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Jacques Giacomoni

Publications and source records attributed to Jacques Giacomoni.

41 records · Page 3Linked to original sources
arXiv2015

Critical exponent for half-Laplacian in the whole space

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-Δ)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1 2,\;1<β\leq2\;, λ>0, K(x)>0, f$ is continuous and sign changing. where $h$ is a real valued function that behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $V(x)$ is a positive, continuous unbounded function. Here $(-Δ)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near $t=0$. We also study the corresponding critical exponent problem for the Kirchhoff equation \[ m\left(\int_{\mathbb R}|(-Δ)^{\frac{1}{2}}u|^2 dx+ \int_\mb R u^2 V(x)dx\right)\left((-Δ)^{\frac{1}{2}} u+ V(x) u\right)= f(u)\;\, \text{in}\, \mathbb R \] where $f(u)$ behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $f(u)\sim u^3$ as $u\rightarrow 0$.

math.AP↗
arXiv2015

Uniqueness of positive solutions of a $n$-Laplace equation in a ball in $\mathbb{r}^n$ with exponential nonlinearity

Let $n \geq 2$ and $Ω\subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(Ω)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $γ_0 > 0$, and a corresponding $R_0(γ_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$ with $\|u\|_{\infty}\geq γ_0$.

math.AP↗
arXiv2013

Quasilinear and singular elliptic systems

In this paper, we investigate a general quasilinear elliptic and singular system. By monotonicity methods, we give some existence and uniqueness results. Next, we give some applications to biological models.

math.AP↗
arXiv2011

A Singular Parabolic Equation: Existence, Stabilization

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{\rm P$_t$} \{{aligned} & u_t-Δ_p u =\frac{1}{u^δ}+f(x,u)\;\text{in}\,(0,T)\timesΩ, & u =0\,\text{on} \;(0,T)\times\partialΩ,\quad u>0 \text{in}\, (0,T)\timesΩ, &u(0,x) =u_0(x)\;\text{in}Ω, {aligned}. {equation} % where $Ω$ is an open bounded domain with smooth boundary in $\R^{\rm N}$, $1 < p< \infty$, $0<δ$ and $T>0$. We assume that $(x,s)\inΩ\times\R^+\to f(x,s)$ is a bounded below Caratheodory function, locally Lipschitz with respect to $s$ uniformly in $x\inΩ$ and asymptotically sub-homogeneous, i.e. % {equation} \label{sublineargrowth} 0 \leq\displaystyle\lim_{t\to +\infty}\frac{f(x,t)}{t^{p-1}}=α_f<λ_1(Ω), {equation} % (where $λ_1(Ω)$ is the first eigenvalue of $-Δ_p$ in $Ω$ with homogeneous Dirichlet boundary conditions) and $u_0\in L^\infty(Ω)\cap W^{1,p}_0(Ω)$, satisfying a cone condition defined below. Then, for any $δ\in (0,2+\frac{1}{p-1})$, we prove the existence and the uniqueness of a weak solution $u \in{\bf V}(Q_T)$ to $({\rm P_t})$. Furthermore, $u\in C([0,T], W^{1,p}_0(Ω))$ and the restriction $δ<2+\frac{1}{p-1}$ is sharp. Finally, in the last section we analyse the case $p=2$. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to $({\rm P}_t)$ for any $δ>0$ in $C([0,T], L^2(Ω))\cap L^\infty(Q_T)$ and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in $L^\infty(Ω)\cap H^1_0(Ω)$ when $δ<3$.

math.AP↗