I've read a lot of papers for differential equations, which study the existence , multiplicity, and/or nonexistence of solutions according to parameter $\lambda$. For example, consider the following elliptic differential equation $\Delta u=\lambda f(u),~x \in \Omega$ $u=0,~\partial \Omega,$ where $\Omega$ is a smooth bounded domain and $ \lambda$ is a parameter.
What is the meaning or role of the parameter $\lambda$? Please let me know if you have any idea, physical example or comment for it?
Thanks in advance!
Any physical meaning will depend on the physical context. Mathematically, each value of $\lambda$ gives you a different equation (though they may be related), and the solutions will depend somehow on $\lambda$. Investigation of how it depends on $\lambda$ may be worthwhile.
Sometimes, the assumption that $\lambda$ is very small or very large may allow a simplification to a solvable model, and the solution for general $\lambda$ might then be written as a series in positive or negative powers of $\lambda$.