Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potenti... more Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potential... more The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, point-wise upper estimate are derived.
We find for small ε positive solutions to the equation−div(|x|−2a∇u)−λ|x|2(1+a)u=(1+εk(x))up−1|x|... more We find for small ε positive solutions to the equation−div(|x|−2a∇u)−λ|x|2(1+a)u=(1+εk(x))up−1|x|bpin RN, which branch off from the manifold of minimizers in the class of radial functions of the corresponding Caffarelli–Kohn–Nirenberg-type inequality. Moreover, our analysis highlights the symmetry-breaking phenomenon in these inequalities, namely the existence of non-radial minimizers.
A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian mani... more A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a C 2 function H to be the mean curvature of some conformal scalar flat metric is that H is positive somewhere. We show that, when the boundary is umbilic and the function H is positive everywhere, all such metrics stay in a compact set with respect to the C 2 norm and the total degree of all solutions is equal to −1.
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neum... more We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.
Equations Near an Isolated Singularity of the Electromagnetic Potential
Asymptotics of solutions to Schrodinger equations with singular magnetic and elec- tric potential... more Asymptotics of solutions to Schrodinger equations with singular magnetic and elec- tric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.
Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials... more Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.
Calculus of Variations and Partial Differential Equations, 2006
Supported by Italy MIUR, national project "Variational Methods and Nonlinear Differential Equatio... more Supported by Italy MIUR, national project "Variational Methods and Nonlinear Differential Equations". 2000 Mathematics Subject Classification. 35J60, 35J20, 35B33.
The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with ... more The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case relate to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.
In this paper we study the asymptotic behavior of solutions to an elliptic equation near the sing... more In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgrentype monotonicity formula is used to determine the exact asymptotic behavior of solutions. Date: September 21, 2012. 2010 Mathematics Subject Classification. 35J75, 35B40, 35B45. Key words and phrases. Hardy's inequality, singular elliptic operators, asymptotic behavior of solutions. V.
Journal of the European Mathematical Society, 2000
We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V (x) ∼ |x| −α , 0 ... more We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V (x) ∼ |x| −α , 0 < α < 2, and K(x) ∼ |x| −β , β > 0. Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1,2 (R N ) is proved under the assumption that σ < p < (N + 2)/(N − 2) for some σ = σ N,α,β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of A = V θ K −2/(p−1) , where θ = (p + 1)/(p − 1) − 1/2.
Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operato... more Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.
Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichle... more Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.
Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichle... more Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.
Communications in Partial Differential Equations, 2014
Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied i... more Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations. R N |ξ| 2s v(ξ) u(ξ) dξ,
Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potenti... more Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potential... more The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, point-wise upper estimate are derived.
We find for small ε positive solutions to the equation−div(|x|−2a∇u)−λ|x|2(1+a)u=(1+εk(x))up−1|x|... more We find for small ε positive solutions to the equation−div(|x|−2a∇u)−λ|x|2(1+a)u=(1+εk(x))up−1|x|bpin RN, which branch off from the manifold of minimizers in the class of radial functions of the corresponding Caffarelli–Kohn–Nirenberg-type inequality. Moreover, our analysis highlights the symmetry-breaking phenomenon in these inequalities, namely the existence of non-radial minimizers.
A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian mani... more A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a C 2 function H to be the mean curvature of some conformal scalar flat metric is that H is positive somewhere. We show that, when the boundary is umbilic and the function H is positive everywhere, all such metrics stay in a compact set with respect to the C 2 norm and the total degree of all solutions is equal to −1.
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neum... more We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.
Equations Near an Isolated Singularity of the Electromagnetic Potential
Asymptotics of solutions to Schrodinger equations with singular magnetic and elec- tric potential... more Asymptotics of solutions to Schrodinger equations with singular magnetic and elec- tric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.
Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials... more Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.
Calculus of Variations and Partial Differential Equations, 2006
Supported by Italy MIUR, national project "Variational Methods and Nonlinear Differential Equatio... more Supported by Italy MIUR, national project "Variational Methods and Nonlinear Differential Equations". 2000 Mathematics Subject Classification. 35J60, 35J20, 35B33.
The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with ... more The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case relate to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.
In this paper we study the asymptotic behavior of solutions to an elliptic equation near the sing... more In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgrentype monotonicity formula is used to determine the exact asymptotic behavior of solutions. Date: September 21, 2012. 2010 Mathematics Subject Classification. 35J75, 35B40, 35B45. Key words and phrases. Hardy's inequality, singular elliptic operators, asymptotic behavior of solutions. V.
Journal of the European Mathematical Society, 2000
We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V (x) ∼ |x| −α , 0 ... more We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V (x) ∼ |x| −α , 0 < α < 2, and K(x) ∼ |x| −β , β > 0. Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1,2 (R N ) is proved under the assumption that σ < p < (N + 2)/(N − 2) for some σ = σ N,α,β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of A = V θ K −2/(p−1) , where θ = (p + 1)/(p − 1) − 1/2.
Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operato... more Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.
Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichle... more Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.
Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichle... more Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.
Communications in Partial Differential Equations, 2014
Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied i... more Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations. R N |ξ| 2s v(ξ) u(ξ) dξ,
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