Louis Jeanjean

On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝ N

Using the “monotonicity trick” introduced by M. Struwe we derive a generic theorem. It says that for a wide class of functionals, having a Mountain-Pass geometry, almost every functionals in this class has a bounded Palais-Smale sequence at the Mountain-Pass level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties which help to insure its convergence. Subsequently these abstract results are applied to prove the existence of a positive solution for a problem of the form −∆u + Ku = f(x, u) u ∈ H1(IR ),K > 0. } (P ) We assume that the functional associated to (P) has a Mountain-Pass geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈]0,∞] as s → +∞ and (ii) f(x, s)s−1 is non decreasing as a function of s ≥ 0, a.e. x ∈ IR .

A remark on least energy solutions in RN

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R N : -Δu=g(u), u ∈ H 1 (R N ), where N > 2. Without the assumption of the monotonicity of t → g(t)/t, we show that the mountain pass value gives the least energy level.

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε2Δu − V(x) u + f(u) = 0, x ∈ RN, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal.

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

We study a class of quasi-linear Schrodinger equations arising in the theory of superfluid film in plasma physics. 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.

Bounded Palais-Smale mountain-pass sequences

Abstract Let I (λ, ·), λ e ℝ, be a family of C 1 -functionals having mountain-pass geometry. Under hypotheses which do not ensure that the mountain-pass level c (λ) is a monotone function of λ, it is shown that I (λ) has a bounded Palais-Smale sequence at level c (λ), for almost every λ.

Local conditions insuring bifurcation from the continuous spectrum

We consider a family of equations −∆u(x) + λu(x) = f(x, u(x)), λ > 0, x ∈ R , (1)λ where the nonlinearity f : RN ×R → R satisfiesf(x, 0) = 0, a.e.x ∈ RN . We say thatλ = 0 is a bifurcation point for(1)λ if there exists a sequence {(λn, un)} ⊂ R+ × H1(RN ) of nontrivial solutions of(1)λn with λn → 0 and ||un||H1(RN ) → 0. In this case{(λn, un)} is called a bifurcating sequence. The aim of the paper is to show that weak conditions on f(x, .) around zero suffice to guarantee that λ = 0 is a bifurcation point for(1)λ. More precisely suppose there is δ > 0 such that (H1) f : RN × [−δ, δ] → R is Caratheodory. (H2) lim |x|→∞ f(x, s) = 0 uniformly for s ∈ [−δ, δ].

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007).

EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R 2 FOR A CLASS OF PERIODIC ALLEN–CAHN EQUATIONS

ABSTRACT We consider a class of periodic Allen–Cahn equations where is an even, periodic, positive function and is modeled on the classical two well Ginzburg–Landau potential . We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of 1 asymptotic as to the pure states ±b, i.e., solutions satisfying the boundary conditions In fact, we prove the existence of solutions of 1-2 which are periodic in the y variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of 1-2 asymptotic to different periodic solutions as .

A note on a mountain pass characterization of least energy solutions

Abstract We consider the equation -uʺ = g(u), u(x) ∈ H1(ℝ). (0.1) Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on ℝN.

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.