On the double critical Maxwell equations

TL;DR

This paper investigates the existence, nonexistence, and asymptotic behavior of solutions for the double critical Maxwell equations involving Hardy and Sobolev exponents, using variational methods and inequalities.

Contribution

It provides new existence and nonexistence results for critical Maxwell equations, including explicit thresholds for parameters and asymptotic analysis of solutions.

Findings

Existence of ground state solutions in certain parameter regimes.

Identification of a negative constant threshold for nonexistence.

Asymptotic behavior of solutions as coefficients approach zero.

Abstract

In this paper, we focus on (no)existence and asymptotic behavior of solutions for the double critical Maxwell equation involving with the Hardy, Hardy-Sobolev, Sobolev critical exponents. The existence and noexistence of solutions completely depend on the power exponents and coefficients of equation. On one hand, based on the concentration-compactness ideas, applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the critical Maxwell equation for three different scenarios. On the other hand, for the case $\lambda<0$ and $0\leq s_2<s_1<2$, which is a type open problem raised by Li and Lin. Draw support from a changed version of Caffarelli-Kohn-Nirenberg inequality, we find that there exists a constant $\lambda^*$ which is a negative number having explicit expression, such that the problem has no nontrivial solution as the coefficient $\lambda<\lambda^*$. Moreover, there exists a constant $\lambda^*<\lambda^{**}<0$ such that, as $\lambda^{**}<\lambda<0$, the equation has a nontrivial solution using truncation methods. Furthermore, we establish the asymptotic behavior of solutions of equation as coefficient converges to zero for the all cases above.