Consider the functional $$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$ where $\Omega$ is a bounded smooth domain.
The problem has been solved for example if $F$ is (1) subquadratic, or (2) superquadratic and subcritical (in the sense of Sobolev embedding exponents), or (3) asymptotically quadratic.
Heuristically, what makes these cases different? And what are the differences among them?
Where can I find a nicely written discussion on the difference among these cases?
Is it true that the asymptotically quadratic case separates the case where you can prove existence of a minimizer (subquadratic case) and the one where you can prove that only non-minimizer critical points exist (asymptotically quadratic and superquadratic)?
