Bénédicte Alziary, Naziha Besbas
Anti-Maximum Principle for a Schrödinger Equation in $\mathbb{R}^N$, with a non radial potential
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 59, 51-62(2005)
MSC:
35B50 Maximum principles
35P30 Nonlinear eigenvalue problems, nonlinear spectral theory for PDO
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Abstract: Anti-maximum for the Schr\"odinger equation $-\Delta u + q(x)u-\lambda u=f(x)$ in $L ^2(\mathbb{R} ^N)$ is extended to potentials $q$ non
necessarily radial. The anti-maximum is proved in the following form:
Let $\varphi_1$ denote the positive eigenfunction associated with the
principal eigenvalue $\lambda _1$ of the Schr\"odinger operator ${\cal A}=-\Delta + q(x)\bullet$ in $L ^2(\mathbb{R} ^N)$. Assume the potential
$q(x)$ grows fast enough near infinity, and the function $f$ satisfy
$f\not\equiv 0$ and $0\leq f/\varphi_1\leq C\equiv const$ a.e. in $\mathbb{R} ^N$.
Then there exists a positive number $\delta$
(depending upon $f$)
such that, for every
$\lambda\in (\lambda_1, \lambda_1 + \delta)$,
the inequality
$u\leq -c\varphi_1$
holds a.e.\ in $\mathbb{R} ^N$,
where $c$ is a positive constant depending upon $f$ and $\lambda$.
Keywords: Positive or negative solutions; pointwise bounds; principal eigenvalue; positive eigenfunction; strong maximum and anti-maximum principles