Multiple solutions for a class of nonlinear equations involving a duality mapping. (Q621532)

The paper studies the existence of multiple solutions to abstract equation \(J_{p}u=N_{f}u\), where \(J_{p}\) is the duality mapping on a real reflexive and smooth Banach space \(X\), corresponding to the gauge function \(\varphi (t)=t^{p-1}\), \(1<p<+\infty \). It is assumed, that \(X\) is compactly imbedded in a Lebesgue space \(L^{q}(\Omega )\), \(p\leq q< p^{*}\) and continuously imbedded in \(L^{p^{*}}(\Omega )\), \(p^{*}\) being the Sobolev conjugate exponent, \(N_{f}\: L^{q}(\Omega )\to L^{q'}(\Omega )\), \(1/q+1/q'=1\), being the Nemytskii operator generated by a function \(f\in \mathcal {C}(\bar \Omega \times \mathbf {R},\mathbf {R})\), which satisfies some appropriate conditions. These assumptions allow the use of many procedures appearing essentially in the paper by \textit{G. B. Li} and \textit{H. S. Zhou} [J. Lond. Math. Soc., II. Ser. 65, No. 1, 123--138 (2002; Zbl 1171.35384)]. The results for the abstract problem are applied to Dirichlet and Neumann problems for \(p\)-Laplacian.