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 $u^*$?
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asked Jan 10, 2015 at 3:52
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$\begingroup$ Can't you just take $u=\frac{\alpha}{\alpha^*}u^*$ if $\alpha^*\neq0$? $\endgroup$– Joonas IlmavirtaCommented Jan 10, 2015 at 11:39
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1 Answer
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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.
answered Jan 10, 2015 at 17:55