arXiv subjects

Alberto Saldaña

Publications and source records attributed to Alberto Saldaña.

At least 37 records · Page 2Linked to original sources

arXiv2023

Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik-Schnirelman theory, and new estimates on the category of suitable configuration spaces.

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arXiv2023

An asymptotic relationship between Lane-Emden systems and the 1-bilaplacian equation

Consider the following Lane-Emden system with Dirichlet boundary conditions: \[ -ΔU = |V|^{β-1}V,\ -ΔV = |U|^{α-1}U \text{ in }Ω,\qquad U=V= 0 \text{ on }\partial Ω, \] in a bounded domain $Ω$, for $(α,β)$ subcritical. We study the asymptotic behavior of least-energy solutions when $β\to \infty$, for any fixed $α$ which, in the case $N\geq 3$, is smaller than $2/(N-2)$. We show that these solutions converge to least-energy solutions of a semilinear equation involving the 1-bilaplacian operator, establishing a new relationship between these objects. As a corollary, we deduce the asymptotic behavior of solutions to $p$-bilaplacian Lane-Emden equations as the power in the nonlinearity goes to infinity. For the proof, we rely on the reduction by inversion method and on tools from nonsmooth analysis, considering an auxiliary nonlinear eigenvalue problem. We characterize its value in terms of the Green function, and prove a Faber-Krahn type result. In the case of a ball, we can characterize explicitly the eigenvalue, as well as the limit profile of least-energy solutions to the system as $β\to\infty$.

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arXiv2023

Small order limit of fractional Dirichlet sublinear-type problems

We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol $\ln(|ξ|^2)$. In the case of a logistic-type nonlinearity, our results have the following biological interpretation: in the presence of a toxic boundary, species with reduced mobility have a lower saturation threshold, higher survival rate, and are more homogeneously distributed. As a result of independent interest, we show that sublinear logarithmic problems have a unique least-energy solution, which is bounded and Dini continuous with a log-Hölder modulus of continuity.

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arXiv2023

Existence of solutions to a slightly supercritical pure Neumann problem

We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric domains (e.g. ellipsoids) are also discussed.

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arXiv2023

Exponential decay of the solutions to nonlinear Schrödinger systems

We show that the components of finite energy solutions to general nonlinear Schrödinger systems have exponential decay at infinity. Our results apply to positive or sign-changing components, and to cooperative, competitive, or mixed-interaction systems. As an application, we use the exponential decay to derive an upper bound for the least possible energy of a solution with a prescribed number of positive and nonradial sign-changing components.

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arXiv2022

Classical solutions to integral equations with zero order kernels

We show global and interior higher-order log-Hölder regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with a singularity at the origin that is weaker than that of any fractional Laplacian. As a consequence, under mild regularity assumptions on the right hand side, we show the existence of classical solutions of Dirichlet problems involving the logarithmic Laplacian and the logarithmic Schrödinger operator.

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arXiv2022

Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods

We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.

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arXiv2022

Small order asymptotics for nonlinear fractional problems

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order $2s$ when the parameter $s$ tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e., the pseudodifferential operator with Fourier symbol $\ln(|ξ|^2)$. These results are motivated by some applications of nonlocal models where a small value for the parameter $s$ yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.

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arXiv2021

Existence and convergence of solutions to fractional pure critical exponent problems

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical problem \begin{equation*} (-Δ)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(Ω),\quad 2^\star_s:=\frac{2N}{N-2s}, \end{equation*} where $s$ is any positive number, $Ω$ is either $\mathbb{R}^N$ or a smooth symmetric bounded domain, and $D^s_0(Ω)$ is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign changing. We show that, up to a subsequence, a l.e.s.s. $u_s$ converges to a l.e.s.s. $u_{t}$ as $s$ goes to any $t>0$. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order $t-\varepsilon$. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If $t$ is an integer, these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any $s>1$.

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arXiv2021

Critical polyharmonic systems and optimal partitions

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in R^N which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to $-\infty$. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of R^N. We give a detailed description of the shape of these domains.

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arXiv2021

Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domains

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.

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arXiv2021

On the least-energy solutions of the pure Neumann Lane-Emden equation

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -Δu = |u|^{p-1} u \text{ in }Ω, \qquad \partial_νu=0 \text{ on }\partial Ω, \] in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as $p\searrow 0$; to a l.e.n.s. of the critical problem as $p\nearrow 2^*$ (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent $2^*$); and we show that the limit as $p\to 1$ depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.

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arXiv2020

Fractional Laplacians on ellipsoids

We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of $s$-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for $s\in(1,\sqrt{3}+3/2)$ in any dimension $n\geq 2$. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not satisfy positivity preserving properties and we give some examples.

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arXiv2020

A solution to a slightly subcritical elliptic problem with non-power nonlinearity

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.

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arXiv2020

On fractional higher-order Dirichlet boundary value problems: between the Laplacian and the bilaplacian

The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to these type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of the Dirichlet boundary conditions for the fractional setting. Solutions are obtained variationally and, in the case of the ball, via explicit kernels. The validity of maximum principles for these intermediate problems is also discussed as well as the limiting behavior of solutions when approaching the Laplacian or the bilaplacian case.

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arXiv2019

Phase separation, optimal partitions, and nodal solutions to the Yamabe equation on the sphere

We study an optimal M-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least-energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely M nodal domains. The existence of an optimal partition is established through the study of the limit profiles of least-energy solutions to a weakly coupled competitive elliptic system on the sphere.

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arXiv2019

Entire nodal solutions to the critical Lane-Emden system

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-Δu=|v|^{q-2}v,\quad -Δv=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical hyperbola $\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}$. These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.

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arXiv2018

Integral representation of solutions to higher-order fractional Dirichlet problems on balls

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if $s$ is a natural number. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of $s$-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.

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