Multiple solutions for a class of nonlinear equations involving a duality mapping

Abstract

The paper studies the existence of multiple solutions to abstract equation \begin{eqnarray*} J_p u = N_f u, \end{eqnarray*} 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, $N_f :L^q(\Omega) \rightarrow L^{q'}(\Omega)$ , ${\frac{1}{q}+ \frac{1}{q'}=1 }$, 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 Li and Zhou [9]. As applications we obtain in a unitary manner the multiplicity results already given in [9] for a Dirichlet problem with $p$-Laplacian as well as some new multiplicity results for the Neumann problem \begin{eqnarray*} - \Delta_{p} u + | {u} | ^{p-2} u & =& f(x,u) \qquad \textrm{in}\quad \Omega,\\ | {\nabla u} | ^{p-2}\frac{\partial u}{\partial n} & =& 0 \qquad \textrm{on}\quad \partial \Omega. \end{eqnarray*}