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Andrzej Szulkin

Publications and source records attributed to Andrzej Szulkin.

At least 19 recordsLinked to original sources
arXiv2026

On a critical curl-curl system of Lane-Emden type

We study the system $\nabla\times\nabla\times u = |v|^{q-2}v$, $\nabla\times\nabla\times v = |u|^{p-2}u$ of Lane-Emden type in $\mathbb R^3$, with exponents $p,q>3$ on the critical hyperbola $1/p+1/q=1/3$. We prove the existence of a ground state solution by reducing the system to a quasilinear fourth-order curl-curl equation and minimizing the associated Sobolev-curl quotient. A concentration-compactness argument shows there are finitely many concentration points for a Palais-Smale sequence $(u_n)$. This yields a.e. convergence of $\nabla\times\nabla\times u_n$ to $\nabla\times\nabla\times u$ where $u$ is the weak limit of $(u_n)$. We also prove that the optimal Sobolev-curl constant is attained and is strictly larger than the usual second-order Sobolev constant. If $p=q$, we show $u$ and $v$ are necessarily equal for every solution. In an appendix we show that our concentration argument also applies to a scalar quasilinear fourth order equation.

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arXiv2026

Optimizers in Sobolev-curl inequalities

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer $u:\mathbb{R}^3\to \mathbb{R}^3$ which yields a solution to the $p$-curl-curl equation in the critical case $$ \nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\quad \hbox{in }\mathbb{R}^3.$$ The cases $p=2$ and $p=3/2$ are motivated, respectively, by nonlinear Maxwell equations and zero modes of three-dimensional Dirac operators. The infinite-dimensional kernel of curl and the critical exponent prevent a direct application of standard compactness arguments. Our proof combines local compactness under the nonlinear constraint $\operatorname{div}(|u|^{p^*-2}u)=0$ and a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also establish existence and compactness results in axially symmetric classes, including two types of solutions for $p=3/2$ distinct from the explicit Loss--Yau fields. Finally, the same minimization strategy gives a new proof of compactness modulo translations and dilations for minimizing sequences in the classical critical Sobolev inequality.

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arXiv2025

A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core

We consider the equation $$-Δu+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the existence of a least energy solution for each $\varepsilon\in(0,\frac{1}{2})$ and show that their $H^1$ and $L^p$ norms concentrate locally at points of $\mathbb{Z}^N$ as $\varepsilon\to 0$.

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arXiv2025

Multiple normalized solutions to a system of nonlinear Schrödinger equations

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{array}{l} -Δu_i+ λ_i u_i = \sum_{j=1}^Kβ_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2} \quad \mathrm{in} \, \mathbb{R}^3,\newline u_i \in H^1_{rad}(\mathbb{R}^3),\newline \int_{\mathbb{R}^3} |u_i|^2 \, dx = ρ_i^2 \quad \text{for }i=1,\ldots, K, \end{array} \right. \end{equation*} where $ρ=(ρ_1,\ldots,ρ_K)\in(0,\infty)^K$ is prescribed, $(λ,u) \in \mathbb{R}^K\times H^1(\mathbb{R}^3)^K$ are the unknown and $4\leq p<6$. In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.

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arXiv2024

On a Schrödinger system with shrinking regions of attraction

In this paper we consider a competitive weakly coupled elliptic system in which each species is attracted to a small region and repelled from its complement. In this setting, we establish the existence of infinitely many solutions and of a nonnegative least energy solution. We show that, as the regions of attraction shrink, least energy solutions of the system concentrate. We study this behavior and characterize their limit profile. In particular, we show that if each component of a least energy solution is attracted to a different region, then the components decouple in the limit, whereas if all the components are attracted to the same region, they remain coupled.

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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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arXiv2022

Normalized solutions to a non-variational Schrödinger system

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.

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arXiv2021

A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent

Let $Ω\subset \mathbb{R}^3$ be a Lipschitz domain and let $S_\mathrm{curl}(Ω)$ be the largest constant such that $$ \int_{\mathbb{R}^3}|\nabla\times u|^2\, dx\geq S_{\mathrm{curl}}(Ω) \inf_{\substack{w\in W_0^6(\mathrm{curl};\mathbb{R}^3)\\ \nabla\times w=0}}\Big(\int_{\mathbb{R}^3}|u+w|^6\,dx\Big)^{\frac13} $$ for any $u$ in $W_0^6(\mathrm{curl};Ω)\subset W_0^6(\mathrm{curl};\mathbb{R}^3)$ where $W_0^6(\mathrm{curl};Ω)$ is the closure of $\mathcal{C}_0^{\infty}(Ω,\mathbb{R}^3)$ in $\{u\in L^6(Ω,\mathbb{R}^3): \nabla\times u\in L^2(Ω,\mathbb{R}^3)\}$ with respect to the norm $(|u|_6^2+|\nabla\times u|_2^2)^{1/2}$. We show that $S_{\mathrm{curl}}(Ω)$ is strictly larger than the classical Sobolev constant $S$ in $\mathbb{R}^3$. Moreover, $S_{\mathrm{curl}}(Ω)$ is independent of $Ω$ and is attained by a ground state solution to the curl-curl problem $$ \nabla\times (\nabla\times u) = |u|^4u $$ if $Ω=\mathbb{R}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $Ω$ $$\nabla\times (\nabla\times u) +λu = |u|^4u\quad\hbox{in }Ω$$ with the so-called metallic boundary condition $ν\times u=0$ on $\partialΩ$, where $ν$ is the exterior normal to $\partialΩ$.

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arXiv2021

Non-variational weakly coupled elliptic systems

We establish the existence of a nonnegative fully nontrivial solution to a non-variational weakly coupled competitive elliptic system. We show that this kind of solutions belong to a topological manifold of Nehari-type, and apply a degree-theoretical argument on this manifold to derive existence.

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arXiv2020

Solutions to indefinite weakly coupled cooperative elliptic systems

We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$, $κ_1,κ_2\in\mathbb{R}$, $μ_1,μ_2,λ>0$, $α,β>1$, and $α+ β= p\le 2^*:=\frac{2N}{N-2}$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $λ$ sufficiently large. If $p=2^*$ and $κ_1,κ_2>0$ we establish the existence of a ground state for $λ$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $κ_1$ nor $κ_2$ are Dirichlet eigenvalues of $-Δ$ in $Ω$.

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arXiv2019

Multiple solutions to a nonlinear curl-curl problem in $\mathbb{R}^3$

We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is controlled by an $N$-function $Φ:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}Φ(s)/s^6=\lim_{s\to+\infty}Φ(s)/s^6=0$. We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results for our problem have not been studied so far in $\mathbb{R}^3$ and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations.

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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.

math.AP↗
arXiv2019

A simple variational approach to weakly coupled competitive elliptic systems

The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a $\mathcal{C}^1$-functional $Ψ:\mathcal{U}\to\mathbb{R}$ defined in an open subset $\mathcal{U}$ of the product $\mathcal{T}:=S_1\times\cdots\times S_M$ of unit spheres $S_i$ in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).

math.AP↗
arXiv2017

On some weakly coercive quasilinear problems with forcing

We consider the forced problem $-Δ_p u - V(x)|u|^{p-2} u = f(x)$, where $Δ_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $Ω\subset \mathbb{R}^N$, $V\ge 0$ and $Q_V (u) := \int_Ω|\nabla u|^p\, dx - \int_ΩV|u|^p\,dx$ satisfies the condition (A) stated at the beginning of the paper. We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)=λ_{1,p}(Ω)$, where $λ_{1,p}(Ω)$ is the Poincaré constant on an infinite strip.

math.AP↗
arXiv2016

Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term

We study the Schrödinger equations $-Δu + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-Δu - λu = f(x,u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)/|u|$ is nondecreasing. In $\mathbb{R}^N$ we also assume that $V$ and $f$ are periodic in $x_1,\ldots,x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those in \cite{sw1} where $u\mapsto f(x,u)/|u|$ was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.

math.AP↗
arXiv2015

A logarithmic Schrödinger equation with asymptotic conditions on the potential

In this paper we consider a class of logarithmic Schrödinger equations with a potential which may change sign. When the potential is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem, and in the case of bounded potential we obtain a ground state solution, i.e. a nontrivial solution with least possible energy. The functional corresponding to the problem is the sum of a smooth and a convex lower semicontinuous term.

math.AP↗
arXiv2015

Ground states of critical and supercritical problems of Brezis-Nirenberg type

We study the existence of symmetric ground states to the supercritical problem \[ -Δv=λv+\left\vert v\right\vert ^{p-2}v\text{ \ in }Ω,\qquad v=0\text{ on }\partialΩ, \] in a domain of the form \[ Ω=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \inΘ\}, \] where $Θ$ is a bounded smooth domain such that $\overlineΘ \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $λ\in\mathbb{R},$ and $p=\frac{2(N-k)}{N-k-2}$ is the $(k+1)$-st critical exponent. We show that symmetric ground states exist for $λ$ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval $(-\infty,λ_{\ast})$ with $λ_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem \[ -\text{div}(a(x)\nabla u)=λb(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ Θ,\qquad u=0\quad\text{on}\ \partialΘ, \] where $a,b,c$ are positive continuous functions on $\overlineΘ.$ We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of $λ,$ and obtain a bifurcation result for ground states.

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