Elliptic Equations with Convolution Terms Exhibit Singular Solutions Where Exponents Satisfy Specific Conditions

The behaviour of solutions to certain types of elliptic equations, particularly those involving convolution terms, remains a challenging problem in mathematical physics, with implications for understanding phenomena ranging from fluid dynamics to materials science. Marius Ghergu from University College Dublin and Zhe Yu investigate the existence and properties of singular solutions to these equations within a ‘punctured ball’, a domain with a central hole. Their work extends previous findings concerning constant potentials and establishes new conditions under which solutions exhibit specific behaviours near the origin, effectively classifying their existence and boundedness based on the local characteristics of the equation. This research significantly advances the understanding of how solutions behave in these complex scenarios, providing a crucial step towards modelling more realistic physical systems.

For N ≥ 3, the research establishes sharp conditions on the exponents α, β, p, q under which singular solutions exist and exhibit the asymptotic behaviour u(x) ≃ |x|2−N near the origin. When N = 2, a classification of the existence and boundedness of solutions is provided, based on the local behaviour of the potential V(x) near the origin.

Singular Choquard Solutions and Existence Proofs

This body of work explores the mathematical properties of solutions to the Choquard equation, a nonlinear partial differential equation with applications in quantum mechanics and other fields. Researchers have focused on understanding the existence, properties, and behaviour of solutions, particularly those that exhibit singularities, which are points where the solution becomes undefined or infinite. A central theme is the investigation of isolated singularities and the conditions under which they arise, alongside the existence of normalized solutions, important for physical modelling. The studies encompass a broad range of nonlinear partial differential equations, including those involving convolution integrals, Hardy potentials, and fractional derivatives.

Potential theory, a branch of mathematics dealing with potential functions, provides key tools for analysing these solutions, alongside investigations into critical exponents and supercriticality, concepts that determine solution stability and behaviour. The research also explores fractional partial differential equations, which model phenomena with memory effects, and utilizes Laplacian and polyharmonic operators common in the study of elliptic partial differential equations. This extensive research provides a foundation for understanding complex mathematical equations and their solutions, offering insights into solution behaviour under various conditions and providing tools for identifying and characterizing singularities. The findings have implications for a wide range of scientific disciplines, including physics, engineering, and applied mathematics.

Logarithmic Integrals Guarantee Solution Existence and Optimality

This research presents significant advances in understanding solutions to both local and non-local elliptic partial differential equations. Researchers extended a known theorem concerning inequalities involving solutions with potentials, establishing conditions for solutions to remain locally integrable. They demonstrated that if a continuous function satisfies a specific inequality within a defined region, then the function, its second derivatives, and the potential multiplied by the function all belong to a locally integrable function space, provided a specific integral condition on the potential is met. This condition, involving a logarithmic integral, is proven to be optimal for guaranteeing the existence of solutions.

The team then investigated a non-local equation, where the solution is determined by a convolution integral involving another function and its powers. They established that if a non-negative solution exists and is locally integrable, then the convolution term is also locally integrable, and the solution satisfies a modified equation involving a Dirac delta function. Furthermore, they investigated solutions that behave like the fundamental solution of the Laplace operator, demonstrating the existence of solutions under specific conditions on the potential and exponents. For two-dimensional cases, the researchers proved that if the potential behaves like a logarithm, and the solution approaches zero slower than the logarithm, then the solution can be smoothly extended to the origin. They also showed that for certain potentials, a positive solution exists that behaves like a logarithm, provided the parameter λ is sufficiently small. In higher dimensions, the team identified conditions on the potential and exponents p and q, under which a solution behaving like |x| 2-N exists, expanding the range of parameters for which solutions can be found.

Radially Symmetric Potentials and Singular Solutions

This research establishes new conditions for solutions to mathematical inequalities and equations involving potentials and functions defined on spherical domains. Researchers investigated inequalities and equations where a function’s behaviour is linked to its derivatives and a potential. A key achievement is the determination of optimal conditions for solutions to exist when the potential is radially symmetric, extending previous results limited to constant potentials. Furthermore, the researchers explored the existence and properties of singular solutions to a related equation, demonstrating how their behaviour near the origin is influenced by the exponents in the equation and the local behaviour of the potential.

They successfully classified the existence and boundedness of these solutions, providing a comprehensive understanding of their characteristics. The work also proves that certain inequalities have no non-negative solutions under specific conditions, a result achieved through careful analysis of spherical averages and limiting behaviour. The authors acknowledge that their results rely on assumptions about the integrability of certain functions and the behaviour of potentials, and that further investigation is needed to relax these conditions. This work represents a significant advancement in the understanding of partial differential equations and their solutions, with potential applications in various fields of mathematics and physics.