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In historical mathematics and physics, the Kardar-Parisi-Zhang equation or aquasilinear stationary version of a time-dependent viscous Hamilton-Jacobiequation in growing interface and universality classes, is also known by thedifferent name as the quasilinear Riccati type equation. The existence ofsolutions to this type of equation under some assumptions and requirements,still remains an interesting open problem at the moment. In our previousstudies \cite{MP2018, MPT2019}, we obtained the global bounds and gradientestimates for quasilinear elliptic equations with measure data. There have beenmany applications are discussed related to these works, and main goal of thispaper is to obtain the existence of a renormalized solution to the quasilinearstationary solution to the degenerate diffusive Hamilton-Jacobi equation withthe finite measure data in Lorentz-Morrey spaces.

An application of global gradient estimates in Lorentz-Morrey spaces:  The existence of stationary solution to degenerate diffusive Hamilton-Jacobi  equations