On a Dirichlet problem with \(p\)-Laplacian and asymmetric nonlinearity (Q2264060)

The authors study the following homogeneous Dirichlet problem \[ \begin{cases} -\Delta_p u= f(x, u) &\text{in}\,\, \Omega,\\ u=0 &\text{on}\,\, \partial \Omega,\end{cases}\tag{1} \] where \(\Omega \) is a bounded domain of \(\mathbb R^N\,(N\geq 3),\) with smooth boundary \(\partial\Omega. \;\Delta_p u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplace differential operator with \(p>1.\) \(f: \Omega \times \mathbb R \rightarrow \mathbb R\) is a Carathéodory function such that \(f(x, 0)=0.\) Under some hypotheses on \(f \,((p-1)-\)linear and asymmetric at \(\pm\infty\)) and using some variational and truncation techniques, the authors prove the existence of at least two nonnegative smooth solutions to problem (1). The existence of a third nontrivial smooth solution of problem (1), in the case \(p=2,\) is also investigated using Morse's theory.