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Delia Schiera

Publications and source records attributed to Delia Schiera.

16 recordsLinked to original sources
arXiv2026

On Neumann $p$-Laplacian Lane-Emden equations and their asymptotic relationship with relative isoperimetric problems

We consider a family of pure Neumann $p$-Laplacian problems, including eigenvalue problems, Lane-Emden type equations, and extremal cases such as sign nonlinearities and the $1$-Laplacian. Using variational methods, we develop a unified framework that establishes existence of solutions and characterizes their asymptotic behavior as the parameters vary. This approach reveals a natural asymptotic connection between pure Neumann $p$-Laplacian equations and a relative isoperimetric problem known as the Neumann-Cheeger problem. We describe the shape of minimizers in domains with different geometries and obtain results on regularity, uniqueness, multiplicity, symmetry, and symmetry breaking phenomena.

math.AP↗
arXiv2026

Analysis of an Emden-Fowler-Hénon type equation

This work investigates a Hénon-type equation with the cap-shaped weight $V_α(|x|)=(4|x|(1-|x|))^α$, which concentrates on the sphere of radius $1/2$ as $α\to\infty$. Particular attention is devoted to ground state solutions and the phenomenon of symmetry breaking in the large-$α$ regime. A complete description of the asymptotic behavior of ground state radial solutions is obtained as $α\to\infty$. The analytical results are further supported by numerical simulations of the radial ground states.

math.AP↗
arXiv2026

Sublinear elliptic equations with a sharp change of sign in the nonlinearity

We study the semilinear indefinite elliptic problem \[ -Δu = Q_Ω|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_Ω= χ_Ω- χ_{\mathbb{R}^N \setminus Ω}$, $Ω\subset \mathbb{R}^N$ is a bounded smooth subset, $N \geq 3$, and $1 \leq p < 2$, with $p=1$ corresponding to the sign nonlinearity. Using a variational approach, we investigate the uniqueness or multiplicity of nonnegative solutions depending on the shape of $Ω$ and the existence of different types of nodal solutions. We also show that all solutions have compact support and analyze how the support of the ground state depends on $p$, proving convergence to the whole space as $p\to 2^{-}$ and identifying some qualitative features such as starshapedness and Lipschitz regularity of the support. We also establish a link between these problems and a two-phase Serrin-type torsion overdetermined problem.

math.AP↗
arXiv2025

Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions

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} = 0 \ \text{on } \partial Ω, \end{equation*} where $Ω\subset \mathbb{R}^N $ is a bounded $ C^2$ domain with $ N \leq 4 $, and $ ν$ denotes the outward unit normal on the boundary. We investigate the existence of non-constant least energy solutions in both the cooperative ($β> 0 $) and the competitive ($ β< 0 $) regimes, considering both the definite and the indefinite case, namely $λ_1,λ_2\in\mathbb R$. We emphasize that our analysis includes both the subcritical case $ N \leq 3 $ and the critical case $ N = 4 $. Depending on the values of $β,λ_1,λ_2$, the least energy solution is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on the set of all non-trivial weak solutions. Our results and techniques can be also adapted to cover some previously untreated cases for Dirichlet conditions.

math.AP↗
arXiv2024

On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity

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 domain of $\mathbb{R}^N$ with $N \ge 1$. We study the multiplicity of solutions and the convergence of least energy (nodal) solutions (l.e.s.) as the exponents $p, q > 0$ vary in the subcritical regime $1/(p+1) + 1/(q+1) > (N-2)/N$, or in the critical case $1/(p+1) + 1/(q+1) =(N-2)/N$ with some additional assumptions. We consider, for the first time in this setting, the cases where one or two exponents tend to zero, proving that l.e.s. converge to a problem with a sign nonlinearity. Our approach is based on an alternative characterization of least energy levels in terms of the nonlinear eigenvalue problem \[ Δ(|Δu|^{\frac 1 q -1} Δu) = λ|u|^{p-1} u, \quad \partial_νu=\partial_ν(|Δu|^{\frac 1 q -1} Δu)=0 \text{ on } \partial Ω. \] As an application, we show a symmetry breaking phenomenon for l.e.s. of a bilaplacian equation with sign nonlinearity and for other equations with nonlinear higher-order operators.

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arXiv2024

On a critical Hamiltonian system with Neumann boundary conditions

We consider the Hamiltonian system with Neumann boundary conditions: \[ -Δu + μu=v^{q }, \quad -Δv+ μv=u^{p} \quad \text{ in $Ω$}, \qquad u, v >0 \quad \text{ in $Ω$,} \qquad \partial_νu= \partial_νv=0 \quad \text{ on $\partial Ω$, } \] where $μ>0$ is a parameter and $Ω$ is a smooth bounded domain in $\mathbb R^N .$ When $(p, q)$ approaches from below the critical hyperbola $N/(p+1) + N/(q+1)=N-2$, we build a solution which blows-up at a boundary point where the mean curvature achieves its minimum and negative value.

math.AP↗
arXiv2023

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

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 $\mathbb{R}^N$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $Ω$ is a ball or an annulus.

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arXiv2023

Spectral optimization for weighted anisotropic problems with Robin conditions

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 associated with a positive and a negative eigenfunction. Next, we analyze the minimization of $λ^{\pm}$ with respect to the sign-changing weight, showing that the optimal eigenvalues $Λ^{\pm}$ are equal and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of $Ω$ satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optmization problem for $λ^{+}$ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in a heterogeneous habitat.

math.AP↗
arXiv2023

A family of nonlocal degenerate operators: maximum principles and related properties

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, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases.

math.AP↗
arXiv2021

Principal spectral curves for Lane-Emden fully nonlinear type systems and applications

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 principal spectral curves on the plane. We also construct a possible third spectral curve related to a second eigenvalue and an anti-maximum principle, which are novelties even for Lane-Emden systems involving linear operators. As applications, we derive a maximum principle in small domains for these systems, as well as existence and uniqueness of positive solutions in the sublinear regime. Most of our results are new even in the scalar case, in particular for a class of Isaac's operators with unbounded coefficients, whose $W^{2,\varrho}$ regularity estimates we also prove.

math.AP↗
arXiv2021

Symmetric positive solutions to nonlinear Choquard equations with potentials

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 consequence, the positive solution found will be invariant under the same action. Power nonlinearities with exponent greater or equal than two or less than two will be handled. Our results include the physical case.

math.AP↗
arXiv2019

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

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 $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain $Ω\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq 1$, $λ\in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(Ω)$, $c_{ij}\geq 0$, $M_i$ satisfies $0<μ_1 I\leq M_i\leq μ_2 I$, and $F_i$ is an uniformly elliptic Isaacs operator. We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

math.AP↗
arXiv2019

Uniqueness results for higher order elliptic equations and systems

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, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet's: the obstruction is apparently a new phenomenon and seems due to some loss of information though far from being clear. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier's boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

math.AP↗
arXiv2017

Existence of solutions to higher order Lane-Emden type systems

We prove existence results for the Lane-Emden type system \[ \begin{cases} \begin{aligned} (-Δ)^α u=\left| v \right|^q \\ (-Δ)^β v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\ \frac{\partial^{r} u}{\partial ν^{r}}=0, \, r=0, \dots, α-1, \text{ on } \partial B_1 \\ \frac{\partial^{r} v}{\partial ν^{r}}=0, \, r=0, \dots, β-1, \text{ on } \partial B_1. \end{cases} \] where $B_1$ is the unitary ball in $\mathbb{R}^N$, $N >\max \{2α, 2β\}$, $ν$ is the outward pointing normal, $α, β\in \mathbb{N}$, $α, β\ge 1$ and $(-Δ)^α= -Δ((-Δ)^{α-1})$ is the polyharmonic operator. A continuation method together with a priori estimates will be exploited. Moreover, we prove uniqueness for the particular case $α=2$, $β=1$ and $p, q>1$.

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