论文标题
$ \ mathbb {r}^n $中非线性schrödinger方程的新型解决方案
New type of solutions for the Nonlinear Schrödinger Equation in $\mathbb{R}^N$
论文作者
论文摘要
我们为非线性schrödinger方程构建了一个新的解决方案的新家庭 \ begin {align*} \ begin {case} -Δu+ v（y）u = u^p，\ quad u> 0，\ quad \ text {in}〜\ m athbb {r}^n， \\ [2mm] u \ in H^1（\ Mathbb {r}^n），\ end {case} \ end end {align*}其中$ p \ in（1，1，\ frac {n+2} {n-2} {n-2} {n-2} {n-2} {n-2} {n-2}）$ and $ n \ n \ geq 3 $，和$ v（y ys unitation ys yise y yes y y y y y y y | y |） = v_0 + \ frac {a} {| y |^m} + o（\ frac {1} {| y |^{m +σ}}），\ quad {\ mbox {as}}}} \ quad | y | y | y | \ to \ infty，$$对于某些固定常数$ v_0，a，σ> 0 $和$ m> 1 $。我们的解决方案具有很强的类比，与Yamabe方程的整个有限能量换入解决方案的构建增加了一倍。
We construct a new family of entire solutions for the nonlinear Schrödinger equation \begin{align*} \begin{cases} -Δu+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases} \end{align*} where $p\in (1, \frac{N+2}{N-2})$ and $N\geq 3$, and $V (y)= V(|y|)$ is a positive bounded radial potential satisfying $$ V(|y|) = V_0 + \frac{a}{|y|^m} + O( \frac{1}{|y|^{m+σ}} ), \quad {\mbox {as}} \quad |y| \to \infty , $$ for some fixed constants $V_0, a, σ>0$, and $m>1$. Our solutions have strong analogies with the doubling construction of entire finite energy sign-changing solution for the Yamabe equation.
