论文标题
在周期性和紧凑的支持半线性椭圆方程的最小能量解决方案中
On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity
论文作者
论文摘要
我们讨论了一个变量中周期性的存在和不存在，在其他变量中对具有非lipschitz非线性的方程式中的最少能量解决方案进行了：$-ΔU=λu=λu=λu=λu=λu=λu^p-u^q $ in $ \ mathbb {r} \ mathbb {r} $。该方法基于nehari歧管方法，该方法是通过合适的pohozaev身份功能给出的单方面约束的补充。该参数$λ$的限值，即适用该方法，对应于一个变量中的周期性存在，并在其他变量中得到紧凑的最低能量解决方案。通过非线性广义雷利商的极端值以及$ p，q $的关键指数的所谓曲线找到该值。得出了解决方案的重要属性，例如它们相对于周期性变量并不微不足道，并且与整个空间$ \ Mathbb {r}^{n+1} $的紧凑型解决方案不一致。
We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: $-Δu=λu^p - u^q$ in $\mathbb{R}^{N+1}$, where $ 0< q < p \leq 1$, $λ\in \mathbb{R}$. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter $λ$, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of $p,q$. Important properties of the solutions are derived, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space $\mathbb{R}^{N+1}$.
