Silvia Cingolani

Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields

Abstract We study the existence of standing waves for a class of nonlinear Schrodinger equations in R n , with both an electric and a magnetic field. Under suitable non-degeneracy assumptions on the critical points of an auxiliary function related to the electric field, we prove the existence and the multiplicity of complex-valued solutions in the semiclassical limit. We show that, in the semiclassical limit, the presence of a magnetic field produces a phase in the complex wave, but it does not influence the location of peaks of the modulus of these waves.

Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity

ABSTRACT The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed.

Semiclassical stationary states of Nonlinear Schrodinger equations with an external magnetic field

Abstract In this paper we obtain multiple solutions u: R N → C of the nonlinear Schrodinger equation with an external magnetic field h i ∇ −A(x) 2 u+(U(x)−E)u=f(x,u), x∈ R N , where N⩾2, A is a real-valued vector magnetic potential, U is a real electric potential function and the nonlinear term f(x,t) grows subcritically in t. The number of solutions to the equation is shown to be bounded below by some number which depends on the category of a set defined by some properties of V and the coefficients of the nonlinear term. We perform appropriate changes of gauges which are made on functions which are concentrated around points lying in some well-defined manifold.

Critical groups computations on a class of Sobolev Banach spaces via Morse index

Abstract In this paper we deal with critical groups estimates for a functional f :W 0 1,p ( Ω )→ R (p>2), Ω bounded domain of R N , defined by setting f(u)= 1 p ∫ Ω |∇u| p dx+ 1 2 ∫ Ω |∇u| 2 dx+ ∫ Ω G(u) dx where G(t)=∫ 0 t g(s) ds and g is a smooth real function on R , growing subcritically. We remark that the second derivative of f in each critical point u is not a Fredholm operator from W 1,p 0 ( Ω ) to its dual space, so that the generalized Morse splitting lemma does not work. In spite of the lack of an Hilbert structure, we compute the critical groups of f in u via its Morse index.

Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation

We consider the magnetic NLS equation where N ? 3, 2 < p < 2* := 2N/(N ? 2), is a magnetic potential and is a bounded electric potential. We consider a group G of orthogonal transformations of , and we assume that A(gx) = gA(x) and V(gx) = V(x) for any g G, . Given a group homomorphism into the unit complex numbers, we show the existence of semiclassical solutions to problem (0.1), which satisfy for all g G, . Moreover, we show that there is a combined effect of the symmetries and the electric potential V on the number of solutions of this type.

Ground states for the pseudo-relativistic Hartree equation with external potential

We prove the existence of positive ground state solutions to the pseudo-relativistic Schrodinger equation where N ≥ 3, m > 0, V is a bounded external scalar potential and W is a radially symmetric convolution potential satisfying suitable assumptions. We also provide some asymptotic decay estimates of the found solutions.

On the Poincaré-Hopf theorem for functionals defined on Banach spaces

Abstract Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S)+. We prove that each isolated critical point of f has critical groups of finite type and that the Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result.

Morse Index Computations for a Class of Functionals Defined in Banach Spaces

In Morse theory the behavior of a C2Euler functionalF, defined on a Hilbert spaceH,near its critical points can be described by the estimates of thecritical groupsin the critical points. For convenience of the reader we recall the definition of critical group. For any a e R, we denote. Moreover let u be a critical point ofF, at levelc = F(u).We call the qth critical group ofFat u,q =0, 1, 2,…, whereHq (A,B)stands for the qth Alexander-Spanier cohomology group of the pair (A,B)with coefficients in 1K (cf. [2]).

Asymmetric modes on symmetric nonlinear optical waveguides

We study a symmetric nonlinear value problem in all R, arising in nonlinear optics from the study of propagation of electromagnetic guided waves through a layered medium with a nonlinear response. By variational arguments, we prove the existence of a positive asymmetric solution of the problem, corresponding to an asymmetric guided wave.

Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of nonconstant solutions u ∈ H1,p(M) to the quasilinear problem for e > 0 small enough, where 2 ≤ p < n, p < q < p*, p* = np/(n - p) and is the p-laplacian associated to g of u (note that Δ2,g = Δg) and denotes the Poincare polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pe).