<p>This article presents three characterizations of regular-global-inf, quasi-regular-global-inf, and transfer weakly lower continuous functions by examining the relationship between their minimum points and those of their corresponding lower envelopes. Building upon this groundwork, we investigate two essential coercivity and asymptotic conditions that have recently been utilized to determine the existence of solutions for optimization problems. We demonstrate that these conditions are equivalent in finite-dimensional spaces. Furthermore, we present the necessary and sufficient conditions for the existence of optimal solutions, thereby extending and generalizing recent results in the field. Ultimately, we provide a new characterization of generalized Tykhonov well-posedness for regular-global-inf functions, using asymptotic functions. Our results contribute to the overall understanding of optimization theory and enhance existing methodologies.</p>