Students' intentions to persevere and their career choices in science, technology, engineering, and math fields can be impacted by their physics identities. Women are severely underrepresented at all levels in physics and engineering. Physics in particular has stereotypes about being a discipline for brilliant men. Therefore, it is particularly difficult for women who do not fit the description of a stereotypical physicist to develop a physics identity. Thus, understanding the factors underlying physics identity in introductory physics classrooms is important for creating an equitable and inclusive physics learning environment and has the potential to at least partly explain the current underrepresentation of women in physics-related majors and careers. In this study, we examined physics identity and several other motivational constructs of male and female students by administering a survey in introductory calculus-based physics courses at a large research university. We found gender differences in how students identify as a physics person and how their perceived recognition from others, such as their teaching assistants or instructors, peers, or family members relates to their physics identities. We tested separate models by gender that examined how different motivational constructs relate to students' physics identities. We found that the perception of being recognized by influential others such as the course instructor or teaching assistants was differentially related to female and male students' physics self-efficacy and sense of belonging in the physics classroom. These findings call for improving the physics learning environments to make them equitable so that all students have a high sense of belonging and self-efficacy and opportunity to develop a strong physics identity.