Infinitely many solutions of superlinear fourth order boundary value problems
Infinitely many solutions of superlinear fourth order boundary value problems
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DOI：
10.12775/tmna.2002.016
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发表时间：
2002-06
影响因子：
0.7
通讯作者：
B. Rynne
中科院分区：
文献类型：
--
作者：
B. Rynne
We consider the boundary value problem \begin{gather*} u^{(4)}(x) =g(u(x)) + p(x,u^{(0)}(x),\dots,u^{(3)}(x)) , \quad x \in (0,1), \\ u(0) =u(1) =u^{(b)}(0) =u^{(b)}(1) =0, \end{gather*} where: \begin{itemize} \item[(i)] $g \colon \mathbb R \to \mathbb R$ is continuous and satisfies $\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$ ($g$ is superlinear as $|\xi| \to \infty$), \item[(ii)] $p \colon [0,1] \times \mathbb R^4 \to \mathbb R$ is continuous and satisfies $$ |p(x,\xi_0,\xi_1,\xi_2,\xi_3)| \le C + \frac{1}{4} |\xi_0| , \quad x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \mathbb R^4, $$ for some $C> 0$, \item[(iii)] either $b=1$ or $b=2$. \end{itemize} We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.
We consider the boundary value problem \begin{gather*} u^{(4)}(x) =g(u(x)) + p(x,u^{(0)}(x),\dots,u^{(3)}(x)) , \quad x \in (0,1), \\ u(0) =u(1) =u^{(b)}(0) =u^{(b)}(1) =0, \end{gather*} where: \begin{itemize} \item[(i)] $g \colon \mathbb R \to \mathbb R$ is continuous and satisfies $\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$ ($g$ is superlinear as $|\xi| \to \infty$), \item[(ii)] $p \colon [0,1] \times \mathbb R^4 \to \mathbb R$ is continuous and satisfies $$ |p(x,\xi_0,\xi_1,\xi_2,\xi_3)| \le C + \frac{1}{4} |\xi_0| , \quad x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \mathbb R^4, $$ for some $C> 0$, \item[(iii)] either $b=1$ or $b=2$. \end{itemize} We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.