Jacques Giacomoni

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A SINGULAR AND CRITICAL ELLIPTIC PROBLEM IN ℝ2

In this paper, we are interested in the following singular problem: where 0 0, 0 < δ < 3, Ω a smooth bounded domain. We show on suitable conditions on h there exist two solutions in . Investigating the radial case, we are able to prove that the condition at infinity for h is optimal for getting the existence of the second solution.

Fractional elliptic equations with critical exponential nonlinearity

Abstract We study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

Quasilinear and singular elliptic systems

Abstract. In this paper, we investigate the following quasilinear elliptic and singular system : where is an open bounded domain with smooth boundary in , , and are two positive functions. Under suitable conditions on f1 and f2, we first give a general result on the existence of positive weak solutions pairs to . Next, we give some applications to Biology.

Positive solutions of fractional elliptic equation with critical and singular nonlinearity

Abstract In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity: ( - Δ ) s ⁢ u = u - q + λ ⁢ u 2 s * - 1 , u > 0   in ⁢ Ω , u = 0   in ⁢ ℝ n ∖ Ω , (-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }% \Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} , n > 2 ⁢ s {n>2s} , s ∈ ( 0 , 1 ) {s\in(0,1)} , λ > 0 {\lambda>0} , q > 0 {q>0} and 2 s * = 2 ⁢ n n - 2 ⁢ s {2^{*}_{s}=\frac{2n}{n-2s}} . We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.

Existence of mild solutions for a singular parabolic equation and stabilization

Abstract In this paper, we study the existence and the uniqueness of a positive mild solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: (St) ∂tu - Δpu = u -δ + f(x,u) in (0,T) × Ω =: QT, u = 0 on (0,T) × ∂Ω, u > 0 in QT, u(0,x) = u0 ≥ 0 in Ω, where Ω stands for a regular bounded domain of ℝN, Δpu is the p-Laplacian operator defined by Δpu = div(|∇u|p-2|∇u|) 1 < p < ∞, δ > 0 and T > 0. The nonlinear term f : Ω × ℝ → ℝ is a bounded below Carathéodory function and nonincreasing with respect to the second variable (for a.e. x ∈ Ω). We prove for any initial positive data u0 ∈ 𝒟(A) ¯ L ∞

Quasilinear parabolic problem with p(x)-laplacian: existence, uniqueness of weak solutions and stabilization

\overline{{\mathcal {D}}(A)}^{L^\infty }

Some results about an anisotropic -Laplace–Barenblatt equation

the existence of a mild solution to (St). Then, we deduce some stabilization results for problem (St) in L∞(Ω) when p ≥ 2. This complements some results obtained in [J. Differential Equations 252 (2012), 5042–5075] stated with the additional restriction δ < 2 + 1/(p - 1).

Quasilinear parabolic problem with variable exponent: Qualitative analysis and stabilization

AbstractWe discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation

Existence of compact support solutions for a quasilinear and singular problem

\left\{ \begin{array}{ll} u_t-\Delta _{p(x)}u = f(x,u)&\quad \text{in } \quad Q_T \stackrel{{\rm{def}}}{=} (0,T)\times\Omega,\\ u = 0 & \quad\text{on} \quad \Sigma_T\stackrel{{\rm{def}}}{=} (0,T)\times\partial\Omega,\\ u(0,x)=u_0(x)& \quad \text{in} \quad \Omega \end{array} \right. \quad\quad (P_{T})