Publications (49)47.11 Total impact
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ABSTRACT: This paper deals with asymptotic bifurcation, first in the abstract setting of an equation G(u)=λuG(u)=λu, where G acts between real Hilbert spaces and λ∈Rλ∈R, and then for squareintegrable solutions of a second order nonlinear elliptic equation on RNRN. The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the Hasymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the Hasymptotic derivative if it is too close to the essential spectrum.  [Show abstract] [Hide abstract]
ABSTRACT: For a mapping between Banach spaces, two weaker variants of the usual notion of asymptotic linearity are defined and explored. It is shown that, under inversion through the unit sphere, they correspond to Hadamard and weak Hadamard differentiability at the origin of the inversion. Nemytskii operators from Sobolev spaces to Lebesgue spaces over RNRN share these weaker properties but they are not asymptotically linear in the usual sense.  [Show abstract] [Hide abstract]
ABSTRACT: For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C 1–functional on H01(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type. Mathematics Subject Classification (2010)Primary 35J62–Secondary 35J60  [Show abstract] [Hide abstract]
ABSTRACT: We consider an eigenvalue problem for a certain type of quasilinear secondorder differential equation on the interval (0, ∞). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a selffocusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.  [Show abstract] [Hide abstract]
ABSTRACT: We consider secondorder quasilinear elliptic systems on unbounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C1Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabier's approach to proving exponential decay to zero at infinity of solutions. 
Article: Global Continuation for Quasilinear Elliptic Systems on RN and the Equations of Elastostatics
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ABSTRACT: We consider quasilinear systems of second order elliptic equations on IRN. Using a continuation theorem based on the topological degree for C1 Fredholm maps, we derive global properties of a maximal connected set of solutions which decay exponentially to zero at infinity. These results are used to treat a problem concerning the equilibrium of an elastic body occupying the whole space and subjected to a one parameter family of localized external forces.  [Show abstract] [Hide abstract]
ABSTRACT: In the first part of these notes, we deal with first order Hamiltonian systems in the form $Ju\prime(t) = \bigtriangledown H(u(t))$Ju\prime(t) = \bigtriangledown H(u(t)) where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian $H \,\epsilon\, C^{1}(X,{\mathbb{R}})$H \,\epsilon\, C^{1}(X,{\mathbb{R}}) is required to be invariant with respect to the action of a group In the first part of these notes, we deal with first order Hamiltonian systems in the form $Ju\prime(t) = \bigtriangledown H(u(t))$Ju\prime(t) = \bigtriangledown H(u(t)) where the phase space X may be infinite dimensional so as to accommodate some partial differential equations. The Hamiltonian $H \,\epsilon\, C^{1}(X,{\mathbb{R}})$H \,\epsilon\, C^{1}(X,{\mathbb{R}}) is required to be invariant with respect to the action of a group {etA : t e  [Show abstract] [Hide abstract]
ABSTRACT: We study the bifurcation points of an equation of the form F(u)=λuF(u)=λu in a real Hilbert space. Since FF is only required to be Hadamard, but not Fréchet, differentiable at u=0u=0, bifurcation points need not belong to the spectrum of F′(0)F′(0). The abstract results are illustrated in the case of a nonlinear Schrödinger equation.  [Show abstract] [Hide abstract]
ABSTRACT: For N\geq3 and p>1, we consider the nonlinear Schrödinger equation i\partial_{t}w+\Delta_{x}w+V(x) w ^{p1}w=0 where w=w(t,x):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C} with a potential V that decays at infinity like  x^{b} for some b\in (0,2). A standing wave is a solution of the form w(t,x)=e^{i\lambda t}u(x) where \lambda>0 and u:\mathbb{R}^{N}\rightarrow\mathbb{R}. For 1 < p < 1+(42b)/(N2), we establish the existence of a C^1branch of standing waves parametrized by frequencies \lambda in a right neighbourhood of 0. We also prove that these standing waves are orbitally stable if 1 < p < 1+(42b)/N and unstable if 1+(42b)/N < p < 1+(42b)/(N2).  [Show abstract] [Hide abstract]
ABSTRACT: In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form  [Show abstract] [Hide abstract]
ABSTRACT: Starting from Maxwell's equations for a stratified optical medium with a nonlinear refractive index, we derive the equations for monochromatic planar TE modes. It is then shown that TE modes in which the electromagnetic fields are travelling waves correspond to solutions of these reduced equations in the form of standing waves. The equations of the paraxial approximation are then formulated and the stability of the travelling waves is investigated in that context. © The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the nonlinear boundaryvalue problem in ℝN (N ≥ 3), ∇·{C(x)∇u(x)} = λf(u(x)) for x < 1 u(x) = 0 for x = 1, where C ∈C1([0,1]) with C(r) > 0 for all r ∈(0,1], C(0) = 0 and limr→0 C(r)/r2 = 1 and, for some T > 0, f ∈ C1 ([T, T]) is an odd function that is strictly concave on [0, T] with f(0) = f(T) = 0 and f′(0) = 1. We prove that there is vertical bifurcation of radial solutions at every λ > N2/4 in the sense that there exists a sequence {(λ, wn)} of solutions such that wnLp → 0 as n → ∞ for all p ∈[1, ∞). These solutions concentrate at 0 in the sense that wn(0) = T for all n but wn converges uniformly to zero on all compact subsets that do not contain zero. 
Article: Global branch of solutions for nonlinear Schrödinger equations with deepening potential well
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ABSTRACT: We consider the stationary nonlinear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{s \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{x \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero. 


Article: On differentiability and bifurcation
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ABSTRACT: For a function acting between Banach spaces, we recall the notions of Hadamard and wHadamard differentiability and their relation to the common notions of Gâteaux and Fréchet differentiability. We observe that even for a function F: H → H that is both Hadamard and wHadamard differentiable but not Fréchet differentiable at 0 on a real Hilbert space H, there may be bifurcation for the equation F(u) = λu at points λ which do not belong to the spectrum of F′(0). We establish some necessary conditions for λ to be a bifurcation point in such cases and we show how this result can be used in the context of partial differential equations such as  Du( x ) + q( x )u( x ) = l( e x  u( x ) ) for u Î H2 ( \mathbbRN )  \Delta u\left( x \right) + q\left( x \right)u\left( x \right) = \lambda \left( {e^{\left x \right} u\left( x \right)} \right) for u \in H^2 \left( {\mathbb{R}^N } \right) where this situation occurs.  [Show abstract] [Hide abstract]
ABSTRACT: The paper considers the eigenvalue problem where ∞, λ ∈ and for some bounded open set Ω∈RN. Given α>0, does there exist a value of λ>0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in a certain interval (Γ,ξ1) and that in this case the value of λ is unique, λ=Λ(α). The properties of the function Λ(α) are also discussed.  [Show abstract] [Hide abstract]
ABSTRACT: This paper addresses the existence of solutions u ∈ H 1(ℝ+; ℝN) of ODE systems u̇+ F(t, u) = f ∈ L2(ℝ+; ℝN), with boundary condition u1(0) = ξ, where u1 is a (vector) component of u. Under general conditions, the problem corresponds to a, functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H1(ℝ+; ℝN). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.  [Show abstract] [Hide abstract]
ABSTRACT: For a large class of subsets $\varOmega\subset\mathbb{R}^{N}$ (including unbounded domains), we discuss the Fredholm and properness properties of secondorder quasilinear elliptic operators viewed as mappings from $W^{2,p}(\varOmega;\mathbb{R}^{m})$ to $L^{p}(\varOmega;\mathbb{R}^{m})$ with $N\ltp\lt\infty$ and $m\geq1$. These operators arise in the study of elliptic systems of $m$ equations on $\varOmega$. A study in the case of a single equation ($m=1$) on $\mathbb{R}^{N}$ was carried out by Rabier and Stuart.AMS 2000 Mathematics subject classification: Primary 35J45; 35J60. Secondary 47A53; 47F05 
Article: Existence and nonexistence of Schwarz symmetric ground states for elliptic eigenvalue problems
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ABSTRACT: We determine a class of Carathodory functions G for which the minimum formulated in the problem (1.1) below is achieved at a Schwarz symmetric function satisfying the constraint. Our hypotheses about G seem natural and, as our examples show, they are optimal from some points of view.
Publication Stats
611  Citations  
47.11  Total Impact Points  
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Institutions

19962014

École Polytechnique Fédérale de Lausanne
 Mathematics Section
Lausanne, Vaud, Switzerland


2010

Eawag: Das WasserforschungsInstitut des ETHBereichs
Duebendorf, Zurich, Switzerland


2002

Ecole polytechnique fédérale de Lausanne
Lausanne, Vaud, Switzerland


1999

Academia Sinica
T’aipei, Taipei, Taiwan
