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Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$: $$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ u\neq 0} \frac{\int_\Omega |\nabla u|^2 \text{d} x}{\int_\Omega |u|^2 \text{d} x} >0\; .$$ Then a simple scaling argument can be used to show that one has: $$\lambda_1(t\ \Omega) =\frac{1}{t^2}\ \lambda_1(\Omega)$$ for any $t>0$; therefore $t\mapsto \lambda_1(t\ \Omega) \in ]0,+\infty[$ is a continuous function whith $\displaystyle \lim_{t\to 0^+}\lambda_1 (t\ \Omega)= +\infty$ and $\displaystyle \lim_{t\to +\infty}\lambda_1 (t\ \Omega)= 0$, thus all positive numbers are attained values for $t\mapsto \lambda_1(t\ \Omega)$.

Now, let $w:\Omega \to \mathbb{R}$ a positive, radial function (the latter hypothesis can be removed) in $L^r (\Omega)$ for $r>N/2$. It is possible to prove that the weighted Dirichlet Laplacian, i.e. the operator associated to the PDE: $$\begin{cases} -\Delta u =\lambda\ w(x)\ u &\text{, in } \Omega\\ u=0 &\text{, on }\partial \Omega\end{cases}$$ has a first (or lowest) eigenvalue, say $\lambda_1 (\Omega,w)$, given by: $$\lambda_1 (\Omega, w) := \min_{u\in H_0^1(\Omega),\ u\neq 0} \frac{\int_\Omega |\nabla u|^2\ \text{d} x}{\int_\Omega w(x)\ |u|^2\ \text{d} x}\; .$$ I'd like to prove that:

the function $]0,1[\ni t\mapsto\lambda_1(t\ \Omega ,w)$ takes its values in $]\lambda_1(\Omega ,w), +\infty[$ and that each number in $]\lambda_1(\Omega ,w), +\infty[$ is attained.

As far as I can see, the scaling argument does not applies in this case (because of the presence of the weigth $w$, which does not rescale properly)...

On the other hand, I know that $\lambda_1 (\cdot ,w)$ (as a function of the set) is weakly decreasing w.r.t. inclusion, i.e.: $$\Omega^{\prime \prime} \subseteq \Omega^\prime \subseteq \Omega \quad \Rightarrow \quad \lambda_1 (\Omega^{\prime \prime},w)\geq \lambda_1 (\Omega^\prime ,w)$$ (inequality follows from the fact that each $u\in H_0^1(\Omega^{\prime \prime})$ extends to a function of $H_0^1(\Omega^\prime)$ which is zero outside $\Omega^{\prime \prime}$); therefore $t\mapsto \lambda_1(t\ \Omega,w)$ weakly decreases in $]0,1[$.

Any idea or hint?

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