1 e^{u}}{ \int_{Ω_{\boldsymbolε}} V_1 e^{u} dx } - λ_2τ\dfrac{ V_2 e^{-τu}}{ \int_{Ω_{\boldsymbolε}}V_2 e^{ - τu} dx}&\text{in $Ω_{\boldsymbolε}=Ω\setminus \displaystyle \bigcup_{i=1}^m \overline{B(ξ_i,ε_i)}$}\\ \ \ u=0 &\text{on $\partial Ω_{\boldsymbolε}$}, \end{array} \right. $$ where $B(ξ_i,ε_i)$ is a ball centered at $ξ_i\inΩ$ with radius $ε_i$, $τ$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $λ_1>8πm_1$ and $λ_2 τ^2>8π(m-m_1)$ with $m_1 \in \{0,1,\dots,m\}$, there exist radii $ε_1,\dots,ε_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $ξ_1,\dots,ξ_{m_1}$ and $ξ_{m_1+1},\dots,ξ_{m}$, respectively, as the radii approach zero.

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arXiv2019

Critical metrics for Log-determinant functionals in conformal geometry

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula. These correspond to solutions of elliptic equations of Liouville type that are quasilinear, of mixed orders and of critical type. After studying existence, asymptotic behaviour and uniqueness of fundamental solutions, we prove a quantization property under blow-up, and then derive existence results via critical point theory.

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arXiv2017

Sign-Changing Solutions for Critical Equations with Hardy Potential

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schrödinger operator on a smooth bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 3$, with $0 \in Ω$: $$ \left\{ \begin{array}{ll}-Δu-γ\frac{u}{|x|^2}-εu=|u|^{\frac{4}{N-2}}u &\hbox{in }Ωu=0 & \hbox{on }\partial Ω, \end{array}\right. $$ when $ε>0$ is small and $γ< {(N-2)^2\over4}$. Setting $ γ_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $γ\leq \frac{(N-2)^2}{4}-1$ and $γ\neq γ_j$ for any $j$, then for small $ε$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $γ<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $ε$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $γ\neq γ_j$ is not necessary. Indeed, it is known that, if $γ> \frac{(N-2)^2}{4}-1$ and $Ω$ is a ball $B$, then there is no radial positive solution for $ε>0$ small. We complete the picture here by showing that, if $γ\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $ε>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $γ=0$.

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arXiv2015

Equilibria of point-vortices on closed surfaces

We discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface $Σ$. The topological properties of $Σ$ determine the occurrence of three distinct situations, corresponding to $\mathbb{S}^2$, to $\mathbb{RP}^2$ and to $Σ\not=\mathbb{S}^2,\mathbb{RP}^2$. As a by-product, we also obtain new existence results for the singular mean-field equation with exponential nonlinearity.

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arXiv2014

On a quasilinear mean field equation with exponential nonlinearity

The mean field equation involving the $N$-Laplace operator and an exponential nonlinearity is considered in dimension $N\geq2$ on bounded domains with homogenoeus Dirichlet boundary condition. By a detailed asymptotic analysis we derive a quantization property in the non-compact case, yielding to the compactness of the solutions set in the so-called non-resonant regime. In such a regime, an existence result is then provided by a variational approach.

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arXiv2014

Non-topological condensates for the self-dual Chern-Simons-Higgs model

For the abelian self-dual Chern-Simons-Higgs model we address existence issues of periodic vortex configurations -- the so-called condensates-- of non-topological type as $k \to 0$, where $k>0$ is the Chern-Simons parameter. We provide a positive answer to the long-standing problem on the existence of non-topological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as $k \to 0$, a question which is of definite physical interest.

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arXiv2014

Singular mean field equations on compact Riemann surfaces

For a general class of elliptic PDE's in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack.

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arXiv2012

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$Δ_g u+(N-2/4(N-1) S_g+εh)u=u^{N+2/N-2} in M, u>0 in M,$$ where Δ_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,α}(M)$, and $ε$ is a small parameter.

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arXiv2012

The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). We show that a-priori L^\infty-bounds fail for linear perturbations on all manifolds with n\geq 4 as well as a-priori gradient L^2--bounds fail for non-locally conformally flat manifolds with n\geq 6 and for locally conformally flat manifolds with n\geq 7. In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g.

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arXiv2012

Some remarks concerning symmetry-breaking for the Ginzburg-Landau equation

The correlation term, introduced in [13] to describe the interaction between very far apart vortices, governs symmetry-breaking for the Ginzburg-Landau equation in R^2 or bounded domains. It is a homogeneous function of degree (-2), and then for 2π/N-symmetric vortex configurations can be expressed in terms of the so-called correlation coefficient. Ovchinnikov and Sigal [13] have computed it in few cases and conjectured its value to be an integer multiple of π/4. We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequences.

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