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Mathew Gluck

Publications and source records attributed to Mathew Gluck.

8 recordsLinked to original sources
arXiv2026

Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities

This paper concerns a variational system of nonlinear elliptic equations that generalizes the classical Brezis-Nirenberg problem. In addition to considering vector-valued unknown functions in place of scalar-valued unknown functions, the problem under consideration generalizes the Brezis-Nirenberg problem in two directions. First, the nonlinearity is the sum of two homogeneous functions of suitable homogeneity degrees. Second, the Sobolev-critical term in the nonlinearity is nonlocal. We establish conditions under which a nontrivial solution to the system under consideration is guaranteed to exist and conditions under which the system under consideration admits only the trivial solution.

math.AP↗
arXiv2026

Classification of solutions to an $n^{\text{th}}$ order conformally invariant elliptic equation on $\mathbb R^n$ with nonlocal nonlinearity

This paper concerns a conformally invariant elliptic problem on $\mathbb R^n$ driven by $(-\Delta)^{n/2}$ that has a nonlocal exponential nonlinearity of Choquard type. The problem under consideration is a nonlocal generalization of the constant $Q$-curvature problem on $\mathbb R^n$. We classify the asymptotic behavior at infinity of all solutions that satisfy a suitable integrability assumption. Under a growth restriction at infinity and a lower bound on the energy we provide an explicit classification for solutions. The classification is heuristically consistent with the classification of the corresponding local problem.

math.AP↗
arXiv2025

Quantization for sequences of blow-up solutions to an elliptic equation having nonlocal exponential nonlinearity

This work provides a description of the asymptotic behavior of sequences of solutions to an elliptic equation with a nonlocal exponential nonlinearity of Choquard type. The equation under consideration is a nonlocal analog of the classical prescribed Gaussian curvature equation. A concentration-compactness alternative is established for sequences of solutions to the equation under consideration whenever suitable integrability assumptions on the solutions and the curvature functions are satisfied. Under further regularity assumptions on the curvature functions, and when blow-up occurs in the concentration-compactness alternative, an energy quantization result is established.

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arXiv2025

Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity

The existence of an unbounded sequence of solutions to a conformally invariant elliptic equation having nonlocal critical-power nonlinearity is established. The primary obstacle to establishing existence of solutions is the failure of compactness in the Sobolev embedding. To overcome this obstacle, the problem under consideration is lifted to an equivalent problem on the standard sphere so that the symmetries of the sphere can be leveraged. Two classes of symmetries are considered and for each class of symmetries, an unbounded sequence of solutions to the lifted problem with the prescribed symmetries is produced. One class of symmetries always exists and the corresponding solutions are guaranteed to be sign-changing whenever a suitable relationship between the dimension and the nonlocality parameter holds. The other class of symmetries need not always exist but when it exists, the corresponding solutions are guaranteed to be sign-changing.

math.AP↗
arXiv2023

Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities

In this work we consider a system of quasilinear elliptic equations driven by an anisotropic $p$-Laplacian. The lower-order nonlinearities are in potential form and exhibit critical Sobolev growth. We exhibit conditions on the coefficients of the differential operator, the domain of the unknown function, and the lower-order nonlinearities under which nontrivial solutions are guaranteed to exist and conditions on these objects under which a nontrivial solution does not exist.

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arXiv2017

Subcritical approach to conformally invariant extension operators on the upper half space

In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes among others the classical Poisson extension operator and the extension operator with Riesz kernel. We show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions. We also compute the limiting behavior at the boundary of the extensions of the extremal functions.

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arXiv2017

An extension operator on bounded domains and applications

In this paper we study a sharp Hardy-Littlewood-Sobolev (HLS) type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $Ω$ and show that if the extension constant for $Ω$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal functions exist. Using suitable test functions we show that this criterion is satisfied by an annular domain whose hole is sufficiently small. The construction of the test functions is not based on any positive mass type theorems, neither on the nonflatness of the boundary. By using a similar choice of test functions with the Poisson-kernel-based extension operator we prove the existence of an abstract domain having zero scalar curvature and strictly larger isoperimetric constant than that of the Euclidean ball.

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