Multiple positive solutions for classes of \(p\)-Laplacian equations. (English) Zbl 1150.35419

The authors study positive \(C^1(\overline \Omega )\) solutions to classes of boundary-value problems of the form \(-\Delta _p u=\lambda f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\lambda >0\) is a parameter, and \(\Delta _p\) denotes the \(p\)-Laplacian operator defined by \(\Delta _p z:=\operatorname {div}(| \nabla z| ^{p-2}\nabla z)\) \((p>1)\). \(\Omega\) is a bounded domain in \(\mathbb R^N\) \((N\geq 2)\), with \(\partial \Omega \) of connected class \(C^2\). (If \(N=1\), it is assumed that \(\Omega \) is a bounded open interval.) In particular, the existence of three positive solutions for classes of nondecreasing, \(p\)-sublinear functions \(f\) belonging to \(C^1([0,\infty))\) is established.


35J70 Degenerate elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)