论文标题
用于超临界问题的高能解决方案的渐近学
Asymptotics for a high-energy solution of a supercritical problem
论文作者
论文摘要
在本文中，我们处理$ 1 <p <2 $和$ q> p $的方程式\ [ - δ_pu+| u | u |^{p-2} u = | u |^{q-2} u \]，在$ \ mathbb r^n $的单位球中的neumann边界条件下。我们专注于三种正，径向和径向不固定的解决方案，在[13]中证明了它们的存在$ q $。我们将限制概况检测为高能解决方案的$ q \ to \ infty $，并表明，与最小的能源一号不同，它会收敛到常数$ 1 $。证明需要从最小化问题的理论中借用的几种工具，并准确对解决方案的先验估计值进行，这具有独立的利益。
In this paper we deal with the equation \[-Δ_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to\infty$ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant $1$. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest.
