论文标题
分叉成频谱间隙，以实现强烈不确定的choquard方程
Bifurcation into spectral gaps for strongly indefinite Choquard equations
论文作者
论文摘要
我们考虑半连续椭圆方程$$ \ left \ {\ oken {array} {ll}＆-ΔU+v（x）u = \ left（i_α\ ast | ast | u | u |^p \ right）| u |^{p-2} \至0 \ hbox {as} | x | \ to \ infty，\ end {array} \ right。 $$ $i_α$是riesz的潜力，$ p \ in（\ frac {n+α} n，\ frac {n+α} {n-2}）$，$ n \ geq3 $，$ n \ geq3 $，$ v $是连续周期性的。我们假设$ 0 $在于$-δ+ v $的频谱差距$（a，b）$。我们证明，对于每个$λ\ in（a，b）$中的每个$λ\ $ h^1（\ mathbb r^n）$中的许多几何不同解决方案的存在，如果$ \ \ \ \ frac {n+α} n <p <p <p <p <p <p <1+\ frac {2+α} $。此外，$ b $是$ [a，b] $的独特差距 - 爆炸点（零）。当$λ= a $时，我们在$ h^2_ {loc}（\ Mathbb r^n）$中找到了无限的许多几何不同解决方案。关于最终发生的$λ= a $的分叉的最终发言。
We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-Δu+V(x)u=\left(I_α\ast |u|^p\right)|u|^{p-2}u+λu\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right. $$ where $I_α$ is a Riesz potential, $p\in(\frac{N+α}N,\frac{N+α}{N-2})$, $N\geq3$, and $V $ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-Δ+ V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $λ\in(a, b)$, which bifurcate from $b$ if $\frac{N+α}N< p < 1 +\frac{2+α}{N}$. Moreover, $b$ is the unique gap-bifurcation point (from zero) in $[a,b]$. When $λ=a$, we find infinitely many geometrically distinct solutions in $H^2_{loc}(\mathbb R^N)$. Final remarks are given about the eventual occurrence of a bifurcation from infinity in $λ=a$.
