arXiv subjects

Yergen Aikyn

Publications and source records attributed to Yergen Aikyn.

5 recordsLinked to original sources

arXiv2026

Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_\varepsilon$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_\varepsilon$ converge strongly in $H^s(\mathbb{R}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\varepsilon\to0$.

math.AP↗

arXiv2026

Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

math.AP↗

arXiv2026

Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions

In this paper, we investigate existence results for nonlinear nonlocal problems governed by an operator obtained as a superposition of fractional p-Laplacians, subject to Neumann boundary conditions. A spectral analysis of the main operator leads us to apply different variational tools to establish our results. Specifically, we employ either the mountain pass method or the technique of linking over cones. Due to the generality of the setting, the resulting theory applies to a broad class of local-nonlocal models.

math.AP↗

arXiv2026

Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order

The main objective of this paper is to investigate the spectral properties, maximum principles, and shape optimization problems for a broad class of nonlinear ``superposition operators" defined as continuous superpositions of operators of mixed fractional order, modulated by a signed finite Borel measure on the unit interval. This framework encompasses, as particular cases, mixed local and nonlocal operators such as $-Δ_p+(-Δ_p)^s$, finite (possibly infinite) sums of fractional $p$-Laplacians with different orders, as well as operators involving fractional Laplacians with ``wrong" signs. The main findings, obtained through variational techniques, concern the spectral analysis of the Dirichlet eigenvalue problem associated with general superposition operators with special emphasis on various properties of the first eigenvalue and its corresponding eigenfunction. We establish weak and strong maximum principles for positive superposition operators by introducing an appropriate notion of the {\it nonlocal tail} for this class of superposition operators and deriving a logarithmic estimate, both of which are of independent interest. Utilizing these newly developed tools, we further investigate the spectral properties of such superposition operators and prove that the first eigenvalue is isolated and simple. Moreover, we show that the eigenfunctions corresponding to positive eigenvalues are globally bounded and that they change sign when associated with higher eigenvalues. In addition, we demonstrate that the second eigenvalue is well-defined and provide the mountain pass characterization. Finally, we address shape optimization problems, in particular, the Faber--Krahn inequality associated with the principal frequency associated with the superposition operators.

math.AP↗

arXiv2025

Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order

This paper aims to study the Brezis-Nirenberg type problem driven by the nonlinear superposition of operators of the form $$A_{μ, p}u:=\int_{[0,1]}(-Δ)_{p}^{s} u\,\, d μ(s),$$ where $μ$ denotes the signed measure over $[0, 1]$. We consider nonlinear nonlocal equations associated with $A_{μ, p}$, involving critical nonlinearity and lower-order perturbation. Using variational techniques, we establish existence results for the critical problem by employing weak lower semicontinuity arguments under general assumptions on the perturbation term. We discuss the multiplicity results when the perturbation term vanishes at the origin. Additionally, when the lower-order term is a pure power function, we examine the Brezis-Nirenberg-type problem using the mountain pass technique. Furthermore, we address the existence of solutions to subcritical problems associated with $A_{μ, p}.$ Our findings are novel, even in the case of the sum of two distinct fractional $p$-Laplacians or a combination of a fractional $p$-Laplacian with a classical $p$-Laplacian. More generally, our framework is sufficiently broad to accommodate finite sums of different fractional $p$-Laplacians as well as cases involving fractional Laplacians with ``wrong" signs. A key contribution of this study is the development of a unified approach that systematically addresses these problems by incorporating a broad class of operators and lower-order perturbation terms within a common theoretical framework. The results remain new even in the case of linear superposition of fractional operators of different orders.

math.AP↗