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Rabinowitz puts forth the following example from his book "Minimax Methods in Critical Point Theory with Applications to Differential Equations" p.25 I will copy below the statement exactly as written in the book

\begin{array}{ccc} -\Delta u &=& \lambda a(x)u + p(x,u), \ \ \ x\in \Omega \\ u &=& 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \in \partial \Omega \end{array}

where $p$ satisfies $p(x,\xi) \in C(\bar{\Omega}\times \mathbb{R}, \mathbb{R})$ and $|p(x,\xi)| \leq A$ for some $A\geq 0$. If $\lambda$ is not an eigenvalue of \begin{array}{ccc} -\Delta u &=& \lambda a(x)u - p(x,u), \ \ \ x\in \Omega \\ u &=& 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \in \partial \Omega \end{array}
then standard elliptic estimates imply an a priori bound for solutions in $W^{2,\beta}_0(\Omega)$ for e.g. $\beta > n$. Then one can use the linear elliptic theory for such spaces and the Schauder Fixed Point theorem to get a solution.

Can you help me decipher this statement and its proof? Why does not being an eigenvalue matter? By the Schauder Fixed Point theorem, does he really mean Schaefer's Fixed Point theorem? Also, is the switch from $p$ to $-p$ in the two equations a typo? I'm not sure how to apply it to this problem... Any help would be greatly appreciated!

Euler....IS_ALIVE
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1 Answers1

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Two lines before Rabinowitz states Theorem 4.12, He writes: "Thus an interesting question to study is what happen when $\lambda$ is not a eigenvalue of $(2.40)$". Im gonna state $(2.40)$ here. Let $a:\overline{\Omega}\rightarrow\mathbb{R}$ be a positive Lipschitz function. Then, $(2.40)$ is the Sturm-Liouville eigenvalue problem $$\tag {2.40} \left\{ \begin{array}{rl} -\Delta v=\lambda a(x) v &\mbox{ $x\in\Omega$} \\ v=0 &\mbox{ in $\partial\Omega$} \end{array} \right. $$

Let's work with $(2.40)$ instead of $(2.39)$. For $f\in L^2(\Omega)$, consider the problem $$\tag {$2.40_f$} \left\{ \begin{array}{rl} -\Delta v=\lambda a(x) v+f &\mbox{ $x\in\Omega$} \\ v=0 &\mbox{ in $\partial\Omega$} \end{array} \right. $$

If $\lambda$ is not an eigenvalue of $(2.40)$, then $(2.40_f)$ has a unique solution $u$ that by regularity theory is in $H_0^1(\Omega)\cap H^2(\Omega)$. Also, if $f\in L^p$, $$\tag{1}\|u\|_{2,p}\leq C\|f\|_p$$

Moreover, if $f\in L^\infty$, then $u\in C(\overline{\Omega})$. Because all of this, when $\lambda$ is not an eigenvalue of $(2.40)$, then we can associate to $(2.40_f)$ an operator $S:L^2\rightarrow H_0^1\cap H^2$ (solution operator) defined by $Sf=u$, where $u$ is the solution of $(2.40_f)$. Note that $S: C(\overline{\Omega})\to C(\overline{\Omega})$.

Now consider problem $(4.11)$ in page 25 $$\tag {4.11} \left\{ \begin{array}{rl} -\Delta u=\lambda a(x)u +p(x,u) &\mbox{ $x\in\Omega$} \\ u=0 &\mbox{ in $\partial\Omega$} \end{array} \right. $$

Define $G:C(\overline{\Omega})\to C(\overline{\Omega})$ by $Gf(x)=p(x,f(x))$. Define $T:C(\overline{\Omega})\to C(\overline{\Omega})$ by $T=S\circ G$ and note that to solve $(4.11)$ is equivalently to solve the problem $Tu=u$. By $(1)$ we have that for $p>n$ \begin{eqnarray} \|T(f)\|_\infty &\leq& \|T(f)\|_{2,p} \nonumber \\ &\leq& C\|p(\cdot,f)\|_p \nonumber \\ &\le& M \end{eqnarray}

where $M>0$ is a constant. Hence, there is some ball $B\subset C(\overline{\Omega})$ such that $T(B)\subset B$. Also, note that $T:B\to B$ is a compact operator, because for $p>n$ we have that $W^{2,p}$ is compactly embedded in $C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in (0,1)$. From this, you can use the Schauder Fixed Point Theorem.

Euler....IS_ALIVE
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Tomás
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