\(C^1\)-regularity of solutions for \(p\)-Laplacian problems (Q1021819)

Consider the following one-dimensional singular \(p\)-Laplacian problem \[ \varphi_p(u'(t))'+f(t,u(t))=0,\;t\in (0,1),\quad u(0)=u(1)=0, \tag{P} \] where \(\varphi_p(x)=|x|^{p-2}x\), \(p>1\) and \(f\in((0,1)\times \mathbb{R},\mathbb{R})\) satisfies some singular weights. The authors study \(C^1\)-regularity of the solutions of (P) and prove the existence of the solutions. An example is also given to show the multiplicity of positive (or negative) solutions as well as sign-changing solutions especially when the nonlinear term is \(p\)-superlinear.