Infinitely Many Nodal Solutions of Superlinear Third Order Two-Point Boundary Value Problems

We are concerned with the existence of nodal solutions for a third order boundary value problem \(\begin{aligned} \left\{ \begin{array}{ll} u'''(x)=g(u(x))+p(x,u(x),u'(x),u''(x)),~~~~&{}x\in (0,1),\\ u(0)=u(1)=u'(1)=0,\\ \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}^{\prime \prime \prime }(x)=g(u(x))+p(x,u(x),u^{\prime }(x),u^{\prime \prime }(x)),\hfill & x\in (0,1),\hfill \\ u(0)=u(1)=u^{\prime }(1)=0,\hfill \end{array}\right)\end{array}$$ where \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) $g:\mathbb{R}\to \mathbb{R}$ is continuous and satisfies \(\lim _{|\xi |\rightarrow \infty } g(\xi )/\xi =\infty \) ${lim}_{|\xi |\to \infty }g(\xi )/\xi =\infty$ (g is superlinear as \(|\xi |\rightarrow \infty \) $|\xi |\to \infty$ ), \(p:[0,1]\times {\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) $p:[0,1]\times {\mathbb{R}}^3\to \mathbb{R}$ is continuous and satisfies \(|p(x,\xi _0,\xi _1,\xi _2)|\le C+|\xi _0|/3\) $|p(x,{\xi }_0,{\xi }_1,{\xi }_2)|\le C+|{\xi }_0|/3$ , \(x\in [0,1]\) $x\in [0,1]$ , \((\xi _0,\xi _1,\xi _2)\in {\mathbb {R}}^{3}\) $({\xi }_0,{\xi }_1,{\xi }_2)\in {\mathbb{R}}^3$ , for some \(C>0\) $C>0$ . We obtain infinitely many solutions having specified nodal properties. The proof of our main result is based upon bifurcation techniques.