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We study the vector $p$-Laplacian $$\left\}\begin{array}{c}-\left(\right|u^{\text{'}}{{|}^{p-2}{u}^{\text{'}})}^{\text{'}}=\nabla F(t,u)\text{a.e.}t\in [0,T],u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),1<p<\infty .\hfill \end{array}\right.(*)$$ We prove that there exists a sequence $\left(u_n\right)$ of solutions of ($*$) such that $u_n$ is a critical point of $\varphi$ and another sequence $\left(u_n^{*}\right)$ of solutions of $(*)$ such that $u_n^{*}$ is a local minimum point of $\varphi$, where $\varphi$ is a functional defined below.

This paper studies the existence of solutions to the singular boundary value problem $$\left\}\begin{array}{c}-u^{\text{'}\text{'}}=g(t,u)+h(t,u),t\in (0,1),u\left(0\right)=0=u\left(1\right),\hfill \end{array}\right.$$ where $g(0,1)\times (0,\infty )\to ℝ$ and $h(0,1)\times [0,\infty )\to [0,\infty )$ are continuous. So our nonlinearity may be singular at $t=0,1$ and $u=0$ and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.