Abstract: Emotional support conversation (ESC) aims to provide emotional support (ES) to improve one's mental state. Existing works stay at fitting grounded responses and responding strategies (e.g., \textit{question}), which ignore the effect on ES and lack explicit goals to guide emotional positive transition. To this end, we introduce a new paradigm to formalize multi-turn ESC as a process of positive emotion elicitation. Addressing this task requires finely adjusting the elicitation intensity in ES as the conversation progresses while maintaining conversational goals like coherence. In this paper, we propose \textsc{Supporter}, a mixture-of-expert-based reinforcement learning model, and well design ES and dialogue coherence rewards to guide policy's learning for responding. Experiments verify the superiority of \textsc{Supporter} in achieving positive emotion elicitation during responding while maintaining conversational goals including coherence.