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In Ambrosetti, A. and Arcoya, D.,it An Introduction to Nonlinear Functional Analysis and Elliptic Problems there is a theorem that the autors doesn't prove and dont let any reference about it, the theorem is

Let, \begin{equation} \label{equ1.4.1} \left\{ \begin{array}{lrl} -\Delta u = \lambda m(x)u,& &\,\, x\in \Omega\\ \hspace{0.6cm}\, u = 0, & &\,\, x\in\partial\Omega \end{array} \right. \end{equation} Assume that $\Omega\subset{R^{N}} $ is bounded and open, $r\in \left( \frac{N}{2}, +\infty \right)\cap \left(1,+\infty \right)$, and $m\in L^{r}(\Omega)$. Consider the set \begin{eqnarray*} \Omega_{+} & = & \left\{ x\in\Omega ;, m(x) > 0 \right\} \end{eqnarray*} The following assertions hold.

  • $0$ is not an eigenvalue of the equation above.

  • If the Lebesgue measure $\left|\Omega_{+} \right|$ of $\Omega_{+}$ is zero, then the equation has no positive eigenvalue.

  • If $\left|\Omega_{+} \right| > 0$, then the positive eigenvalues of the equation above define a nondecreasing unbounded sequence $\left\{ \lambda_{n} \right\}_{n\in \mathbb{N}} \subset (0, +\infty)$. In addition, $\lambda_{n}$ is characterized by $$ \frac{1}{\lambda_{n}} = \sup_{F\in \mathcal{F}_{n}}\inf\left\{ \displaystyle\int m(x)u^{2}(x)dx : \, \displaystyle\int \left|\nabla u(x)\right|^{2} = 1, \,\,\,\,u\in F \right\} $$ where\, $ {\mathcal{F}}_{n} = \left\{ F\subset H:\, F\,\,\mbox{is a subspace with}\,\,\dim F = n \right\}. $

Has someone already seen this theorem proved anywhere?

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2 Answers 2

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Ad 1. Eigenvalue $0$ would mean that there is a nonzero function $u$ for which \begin{equation} \left\{ \begin{array}{lrl} -\Delta u = 0& &\,\, x\in \Omega,\\ \hspace{0.6cm}\, u = 0 & &\,\, x\in\partial\Omega. \end{array} \right. \end{equation} By the maximum principle for harmonic functions, such $u$ has to be zero.

Ad 2. You're not mentioning the space you're working in, but I assume it's $W^{1,2}(\Omega)$. By Sobolev embedding, we have $u \in L^{\frac{2n}{n-2}}$. The exponent $r$ is chosen so that the equation can be tested with $u$ itself (by Holder's inequality), resulting in $$ \int_\Omega |\nabla u|^2 = - \int_\Omega u \Delta u = \lambda \int_\Omega m u^2. $$ The condition $|\Omega_+| = 0$ means $m \le 0$ a.e. in $\Omega$. If we assume that $\lambda \ge 0$, the RHS is nonpositive, while the LHS is nonnegative. Thus, $\nabla u \equiv 0$ in $\Omega$ and in consequence $u \equiv 0$.

Ad 3. This requires some more work and knowledge. Are you familiar with the case $m=1$, i.e. with the usual spectral theory for $-\Delta$?

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Chapter 4 of this thesis deals with eigenvalues for the weighted Laplacian. It has a few references, so it is a starting point. link This paper deals with the $p$-laplacian but again the references could be useful. link2

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