Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in C_c^{\infty}\left(\mathbf{R}^n\right) $$ Then by Theorem8.38 in Gilbarg&Trudinger's book. In a ball $B_R$, we should have $$-\Delta \phi_R + H(x)\phi_R=\lambda_R\phi_R,$$ where $\lambda_R\ge0$ is the first nonzero eigenvalue, with $\phi_R>0$.

1.If there is no $H(x)$, it will be simple to prove that $\lambda_R$ is a decreasing function when $R$ becomes greater, so there will be a limit for $\lambda_R$ when $R \rightarrow +\infty$, but here we have $H(x)$, so we can't simply let $\phi(x)=\Phi(xR/a)$ to make the ball bigger (with radius $aR$), do we still have the property that the first nonzero eigenvalue is a decreasing function with respect to $R$?

2.what is the limit of $lim_{R\rightarrow+\infty}\lambda_R=\lambda_0$, is it 0?

3.if we let the $R$ tend to $+\infty$, can we also get a limit function $\Phi$, which satisfies that $$-\Delta \Phi + H(x)\Phi=\lambda_0\Phi.$$ And what type is this convergence? Do we need local convergence or?

All these questions have a vague answer in my mind, but I don't know how to figure out the details, if you have any references I would be very glad!

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in C_c^{\infty}\left(\mathbf{R}^n\right) $$ Then by Theorem 8.38 in Gilbarg & Trudinger's book, in a ball $B_R$ we should have $$-\Delta \phi_R + H(x)\phi_R=\lambda_R\phi_R,$$ where $\lambda_R\ge0$ is the first nonzero eigenvalue, with $\phi_R>0$.

1. If there is no $H(x)$, it will be simple to prove that $\lambda_R$ is a decreasing function when $R$ becomes greater, so there will be a limit for $\lambda_R$ when $R \rightarrow +\infty$.
   But here we have $H(x)$, so we can't simply let $\phi(x)=\Phi(xR/a)$ to make the ball bigger (with radius $aR$): do we still have the property that the first nonzero eigenvalue is a decreasing function with respect to $R$?

2. What is the value of $\lim_{R\rightarrow+\infty}\lambda_R=\lambda_0$, is it 0?

3. If we let the $R$ tend to $+\infty$, can we also get a limit function $\Phi$, which satisfies that $$-\Delta \Phi + H(x)\Phi=\lambda_0\Phi, $$ and what type is this convergence? Do we need local convergence or?

All these questions have a vague answer in my mind, but I don't know how to figure out the details, if you have any references I would be very glad!

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Then $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in C_c^{\infty}\left(\mathbf{R}^n\right) $$ Then by Theorem8.38 in Gilbarg&Trudinger's book. In a ball $B_R$, we should have $$-\Delta \phi_R + H(x)\phi_R=\lambda_R\phi_R,$$ where $\lambda_R\ge0$ is the first nonzero eigenvalue, with $\phi_R>0$.

1.If there is no $H(x)$, it will be simple to prove that $\lambda_R$ is a decreasing function when $R$ becomes greater, so there will be a limit for $\lambda_R$ when $R \rightarrow +\infty$, but here we have $H(x)$, so we can't simply let $\phi(x)=\Phi(xR/a)$ to make the ball bigger (with radius $aR$), do we still have the property that the first nonzero eigenvalue is a decreasing function with respect to $R$?

2.what is the limit of $lim_{R\rightarrow+\infty}\lambda_R=\lambda_0$, is it 0?

3.if we let the $R$ tend to $+\infty$, can we also get a limit function $\Phi$, which satisfies that $$-\Delta \phi + H(x)\phi=\lambda_0\phi.$$ And what type is this convergence? Do we need local convergence or?

All these questions have a vague answer in my mind, but I don't know how to figure out the details, if you have any references I would be very glad!

Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in C_c^{\infty}\left(\mathbf{R}^n\right) $$ Then by Theorem8.38 in Gilbarg&Trudinger's book. In a ball $B_R$, we should have $$-\Delta \phi_R + H(x)\phi_R=\lambda_R\phi_R,$$ where $\lambda_R\ge0$ is the first nonzero eigenvalue, with $\phi_R>0$.

1.If there is no $H(x)$, it will be simple to prove that $\lambda_R$ is a decreasing function when $R$ becomes greater, so there will be a limit for $\lambda_R$ when $R \rightarrow +\infty$, but here we have $H(x)$, so we can't simply let $\phi(x)=\Phi(xR/a)$ to make the ball bigger (with radius $aR$), do we still have the property that the first nonzero eigenvalue is a decreasing function with respect to $R$?

2.what is the limit of $lim_{R\rightarrow+\infty}\lambda_R=\lambda_0$, is it 0?

3.if we let the $R$ tend to $+\infty$, can we also get a limit function $\Phi$, which satisfies that $$-\Delta \Phi + H(x)\Phi=\lambda_0\Phi.$$ And what type is this convergence? Do we need local convergence or?

All these questions have a vague answer in my mind, but I don't know how to figure out the details, if you have any references I would be very glad!