论文标题
在准线性不确定问题中，最小化器的独特性和符号特性
Uniqueness and sign properties of minimizers in a quasilinear indefinite problem
论文作者
论文摘要
令$ 1 <q <q <p $和c（\overlineΩ）$的$ a \是签名，其中$ω$是$ \ mathbb {r}^{n} $的有限且平滑的域。我们表明功能\ [i_ {q}（u）：= \int_Ω\ left（\ frac {1} {p} {p} | \ nabla u |^{p} - \ frac {1} {q} {q} {q^} a（x） $ w_ {0}^{1，p}（ω）$或$ w^{1，p}（ω）$）。此外，我们证明$ u_ {q} $是关联的欧拉 - 拉格朗日方程的唯一可能的\ textit {pastic}解决方案，这表明该方程最多具有一个阳性解决方案。此外，我们表明，如果$ q $足够接近$ p $，则$ u_ {q} $是正的，这也可以保证$ i_ {q} $的最小化器不会更改符号。这些结果中有几个是新的，即使对于$ p = 2 $也是如此。
Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which also guarantees that minimizers of $I_{q}$ do not change sign. Several of these results are new even for $p=2$.
