$\{\begin{matrix} {(- Δ)}^{s} u + u - K (x) ϕ | u |^{2_{s}^{*} - 3} u = f_{λ} (x) | u |^{q - 2} u, & x \in ℝ^{3}, \\ {(- Δ)}^{s} ϕ = K (x) | u |^{2_{s}^{*} - 1}, & x \in ℝ^{3} \end{matrix}$

with multiple competing potentials and a critical nonlocal term, where

$s \in (0, 1)$ ,

$q \in (1, 2)$ or

$q \in (4, 2_{s}^{*})$ , and

$2_{s}^{*} = \frac{6}{3 - 2 s}$ is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient

$K$ of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of

$K$ .

Open Access

Nonlocal Lagrange multipliers and transport densities

Assis Azevedo,
José Francisco Rodrigues, and
Lisa Santos

2350014

https://doi.org/10.1142/S1664360723500145

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We prove the existence of generalized solutions of the Monge–Kantorovich equations with fractional $s$ -gradient constraint, $0 < s < 1$ , associated to a general, possibly degenerate, linear fractional operator of the type,

$ℒ^{s} u = - D^{s} \cdot (A D^{s} u + b u) + d \cdot D^{s} u + c u,$

with integrable data, in the space

$Λ_{0}^{s, p} (Ω)$ , which is the completion of the set of smooth functions with compact support in a bounded domain

$Ω$ for the

$L^{p}$ -norm of the distributional Riesz fractional gradient

$D^{s}$ in

$ℝ^{d}$ (when

$s = 1$ ,

$D^{1} = D$ is the classical gradient). The transport densities arise as generalized Lagrange multipliers in the dual space of

$L^{\infty} (ℝ^{d})$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint

$| D^{s} u | \leq g$ . Their existence is shown by approximating the variational inequality through a penalization of the constraint and nonlinear regularization of the linear operator

$ℒ^{s} u$ . For this purpose, we also develop some relevant properties of the spaces

$Λ_{0}^{s, p} (Ω)$ , including the limit case

$p = \infty$ and the continuous embeddings

$Λ_{0}^{s, q} (Ω) \subset Λ_{0}^{s, p} (Ω)$ , for

$1 \leq p \leq q \leq \infty$ . We also show the localization of the nonlocal problems (

$0 < s < 1$ ), to the local limit problem with classical gradient constraint when

$s \to 1$ , for which most results are also new for a general, possibly degenerate, partial differential operator

$ℒ^{1} u$ with coefficients only integrable and bounded gradient constraint.

Open Access

On motives of parabolic Higgs bundles and parabolic connections

Sumit Roy

2450001

https://doi.org/10.1142/S1664360724500012

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Let $X$ be a compact Riemann surface of genus $g \geq 2$ and let $D \subset X$ be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over $X$ with the parabolic structure over $D$ . For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their $E$ -polynomials are the same. We also show that the Voevodsky and Chow motives of these two moduli spaces are also equal. We showed that the Grothendieck motivic classes and the $E$ -polynomials of parabolic Higgs moduli and of parabolic Hodge moduli are closely related. Finally, we considered the moduli spaces with fixed determinants and showed that the above results also hold for the fixed determinant case.

Open Access

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

Yahui Gao,
Xiao Luo, and
Maoding Zhen

2450002

https://doi.org/10.1142/S1664360724500024

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We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

Open Access

Normalized homoclinic solutions of discrete nonlocal double phase problems

Mingqi Xiang,
Yunfeng Ma, and
Miaomiao Yang

2450003

https://doi.org/10.1142/S1664360724500036

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The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian:

$\{\begin{matrix} (- Δ_{𝔻})_{p}^{α} u (k) + μ {(- Δ_{𝔻})}_{q}^{β} u (k) + ω (k) | u (k) |^{p - 2} u (k) \\ = λ | u (k) |^{q - 2} u (k) + h (k) | u (k) |^{r - 2} u (k) & for k \in ℤ, \\ \sum_{k \in ℤ} | u (k) |^{q} = ρ^{q} > 0, \\ u (k) \to 0 & as | k | \to \infty, \end{matrix}$

where

$α, β \in (0, 1)$ ,

$ω : ℤ \to (0, \infty)$ ,

$1 < p \leq q < \infty$ ,

$λ, μ \in ℝ$ ,

$h \in ℓ^{\frac{q}{q - r}} (ℤ)$ if

$1 < r < q$ ,

$h \in ℓ^{\infty} (ℤ)$ if

$r > q$ , and

${(- Δ_{𝔻})}_{κ}^{s}$ (

$s = α$ or

$β$ ,

$κ = p$ or

$q$ ) is the discrete fractional

$κ$ -Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Open Access

The effects of nonlinear perturbation terms on comparison principles for the $p$ -Laplacian

Ahmed Mohammed and
Antonio Vitolo

2450005

https://doi.org/10.1142/S166436072450005X

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In this paper, we investigate various comparison principles for quasilinear elliptic equations of $p$ -Laplace type with lower-order terms that depend on the solution and its gradient. More specifically, we study comparison principles for equations of the following form:

$- Δ_{p} u + H (u, D u) = 0, x \in Ω,$

where

$Δ_{p} u : = div (| D u |^{p - 2} D u)$ is the

$p$ -Laplace operator with

$p > 1$ , and

$H$ is a continuous function that satisfies a structure condition. Many of these results lead to comparison principles for the model equations

$Δ_{p} u = f (u) + g (u) | D u |^{q}, x \in Ω,$

where

$f, g \in C^{0} (ℝ, ℝ)$ are non-decreasing and

$q > 0$ . Our results either improve or complement those that appear in the literature.

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Bulletin of Mathematical Sciences

Open Access

Volume 14, Issue 02 (August 2024)

Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

Nonlocal Lagrange multipliers and transport densities

On motives of parabolic Higgs bundles and parabolic connections

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

Normalized homoclinic solutions of discrete nonlocal double phase problems

The effects of nonlinear perturbation terms on comparison principles for the p<math display="inline" altimg="eq-00001.gif"><mi>p</mi></math>-Laplacian

The effects of nonlinear perturbation terms on comparison principles for the $p$ -Laplacian