We study the existence of standing waves for a class of nonlinear Schrödinger equations in R n, w... more We study the existence of standing waves for a class of nonlinear Schrödinger 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.
In this note we propose a few existence results for solutions to nonlinear elliptic equations dri... more In this note we propose a few existence results for solutions to nonlinear elliptic equations driven by fractional operators. We will focus on two models: a pseudo-relativistic Hartree equation, and an equation involving the fractional laplacian. We refer to the next sections for more details on these problems. Our approach relies on a perturbation technique in Critical Point Theory introduced some years ago by Ambrosetti and his collaborators. It is very useful when dealing with perturbation problems with lack of compactness.
Page 1. arXiv:math/0303202v3 [math.AP] 2 May 2003 On a class of singularly perturbed elliptic equ... more Page 1. arXiv:math/0303202v3 [math.AP] 2 May 2003 On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results A. Pomponio∗ SISSA, via Beirut 2/4 I-34014 Trieste pomponio@sissa.it S. Secchi† Universit`a di Pisa, via F.
By means of a recent variational technique, we prove the existence of radially monotone solutions... more By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the p-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.
The aim of this Ph.d. thesis is to present some very recent results concerning specific problems ... more The aim of this Ph.d. thesis is to present some very recent results concerning specific problems that share a common feature: a certain kind of non– compactness. As explained in the first chapter, while compact problems have a wellestablished existence theory, much less is known for non–compact problems which have attracted a great attention in recent years. Compactness usually breaks down for topological reasons, for example because an equation should be solved on an unbounded domain.
Jx′(t)=∇ H (t, x (t), λ),(1.1) where x∈ H1 (R, R2N), J is a real 2N× 2N− matrix such that JT= J− ... more Jx′(t)=∇ H (t, x (t), λ),(1.1) where x∈ H1 (R, R2N), J is a real 2N× 2N− matrix such that JT= J− 1=− J and the Hamiltonian H: R× R2N× R−→ R is sufficiently smooth. Moreover λ is the bifurcation parameter and∇ H (t, ξ, λ)= DξH (t, ξ, λ) for t∈ R, ξ∈ R2N and λ∈ R. We suppose that x≡ 0 satisfies (1. 1) for all values of the real parameter λ and we study the existence of solutions which are homoclinic to this trivial solution in the sense that lim t→−∞ x (t)= lim t→+∞ x (t)= 0.(1.2)
Fractional scalar field equations have attracted much attention in recent years, because of their... more Fractional scalar field equations have attracted much attention in recent years, because of their relevance in obstacle problems, phase transition, conservation laws, financial market. Strictly speaking, these equations are not partial differential equations, but rather integral equations. Their main feature, and also their main difficulty, is that they are strongly non-local, in the sense that the leading operator takes care of the behavior of the solution in the whole space.
This work is devoted to the Dirichlet problem for the equation −∆u = λu + |x| α |u| 2 * −2 u in t... more This work is devoted to the Dirichlet problem for the equation −∆u = λu + |x| α |u| 2 * −2 u in the unit ball of R N . We assume that λ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided α is small enough. This solution has a variational characterization as a ground state.
We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product... more We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics. To cite this article: S. Secchi et al., CR Acad. Sci. Paris, Ser. I 336 (2003).
Abstract: We study the problem (-\ epsilon\ mathrm {i}\ nabla+ A (x))^{2} u+ V (x) u=\ epsilon^{-... more Abstract: We study the problem (-\ epsilon\ mathrm {i}\ nabla+ A (x))^{2} u+ V (x) u=\ epsilon^{-2}(\ frac {1}{| x|}\ ast| u|^{2}) u, u\ in L^{2}(\ mathbb {R}^{3},\ mathbb {C}),\ text {\\\\}\ epsilon\ nabla u+\ mathrm {i} Au\ in L^{2}(\ mathbb {R}^{3},\ mathbb {C}^{3}), where $ A\ colon\ mathbb {R}^{3}\ rightarrow\ mathbb {R}^{3} $ is an exterior magnetic potential, $ V\ colon\ mathbb {R}^{3}\ rightarrow\ mathbb {R} $ is an exterior electric potential, and $\ epsilon $ is a small positive number.
For the equation− Δu=|| x|− 2| αup− 1, 1<| x|< 3, we prove the existence of two solutions for α l... more For the equation− Δu=|| x|− 2| αup− 1, 1<| x|< 3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2∗= 2N/(N− 2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.
We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation ... more We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.
We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a partic... more We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided α is sufficiently away from zero. We then prove some minimality results for small values of the parameter α.
We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth... more We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.
We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Ou... more We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
Consideriamo il problema seguente: trovare una soluzione positiva u: R→ R dell'equazione differen... more Consideriamo il problema seguente: trovare una soluzione positiva u: R→ R dell'equazione differenziale− u′′+ λu=| u| p− 1u,(1) dove λ> 0e un parametro assegnato ep∈(1,+∞). Quindi l'equazione (1)e un'equazione non lineare. Usando metodi elementari, oppure per verifica a posteriori, possiamo convincerci che l'unica soluzione non nulla di (1) soggetta alla limitazione di tipo omoclino lim| x|→+∞ u (x)= 0(2)e data esplicitamente dalla formula u (x)=
Abstract By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of clas... more Abstract By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $ C^ 1$, we get some necessary conditions for locating the peak-points of a class of singularly perturbed quasilinear elliptic problems in divergence form. More precisely, we show that the points where the concentration occurs, in general, must belong to what we call the set of weak-concentration points. Finally, in the semilinear case, we provide a new necessary condition which involves the Clarke subdifferential of the ground-state function.
In this paper, we prove that a singularly perturbed Neumann problem with potentials admits the ex... more In this paper, we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima o.
We study the existence of standing waves for a class of nonlinear Schrödinger equations in R n, w... more We study the existence of standing waves for a class of nonlinear Schrödinger 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.
In this note we propose a few existence results for solutions to nonlinear elliptic equations dri... more In this note we propose a few existence results for solutions to nonlinear elliptic equations driven by fractional operators. We will focus on two models: a pseudo-relativistic Hartree equation, and an equation involving the fractional laplacian. We refer to the next sections for more details on these problems. Our approach relies on a perturbation technique in Critical Point Theory introduced some years ago by Ambrosetti and his collaborators. It is very useful when dealing with perturbation problems with lack of compactness.
Page 1. arXiv:math/0303202v3 [math.AP] 2 May 2003 On a class of singularly perturbed elliptic equ... more Page 1. arXiv:math/0303202v3 [math.AP] 2 May 2003 On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results A. Pomponio∗ SISSA, via Beirut 2/4 I-34014 Trieste pomponio@sissa.it S. Secchi† Universit`a di Pisa, via F.
By means of a recent variational technique, we prove the existence of radially monotone solutions... more By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the p-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.
The aim of this Ph.d. thesis is to present some very recent results concerning specific problems ... more The aim of this Ph.d. thesis is to present some very recent results concerning specific problems that share a common feature: a certain kind of non– compactness. As explained in the first chapter, while compact problems have a wellestablished existence theory, much less is known for non–compact problems which have attracted a great attention in recent years. Compactness usually breaks down for topological reasons, for example because an equation should be solved on an unbounded domain.
Jx′(t)=∇ H (t, x (t), λ),(1.1) where x∈ H1 (R, R2N), J is a real 2N× 2N− matrix such that JT= J− ... more Jx′(t)=∇ H (t, x (t), λ),(1.1) where x∈ H1 (R, R2N), J is a real 2N× 2N− matrix such that JT= J− 1=− J and the Hamiltonian H: R× R2N× R−→ R is sufficiently smooth. Moreover λ is the bifurcation parameter and∇ H (t, ξ, λ)= DξH (t, ξ, λ) for t∈ R, ξ∈ R2N and λ∈ R. We suppose that x≡ 0 satisfies (1. 1) for all values of the real parameter λ and we study the existence of solutions which are homoclinic to this trivial solution in the sense that lim t→−∞ x (t)= lim t→+∞ x (t)= 0.(1.2)
Fractional scalar field equations have attracted much attention in recent years, because of their... more Fractional scalar field equations have attracted much attention in recent years, because of their relevance in obstacle problems, phase transition, conservation laws, financial market. Strictly speaking, these equations are not partial differential equations, but rather integral equations. Their main feature, and also their main difficulty, is that they are strongly non-local, in the sense that the leading operator takes care of the behavior of the solution in the whole space.
This work is devoted to the Dirichlet problem for the equation −∆u = λu + |x| α |u| 2 * −2 u in t... more This work is devoted to the Dirichlet problem for the equation −∆u = λu + |x| α |u| 2 * −2 u in the unit ball of R N . We assume that λ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided α is small enough. This solution has a variational characterization as a ground state.
We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product... more We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics. To cite this article: S. Secchi et al., CR Acad. Sci. Paris, Ser. I 336 (2003).
Abstract: We study the problem (-\ epsilon\ mathrm {i}\ nabla+ A (x))^{2} u+ V (x) u=\ epsilon^{-... more Abstract: We study the problem (-\ epsilon\ mathrm {i}\ nabla+ A (x))^{2} u+ V (x) u=\ epsilon^{-2}(\ frac {1}{| x|}\ ast| u|^{2}) u, u\ in L^{2}(\ mathbb {R}^{3},\ mathbb {C}),\ text {\\\\}\ epsilon\ nabla u+\ mathrm {i} Au\ in L^{2}(\ mathbb {R}^{3},\ mathbb {C}^{3}), where $ A\ colon\ mathbb {R}^{3}\ rightarrow\ mathbb {R}^{3} $ is an exterior magnetic potential, $ V\ colon\ mathbb {R}^{3}\ rightarrow\ mathbb {R} $ is an exterior electric potential, and $\ epsilon $ is a small positive number.
For the equation− Δu=|| x|− 2| αup− 1, 1<| x|< 3, we prove the existence of two solutions for α l... more For the equation− Δu=|| x|− 2| αup− 1, 1<| x|< 3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent 2∗= 2N/(N− 2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.
We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation ... more We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.
We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a partic... more We study a singular Hamiltonian system with an α-homogeneous potential that contains, as a particular case, the classical N-body problem. We introduce a variational Morse-like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided α is sufficiently away from zero. We then prove some minimality results for small values of the parameter α.
We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth... more We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.
We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Ou... more We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
Consideriamo il problema seguente: trovare una soluzione positiva u: R→ R dell'equazione differen... more Consideriamo il problema seguente: trovare una soluzione positiva u: R→ R dell'equazione differenziale− u′′+ λu=| u| p− 1u,(1) dove λ> 0e un parametro assegnato ep∈(1,+∞). Quindi l'equazione (1)e un'equazione non lineare. Usando metodi elementari, oppure per verifica a posteriori, possiamo convincerci che l'unica soluzione non nulla di (1) soggetta alla limitazione di tipo omoclino lim| x|→+∞ u (x)= 0(2)e data esplicitamente dalla formula u (x)=
Abstract By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of clas... more Abstract By exploiting a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $ C^ 1$, we get some necessary conditions for locating the peak-points of a class of singularly perturbed quasilinear elliptic problems in divergence form. More precisely, we show that the points where the concentration occurs, in general, must belong to what we call the set of weak-concentration points. Finally, in the semilinear case, we provide a new necessary condition which involves the Clarke subdifferential of the ground-state function.
In this paper, we prove that a singularly perturbed Neumann problem with potentials admits the ex... more In this paper, we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima o.
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