Initial value problem for a singular \(n\)-Laplacian equation (Q2779101)

The authors consider the local existence of positive solutions for the initial value problem with the \(n\)-Laplacian operator NEWLINE\[NEWLINE(\phi_n(y'))'= f(t,y,y'),\quad t> 0,\quad y(0)= y'(0)= 0,NEWLINE\]NEWLINE with \(\phi_n(z)=|z|^{n-2}z\), for \(n> 1\), in which \(f\) may be singular. Theorems are proven for each of the cases in which \(f(t,x,y)\) is (i) singular at \(t=0\) and \(y= 0\) (theorem 2), (ii) singular at \(t= 0\) and \(x=0\) (theorem 3) and (iii) singular at \(t= 0\), \(x=0\) and \(y= 0\) (theorem 6). For example, theorem 3 proves the local existence under the conditions (1) \(f\in C((0,\infty)\times \mathbb{R}\setminus\{0\}\times \mathbb{R},\mathbb{R})\), (2) there exists an \(R_0> 0\) such that \(f:(0,\infty)\times (0, R_0]\times [0, R_0]\to [0,\infty)\), (3) \(f(t,x,y)\leq p(t)q(x)m(y)\) for \((t,x,y)\in (0,\infty)\times (0, R_0]\times [0, R_0]\), where \(p\in L^\alpha_{\text{loc}}([0, \infty),(0, \infty))\) for \(\alpha> 1\), \(q\in L^\beta_{\text{loc}}([0, \infty),(0,\infty))\) where \({1\over \alpha}+{1\over\beta}= 1\), \(m\in C([0, R_0], (0,\infty))\) along with a local integrability condition involving both \(p\) and \(q\). The proofs are accomplished using fixed-point theory along with the Ascoli-Arzela theorem.