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Louis Jeanjean

Publications and source records attributed to Louis Jeanjean.

50 records · Page 3Linked to original sources
arXiv2014

Multiple solutions for an indefinite elliptic problem with critical growth in the gradient

We consider the problem $(P)$, $$ -Δu =c(x)u+μ|\nabla u|^2 +f(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω),$$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $μ>0, \, c \in \mathcal{C}(\overlineΩ),$ and $ f \in L^q(Ω)$ for some $ q>\frac{N}{2}$ with $ f\gneqq 0. $ Here $c$ is allowed to change sign. We show that when $c^+ \not \equiv 0$ and $c^+ +μf$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c \leq 0$, where uniqueness holds. To show this multiplicity result we first transform $(P)$ into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so called slow growth at infinity. The key point is to show that the Palais-Smale condition holds.

math.AP↗
arXiv2014

Continuum of solutions for an elliptic problem with critical growth in the gradient

We consider the boundary value problem \begin{equation*} - Δu = λc(x)u+ μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω) \eqno{(P_λ)} \end{equation*} where $Ω\subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(Ω)$ for some $p > N/2$ and that $μ\in L^{\infty}(Ω).$ We explicit a condition which guarantees the existence of a unique solution of $(P_λ)$ when $λ<0$ and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of $(P_0)$. It crosses the axis $λ=0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $λ=0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $μ(x)\geq μ_1>0$ and $h\gneqq 0$, we show that the continuum bifurcates from infinity on the right of the axis $λ=0$ and this implies, in particular, the existence of two solutions for any $λ>0$ sufficiently small.

math.AP↗
arXiv2013

Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions

In this note we present some uniqueness and comparison results for a class of problem of the form \begin{equation} \label{EE0} \begin{array}{c} - L u = H(x,u,\nabla u)+ h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω), \end{array} \end{equation} where $Ω\subset \R^N$, $N \geq 2$ is a bounded domain, $L$ is a general elliptic second order linear operator with bounded coefficients and $H$ is allowed to have a critical growth in the gradient. In some cases our assumptions prove to be sharp.

math.AP↗
arXiv2013

Multiplicity of positive solutions of nonlinear Schrödinger équations concentrating at a potential well

We consider singularly perturbed nonlinear Schrödinger equations \be \label{eq:0.1} - \varepsilon^2 Δu + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $Ω\subset \R^N$ such that \[m_0 \equiv \inf_{x \in Ω} V(x) < \inf_{x \in \partial Ω} V(x) \] and we set $K = \{x \in Ω\ | \ V(x) = m_0\}$. For $\e >0$ small we prove the existence of at least ${\cuplength}(K) + 1$ solutions to (\ref{eq:0.1}) concentrating, as $\e \to 0$ around $K$. We remark that, under our assumptions of $f$, the search of solutions to (\ref{eq:0.1}) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

math.AP↗
arXiv2012

Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.

math.AP↗
arXiv2012

Sharp non-existence results of prescribed L^2-norm solutions for some class of Schrödinger-Poisson and quasilinear equations

In this paper we study the existence of minimizers for $$ F(u) = \1/2\int_{\R^3} |\nabla u|^2 dx + 1/4\int_{\R^3}\int_{\R^3}\frac{| u(x) |^2| u(y) |^2}{| x-y |}dxdy-\frac{1}{p}\int_{\R^3}| u |^p dx$$ on the constraint $$S(c) = \{u \in H^1(\R^3) : \int_{\R^3}|u|^2 dx = c \},$$ where $c>0$ is a given parameter. In the range $p \in [3, 10/3]$ we explicit a threshold value of $c>0$ separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of $F(u)$ restricted to $S(c)$ when $c>0$ is sufficiently small. Finally, as a byproduct of our approaches, we extend some results of \cite{CJS} where a constrained minimization problem, associated to a quasilinear equation, is considered.

math.AP↗
arXiv2012

Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schrödinger-Poisson-Slater equations in $\R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schrödinger-Poisson type equation \label{evolution1} iψ_{t}+ Δψ- (|x|^{-1}*|ψ|^{2}) ψ+|ψ|^{p-2}ψ=0 % \text{in} \R^{3}, when $p \in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| \triangledown u|_{L^{2}(\mathbb{R}^3)}^2+1/4\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-\frac{1}{p}\int_{\mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= \{u \in H^1(\mathbb{R}^3) :|u|_{L^2(\R^3)}^2=c, c>0}.$$ For the values $p \in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0\in H^1(\R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $γ(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

math.AP↗
arXiv2009

Stability and instability results for standing waves of quasi-linear Schrödinger equations

We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. First, using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.

math.AP↗
arXiv2008

Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \R^N \to \C$, under conditions on the nonlinearity which are nearly optimal.

math.AP↗
arXiv2008

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable.

math.AP↗
arXiv2007

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N>2.

math.AP↗