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J. Giacomoni

Publications and source records attributed to J. Giacomoni.

16 recordsLinked to original sources
arXiv2026

Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu |u|^{q-2}u +(I_{\alpha}*|u|^{p^*_{\alpha}})|u|^{p^*_{\alpha}-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & \tau. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p 0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (\max\{0,N-2p\}, N)$, $p^*_{\alpha}=\frac{p}{2}\left(\frac{N+\alpha}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.

math.AP↗
arXiv2025

Existence and Nonexistence Breaking Results For a Weighted Elliptic Problem in Half-Space

In this paper we study the problem $-\mathrm{div}(ρ(x_N)\nabla u)=a|u|^{p-2}u$ in $\mathbb{R}^N_+$, $-\partial u/\partial x_N=b|u|^{q-2}u$ in $\mathbb{R}^{N-1}$ where $a,b \in \mathbb{R}$, $p,q\in (1,\infty)$ and $ρ$ is a positive weight. We establish regularity results for weak solutions and, using a variational approach combined with a new Pohozaev-type identity, we show that the introduction of the weighted operator $-\mathrm{div}(ρ(x_N)\nabla u)$ can reverse the known solvability behavior of the classical Laplacian case. Specifically, we identify regimes where the problem admits solutions despite nonexistence for the corresponding case with $-Δ$, and vice versa, thus inverting the classical existence and nonexistence results.

math.AP↗
arXiv2023

Hölder regularity results for parabolic nonlocal double phase problems

In this article, we obtain higher Hölder regularity results for weak solutions to nonlocal problems driven by the fractional double phase operator \begin{align*} \mc L u(x):=&2 \; {\rm P.V.} \int_{\mathbb R^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps_1}}dy \nonumber &+2 \; {\rm P.V.} \int_{\mathbb R^N} a(x,y) \frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{N+qs_2}}dy, \end{align*} where $1<p\leq q<\infty$, $0<s_2, s_1<1$ and the modulating coefficient $a(\cdot,\cdot)$ is a non-negative bounded function. Specifically, we prove higher space-time Hölder continuity result for weak solutions of time depending nonlocal double phase problems for a particular subclass of the modulating coefficients. Using suitable approximation arguments, we further establish higher (global) Hölder continuity results for weak solutions to the stationary problems involving the operator $\mc L$ with modulating coefficients that are locally continuous.

math.AP↗
arXiv2023

Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical point theory of nonsmooth analysis and the geometry of the energy functional, we show the existence and multiplicity of positive solutions with respect to the parameter $λ$.

math.AP↗
arXiv2023

Fractional Hamiltonian type system on $\mathbb{R}$ with critical growth nonlinearity

This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system: \begin{align*} \begin{cases} (-Δ)^\frac12 u +V_0 u =g(v),~x\in \mathbb{R} (-Δ)^\frac12 v +V_0 v =f(u),~x\in \mathbb{R}, \end{cases} \end{align*} where $(-Δ)^\frac12$ is the square root Laplacian operator, $V_0 >0$ and $f,~g$ have critical exponential growth in $\mathbb{R}$. Using minimization technique over some generalized Nehari manifold, we show that the set $\mathcal{S}$ of ground state solutions is non empty. Moreover for $(u,v) \in \mathcal{S}$, $u,~v$ are uniformly bounded in $L^\infty(\mathbb{R})$ and uniformly decaying at infinity. We also show that the set $\mathcal{S}$ is compact in $H^\frac12(\mathbb{R}) \times H^\frac12(\mathbb{R})$ up to translations. Furthermore under locally lipschitz continuity of $f$ and $g$ we obtain a suitable Pohožaev type identity for any $(u,v) \in \mathcal{S}$. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system \begin{align*} \begin{cases} ε(-Δ)^\frac12 φ+V(x) φ=g(ψ),~x\in \mathbb{R} ε(-Δ)^\frac12 ψ+V(x) ψ=f(φ),~x\in \mathbb{R}, \end{cases} \end{align*} where $ε>0$ and $V \in C(\mathbb{R})$ satisfy the assumption $(V)$ given below (see Section 1). Finally as $ε\rightarrow 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential $V$.

math.AP↗
arXiv2022

A note on the global regularity results for strongly nonhomogeneous $p,q$-fractional problems and applications

In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev versus Hölder minimizer type result, which provides the multiplicity of solutions in the spirit of seminal work \cite{Brezis-Nirenberg}.

math.AP↗
arXiv2021

Some existence and uniqueness results for logistic Choquard equation

We consider the following doubly nonlocal nonlinear logistic problem driven by the fractional $p$-Laplacian \begin{equation*} \pl u = f(x,u) -\cq ~\text{in}~ Ø, ~u=0 ~\text{in}~ \Rn\setminusØ. \end{equation*} Here $ Ø\subset \Rn (N\geq2)$ is a bounded domain with $ C^{1,1}$ boundary $\partial Ø$, $ s \in (0,1) $, $p \in (1,\infty)$ are such that $ps < N$. Also $p_{s,\a}^\#\leq r<\infty$ , where $p_{s,\a}^\#=(2N-\a)/2N$. Under suitable and general assumptions on the nonlinearity $f$, we study the existence, nonexistence, uniqueness, and regularity of weak solutions. As for applications, we treat cases of subdiffusive type logistic Choquard problem. We also consider in the superdiffusive case the Brezis-Nirenberg type problem with logistic Choquard and show the existence of a nontrivial solution for a suitable choice of $ł$. Finally for a particular choice of $f$ viz. $f(x,t)=łt^{q-1}$ with $1<p<2r<q$, we show the existence of at least one energy nodal solution.

math.AP↗
arXiv2021

Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case $\ba<2-\frac{1}{p}$. Subsequently, for the case $\ba\ge p$, we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see \eqref{Prb}). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of $(P)$ for the case $β+δ\geq 1$ that has not been fully answered in former contributions even for $p$-Laplace operators.

math.AP↗
arXiv2019

Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights

In this work, we study the higher order Kirchhoff type Choquard equation $(KC)$ involving a critical exponential non-linearity and singular weights. We prove the existence of solution to $(KC)$ using Mountain pass Lemma in light of Moser-Trudinger and singular Adams-Moser inequalities. In the second part of the paper, using the Nehari manifold technique and minimization over its suitable subsets, we prove the existence of at least two solutions to the Kirchhoff type Choquard equation $(\mathcal{P_{\la,M}})$ involving convex-concave type non-linearity.

math.AP↗
arXiv2018

Nehari Manifold for fractional Kirchhoff system with critical nonlinearity

In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=λf(x)|u|^{q-2}u+ \frac{2α}{α+β}\left|u\right|^{α-2}u|v|^β& \text{in } Ω,\\ \mc L_M(v)&=μg(x)|v|^{q-2}v+ \frac{2β}{α+β}\left|u\right|^α|v|^{β-2}v & \text{in } Ω,\\ u&=v=0 &\mbox{in } \mathbb{R}^{N}\setminus Ω, \end{array} \right. \end{equation} where $\mc L_M(u)=M\left(\displaystyle \int_Ω|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+b t$ with $a, b>0$ and fractional Laplacian $(-Δ)^{s}, s\in(0, 1)$. We consider that $Ω$ is a bounded domain in $\mathbb{R}^N$, {$2s 0$ are {real} parameters, $1<q<2$, $α, β\ge 2$ {and} $α+β=2_s^*={2N}/(N-2s)$ {is a fractional critical exponent}. Using the idea of Nehari manifold technique and a compactness result based on {classical idea of Brezis-Lieb Lemma}, we prove the existence of at least two positive solutions for $(λ, μ)$ lying in a suitable subset of $\mathbb R^2_+$.

math.AP↗
arXiv2018

Radial singular solutions for the N-Laplace Equation with exponential nonlinearities

In this paper, we consider radial distributional solutions of the quasilinear equation $-Δ_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for extending such solutions to the whole domain $B_R$ by preserving the regularity. For a certain class of noninearity $f$ we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.

math.AP↗
arXiv2018

A Global multiplicity result for a very singular critical nonlocal equation

In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2s,\; s \in (0,1),\; \la >0,\; q>0$ satisfies $q(2s-1)<(2s+1)$ and $2^*_s=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(P_\la)$ in $X_0$ when $\la \in (0,\La)$ and no solution when $\la>\La$, where $\La>0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $(P_λ)$ is in $C^α(\R^n)$ with $α=α(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.

math.AP↗
arXiv2017

Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mb R^n$, $n >2s$, $s \in (0,1)$, $(-Δ)^s$ is the well known fractional Laplacian, $μ\in (0,n)$, $2^*_μ= \displaystyle\frac{2n-μ}{n-2s}$ is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, $1 0$ are real parameters. We study the fibering maps corresponding to the functional associated with $(P_{λ,δ})$ and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of $(P_{\la,δ})$ for suitable range of $\la$ and $δ$.

math.AP↗
arXiv2017

Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-Δ)^s u &= u^{-q} + f(x,u), \;u >0\; \text{in}\; (0,T) \times Ω, u &= 0 \; \mbox{in}\; (0,T) \times (\mb R^n \setminusΩ), \quad \quad \quad \quad u(0,x)&=u_0(x) \; \mbox{in} \; {\mb R^n}, \end{split} \quad \right. \end{equation*} where $Ω$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial Ω$, $n> 2s, \;s \in (0,1)$, $q>0$, ${q(2s-1)<(2s+1)}$, $u_0 \in L^\infty(Ω)\cap X_0(Ω)$ and $T>0$. We suppose that the map $(x,y)\in Ω\times \mb R^+ \mapsto f(x,y)$ is a bounded below Carathéodary function, locally Lipschitz with respect to second variable and uniformly for $x \in Ω$ it satisfies \begin{equation}\label{cond_on_f} { \limsup_{y \to +\infty} \frac{f(x,y)}{y}<λ_1^s(Ω)}, \end{equation} where $\la_1^s(Ω)$ is the first eigenvalue of $(-Δ)^s$ in $Ω$ with homogeneous Dirichlet boundary condition in $\mathbb{R}^n \setminus Ω$. We prove the existence and uniqueness of weak solution to $(P_t^s)$ on assuming $u_0$ satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of $(P_t^s)$ when we regularize our initial function $u_0$.

math.AP↗
arXiv2015

Biharmonic equation with singular nonlinearity

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & Δ^2 u = K(x)u^{-α} \quad \mbox{ in }\,Ω, \\ &u> 0\quad \mbox{ in }\,Ω, \;\;u\vert_{\partialΩ}=0, \,Δu\vert_{\partialΩ} = 0. \end{array}\right. \end{eqnarray*} We prove the main existence result: Assume that $α+β<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \begin{eqnarray}\label{behaviour-bound} c_1 ρ(x)\leq u(x)\leq c_2 ρ(x) \end{eqnarray} where $ρ(x)=d(x,\partialΩ)$. This result is sharp: Assume that $α+β\geq 2$. Then, there is no solution to $(P)$.

math.AP↗