Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^\* \in R$ it has a bounded smooth solution $u^\*$ in $R^n$ ($f$ is a nice smooth function). Under what conditoions one can guarantee that for $\alpha$ close to $\alpha^\*$ this equation has a solution $u$ close to $u^\*$? 



Implicit function theorem for elliptic partial differential equations
As your title indicates, you would do this using the implicit function theorem for Banach space applied to an appropriately defined functional. This in turn requires proving that there is a bounded right inverse to the linearized operator.


The second step requires solving a linear elliptic PDE and getting an a priori estimate for the solution in terms of the inhomogeneous term. If you really want to solve this on $\mathbb{R}^n$ and not on a bounded domain, then you will be have make some kind of assumption about the decay of $u$ or its derivatives at infinity. For example, a natural thing to do is to use a Sobolev space. However, if this question arises from some question you're studying, the question often dictates which Banach space you need to use.


