[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
Milan Journal of Mathematics 06/2011; 79(1):327-341. DOI:10.1007/s00032-011-0140-0 · 0.66 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the existence and properties of solutions of the following problem, The main results are Theorems
1 and 2 in section 2, establishing the existence of solutions having any prescribed number of zeros and with the further property
that the zeros of u and u′ interlace. The solutions of (1.1)-(1.2) lead to a description of beams of light which, due to the nonlinearity of the medium
in which they propagate, remain concentrated (self-trapped) near the axis of propagation. In any plane transverse to the axis
of propagation the intensity of illumination is radially symmetric with respect to the axis and the zeros and turning points
of u correspond to circles of zero and maximal intensity. Our analysis and conclusions are similar to those in [1] for another
model for self-trapped light.
Nonlinear Diffusion Equations and Their Equilibrium States, 3, 01/2011: pages 391-405;
[Show abstract][Hide abstract] ABSTRACT: We consider an eigenvalue problem for a certain type of quasi-linear second-order 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 self-focusing 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.
Mathematical Models and Methods in Applied Sciences 09/2010; 20(02). DOI:10.1142/S0218202510004751 · 3.09 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider second-order 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 C1-Fredholm 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.
[Show abstract][Hide abstract] ABSTRACT: We consider quasilinear systems of second order elliptic equations on R-N. Using a continuation theorem based on the topological degree for C-1-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
Milan Journal of Mathematics 11/2008; 76(1):329-399. DOI:10.1007/s00032-008-0089-9 · 0.66 Impact Factor
[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| ^{p-1}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+(4-2b)/(N-2), we establish the existence of a C^1-branch 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+(4-2b)/N and unstable if 1+(4-2b)/N < p < 1+(4-2b)/(N-2).
Discrete and Continuous Dynamical Systems 05/2008; 21:137-186. DOI:10.3934/dcds.2008.21.137 · 0.83 Impact Factor
[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
Proceedings of the Royal Society of Edinburgh Section A Mathematics 11/2007; 137(06):1249 - 1285. DOI:10.1017/S0308210506000424 · 1.01 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider the nonlinear boundary-value 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 |wn|Lp → 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.
[Show abstract][Hide abstract] ABSTRACT: We consider the stationary non-linear 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.
Proceedings of the London Mathematical Society 04/2006; 92(03):655 - 681. DOI:10.1017/S0024611505015637 · 1.11 Impact Factor
Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 01/2006; 17. DOI:10.4171/RLM/471 · 0.52 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: For a function acting between Banach spaces, we recall the notions of Hadamard and w-Hadamard 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 w-Hadamard 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 -Delta u-alpha u+lambda g(x)u = 0 with u is an element of H-1(R-N), u not equal 0, where alpha, lambda is an element of R and g(x) equivalent to 0 on (Omega) over bar, g(x) is an element of (0, 1) on R-N and lim (vertical bar x vertical bar ->+infinity g) g(x) = 1 for some bounded open set Omega is an element of R-N. Given a > 0, does there exist a value of A > 0 for which the problem has a positive solution? It is shown that this occurs if and only if a lies in a certain interval (Gamma, xi(1)) and that in this case the value of lambda is unique, lambda = Lambda (alpha). The properties of the function Lambda(alpha) are also discussed.
Journal of the London Mathematical Society 10/2005; 72. DOI:10.1112/S0024610705006873 · 0.82 Impact Factor
[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 second-order 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
Proceedings of the Edinburgh Mathematical Society 01/2005; 48(01):91 - 124. DOI:10.1017/S0013091504000550 · 0.48 Impact Factor
[Show abstract][Hide abstract] 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.
Annali di Matematica Pura ed Applicata 01/2005; 184(3):297-314. DOI:10.1007/s10231-004-0114-8 · 1.07 Impact Factor