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View article: Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions
Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions Open
We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -Δu + λ_1 u = u^3 + βuv^2, \ -Δv + λ_2 v = v^3 + βu^2 v \ \text{in } Ω,\qquad \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} …
View article: On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity
On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity Open
We consider the following Lane-Emden system with Neumann boundary conditions \[ -Δu= |v|^{q-1}v \text{ in } Ω,\qquad -Δv= |u|^{p-1}u \text{ in } Ω,\qquad \partial_νu=\partial_νv=0 \text{ on } \partial Ω, \] where $Ω$ is a bounded smooth do…
View article: On a critical Hamiltonian system with Neumann boundary conditions
On a critical Hamiltonian system with Neumann boundary conditions Open
We consider the Hamiltonian system with Neumann boundary conditions: \[ -Δu + μu=v^{q }, \quad -Δv+ μv=u^{p} \quad \text{ in $Ω$}, \qquad u, v >0 \quad \text{ in $Ω$,} \qquad \partial_νu= \partial_νv=0 \quad \text{ on $\partial Ω$, } \] wh…
View article: Spectral optimization for weighted anisotropic problems with Robin conditions
Spectral optimization for weighted anisotropic problems with Robin conditions Open
We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains $Ω\subset \mathbb{R}^{N} $, $N\ge1$, under Robin boundary conditions, proving the existence of two positive eigenvalues $λ^{\pm}$ respectively a…
View article: A family of nonlocal degenerate operators: maximum principles and related properties
A family of nonlocal degenerate operators: maximum principles and related properties Open
We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, a…
View article: Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions
Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions Open
We study the following Lane-Emden system \[ -Δu=|v|^{q-1}v \quad \text{ in } Ω, \qquad -Δv=|u|^{p-1}u \quad \text{ in } Ω, \qquad u_ν=v_ν=0 \quad \text{ on } \partial Ω, \] with $Ω$ a bounded regular domain of $\mathbb{R}^N$, $N \ge 4$, an…
View article: Maximum principles and related problems for a class of nonlocal extremal operators
Maximum principles and related problems for a class of nonlocal extremal operators Open
We study the validity of the comparison and maximum principles and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one-dimensional fractional diffusion.
View article: Symmetric positive solutions to nonlinear Choquard equations with potentials
Symmetric positive solutions to nonlinear Choquard equations with potentials Open
Existence results for a class of Choquard equations with potentials are established. The potential has a limit at infinity and it is taken invariant under the action of a closed subgroup of linear isometries of $\mathbb{R}^N$. As a consequ…
View article: Maximum principles and related problems for a class of nonlocal extremal operators
Maximum principles and related problems for a class of nonlocal extremal operators Open
We study the validity of the comparison and maximum principles, and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion.
View article: Positive bound states to nonlinear Choquard equations in the presence of nonsymmetric potentials
Positive bound states to nonlinear Choquard equations in the presence of nonsymmetric potentials Open
The existence of a positive solution to a class of Choquard equations with potential going at a positive limit at infinity possibly from above or oscillating is proved. Our results include the physical case and do not require any symmetry …
View article: Principal spectral curves for Lane-Emden fully nonlinear type systems and applications
Principal spectral curves for Lane-Emden fully nonlinear type systems and applications Open
In this paper we exploit the phenomenon of two principal half eigenvalues in the context of fully nonlinear Lane-Emden type systems with possibly unbounded coefficients and weights. We show that this gives rise to the existence of two prin…
View article: A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth
A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth Open
We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =λc_{i1}(x) u_1 + \cdots + λc_{in}(x) u_n +h_i(x), $$ for …
View article: A priori estimates and multiplicity for systems of elliptic PDE with\n natural gradient growth
A priori estimates and multiplicity for systems of elliptic PDE with\n natural gradient growth Open
We consider fully nonlinear uniformly elliptic cooperative systems with\nquadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)-\n\\langle M_i(x)D u_i, D u_i \\rangle =\\lambda c_{i1}(x) u_1 + \\cdots + \\lambda\nc_{in}(x…
View article: Uniqueness results for higher order elliptic equations and systems
Uniqueness results for higher order elliptic equations and systems Open
In this paper we develop a Gidas-Ni-Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations…
View article: Existence, non existence and uniqueness results for higher order elliptic systems
Existence, non existence and uniqueness results for higher order elliptic systems Open
In this dissertation, we deal with Hamiltonian Lane-Emden type systems where in place of the Laplace operator we take into account the polyharmonic operator. Recallthatthepolyharmonicoperatordoesnotalwayssatisfyamaximumprinciple. Indeed, i…
View article: Existence and non-existence results for variational higher order elliptic systems
Existence and non-existence results for variational higher order elliptic systems Open
Let $α ∈ \mathbb{N}$, $α ≥ 1$ and $(-Δ)^{α} = -Δ((-Δ)^{α-1})$ be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions $\…