The aim of this paper is to give some oscillation criteria for the following partial differential equation (PDE) with \(p\)-Laplacian: \[ \text{div}(\|\nabla u\|^{p-2}\nabla u)+ c(x)\phi(u)=0,\tag{1} \] where \(p>1\) and \(\phi(u)=|u|^{p-2}u\). The equation (1) is said to be oscillatory if every solution \(u\) is oscillatory, i.e., \(u\) has a zero in \(\Omega(a)= \{x\in \mathbb{R}^n: a\leq\|x\|\}\) for every \(a\in\mathbb{R}^+\). There are introduced the following notations: \[ \begin{aligned} &\Omega(a,b)= \{x\in \mathbb{R}^n: a\leq\|x\|\leq b\};\\ & C_p(t)= {p-1\over t^{p-1}} \int^t_1 s^{p-2} \int_{\Omega(1,s)}\|x\|^{1-n}c(x) dx ds\text{ with the finite limit }C_0= \lim_{t\to\infty} C_p(t);\\ & Q(t)= t^{p-1} \Biggl(C_0- \int_{\Omega(1,t)}\|x\|^{1-n} c(x) dx\Biggr);\\ & H(t)= {1\over t}\int_{\Omega(1, t)}\|x\|^{p- n+1} c(x) dx;\\ & Q_*= \liminf_{t\to\infty} Q(t),\quad Q^*= \limsup_{t\to\infty} Q(t);\\ & H_*= \liminf_{t\to\infty} H(t),\quad H^*= \limsup_{t\to\infty} H(t);\end{aligned} \] \(\omega_n\) is the measure of the \(n\)-dimensional unit sphere in \(\mathbb{R}^n\) and \(q\) is the conjugate number to \(p\), i.e., \(q= {p\over p-1}\). \(A\) denotes the smaller of the zeros of the equation \[ (p-1)\omega^{-q/p}_n|x|^q+ (n-p)x+ (p-1)Q_*= 0, \] and \(B\) denotes the larger of the zeros of the equation \[ (p-1) \omega^{-q/p}_n|x|^q+(n- p)x+ H_*= 0. \] The main results state that each of the following condition is sufficient for equation (1) to be oscillatory: (i) \(\limsup_{t\to\infty} {t^{p-1}\over \ln t} [C_0- C_p(t)]> [{p- n\over p}]^p\omega_n\); (ii) \(\limsup_{t\to\infty} t^{n-1}[C_0- C(t)]= \infty\); (iii) \(Q_*>-\infty\quad \text{and}\quad \limsup_{t\to\infty} \int_{\Omega(1, t)}c(x) dx= \infty\); (iv) \(\limsup_{t\to\infty} [Q(t)+ H(t)]> |{1- n\over p}|^p {\omega_n\over p-1}+ |{p- n+1\over p}|^p \omega_n\). Assuming \[ {(n- 1)- p(p-1)\over p(p-1)} \phi\Biggl({n-1\over p}\Biggr) \omega_n\leq Q_*\leq \Biggl|{n- p\over p}\Biggr|^p {\omega_n\over p-1}\tag{2} \] and (or) \[ {1-n\over p} \phi\Biggl({p-n+1\over p}\Biggr) \omega_n\leq H_*\leq\Biggl|{n- p\over p}\Biggr|^p \omega_n,\tag{3} \] then each of the following conditions also implies oscillation of equation (1): (v) inequality (2) and \(H^*>|{p-n+ 1\over p}|^p \omega_n-A\) hold; (vi) inequality (3) and \(Q^*> {1\over p-1} |{1-n\over p}|^p \omega_n+ B\) hold; (vii) (2), (3) and \(Q^*> Q_*- A+B\) hold; (viii) (2), (3) and \(H^*> H_*- A+B\) hold; (ix) (2), (3) and \(\limsup_{t\to\infty} [Q(t)+ H(t)]> Q_*+ H_*- A+B\) hold. The main tool for the investigation is the Riccati technique combined with suitable a priori bounds.