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In this paper, we explore the feasibility of solar-powered road-side unit (SRSU)-assisted vehicular edge computing (VEC) system, where SRSU is equipped with small cell base station (SBS) and VEC server, both of which are powered solely by solar energy. However, the limited capacity of solar energy, VEC server's computing, and SBS's bandwidth resources may prohibit vehicle users (VUs) from offloading their vehicular applications to VEC server for better service quality. We address this challenge by dynamically determining vehicular task partitioning and offloading, VEC server's system configuration, and vehicular application level adjustment decisions. We aim at minimizing the end-to-end delay of vehicular applications while maximizing their application level performance (e.g., accuracy). We also implement an object detection vehicular application on an edge computing platform and measure the corresponding energy consumption, computation delay, and detection accuracy performance to establish empirical models for the SRSU-assisted VEC system. We then propose a dynamic programming-based heuristic algorithm which jointly makes the task partitioning and offloading, as well as system and application-level adaption decisions in real-time. We build a simulation framework with the above empirical models to evaluate the proposed algorithm. The simulation results show that our proposed approach can significantly reduce the end-to-end delay while maximizing the detection accuracy compared to existing techniques. [ABSTRACT FROM AUTHOR]