On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities
On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities
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DOI：
10.3934/dcdsb.2016066
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发表时间：
2016-09
影响因子：
1.2
通讯作者：
Ruyun Ma;Tianlan Chen;Yanqiong Lu
中科院分区：
文献类型：
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作者：
Ruyun Ma;Tianlan Chen;Yanqiong Lu
设$B_1$是$\mathbb{R}^N$中的单位球，其中$N \geq 2$。令$f\in C^1（[0，\infty），\mathbb{R}）$，$f（0）=0$，$f（\beta）= \beta，\ f（s）s$ for $s \in（\beta，\infty）$ and $f'（\beta）>\lambda^{r}_k$。D.你好，B。Noris和T. Weth [Ann. Inst. H.庞加莱肛门。Non Lineaire 29（4）（2012）]证明了半线性Neumann问题$$ -\Delta u+u=f（u）\ \text{in}\ B_1，\ \\\partial_\nu u=0 \ \text{on}\ \\partial B_1 $$的非减径向正解的存在性，其中$k=2$，他们证明了存在一个与$\beta$有$k$个交点的径向解，只要$f '（\beta）>\lambda^r_k$对于$k>2$，其中$\lambda^r_k$是具有Neumann边界条件的单位球中$\Delta + I$的第k个径向特征值。在本文中，我们表明，答案是肯定的情况下，线性有界的非线性。
Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s) s$ for $s \in (\beta, \infty)$ and $f'(\beta)>\lambda^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincare Anal. Non Lineaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -\Delta u+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_\nu u=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$, where $\lambda^r_k$ is the $k$-th radial eigenvalue of $\Delta + I$ in the unit ball with Neumann boundary conditions. In this paper, we show that the answer is yes in the case of linearly bounded nonlinearities.