Multiplicity results for the 1-dimensional generalized \(p\)-Laplacian
Of concern is the boundary value problem (P) \(- (a (|u' |^p) |u' |^{p - 2} u')' = f(u)\) in \(I = (0,1)\), \(u(0) = u(1) = 0\), where \(p > 1\), \(a \in C (\mathbb{R}^+)\), and \(f\) is a real continuous odd function. A weak solution to (P) is a function \(u \in W^{1, p}_0 (I)\) such that \(a(|u' |^p) |u' |^{p - 2} u' \in L^{p'} (I)\) \((p^{-1} + p'{}^{-1} = 1)\), and \(\int^1_0 [a (|u' |^p) |u' |^{p - 2} u' v' - f(u)v] = 0\), \(\forall v \in W^{1, p}_0 (I)\). The author discusses the existence and multiplicity of weak solutions of (P) by using an energy method.
scientific article; zbMATH DE number 765145