The present article investigates the two-dimensional Euler-Boussinesq system with critical fractional dissipation and general source term. First, we show that this system admits a global solution of Yudovich type, and as a consequence, we treat the regular vortex patch issue.

This paper dealt with the existence of multiple solutions for some singular $p(s)$-Laplacian problems involving the $\phi$-Hilfer derivative. Precisely, we combined the variational method with the Nehari manifold to prove that such a problem admited two nontrivial solutions. An example was presented to illustrate the effectiveness of our main result.