论文标题
准线性不确定问题中的独特性和积极性问题
Uniqueness and positivity issues in a quasilinear indefinite problem
论文作者
论文摘要
我们考虑问题$$（p_λ）\ quad-Δ_{p} u =λu^{p-1}+a（x）u^{q-1}，\ quad u \ geq0 \ geq0 \ qeq0 \ quad \ quad \ mbox {in}ω$$ dirichlet或neumann边界条件下。 $ \ mathbb {r}^{n} $（$ n \ geq1 $），$λ\ in \ mathbb {r} $，$ 1 <q <p $，和$ a \ in C（\edimeplineΩ）$更改符号。这些条件使该问题的死亡核心解决方案可以承认多种非平凡解决方案。我们表明，对于$λ<0 $，功能\ [i_λ（u）：= \int_Ω\ left（\ frac {1} {p} {p} | \ nabla u |^{p} - \ frac {λ} {λ} {p} {p} {p} {p} |^p} | U | ，在$ x = w_ {0}^{1，p}（ω）$或$ x = w^{1，p}（ω）$中定义的]，具有\ textit {恰好}一个非阴性全局最小化器，这是\ textIt {唯一的} $（p_λ）$ at $（p_λ）$} $ a> 0 $）。特别是，这个问题最多具有$λ<0 $的一个积极解决方案。在$ a $上的某种条件下，上述唯一性结果因$λ> 0 $的某些值而失败，除了基态解决方案之外，在$ω__{a}^{+} $中呈阳性。我们还提供$λ$，$ a $和$ q $的条件，以便这些解决方案在$ω$中变为阳性，并分析用于通用解决方案的死核的形成。
We consider the problem $$ (P_λ)\quad -Δ_{p}u=λu^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }Ω$$ under Dirichlet or Neumann boundary conditions. Here $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1<q<p$, and $a\in C(\overlineΩ)$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $λ<0$ the functional \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_{0}^{1,p}(Ω)$ or $X=W^{1,p}(Ω)$, has \textit{exactly} one nonnegative global minimizer, and this one is the \textit{only} solution of $(P_λ)$ being positive in $Ω_{a}^{+}$ (the set where $a>0$). In particular, this problem has at most one positive solution for $λ<0$. Under some condition on $a$, the above uniqueness result fails for some values of $λ>0$ as we obtain, besides the ground state solution, a \textit{second} solution positive in $Ω_{a}^{+}$. We also provide conditions on $λ$, $a$ and $q$ such that these solutions become positive in $Ω$, and analyze the formation of dead cores for a generic solution.
