On nonlinear nonhomogeneous resonant Dirichlet equations (Q372514)

The authors study the nonlinear Dirichlet problem NEWLINE\[NEWLINE-\Delta_pu(z)-\Delta u(z)=f(z,u(z))\text{ in }\Omega,\qquad u|{\partial\Omega}=0,\;2<pNEWLINE\]NEWLINE where \(\Omega\subseteq R^N\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\) and \(\Delta_p\) denotes the \(p\)-Laplacian differential operator defined by NEWLINE\[NEWLINE\Delta_p u(z) = \mathrm{div}(\|Du(z)\|^{p-2} Du(z))\text{ for all }u\in W_0^{1,p}(\Omega).NEWLINE\]NEWLINE They prove the existence of at least five non-trivial smooth solutions and provide sign information for all of them, using variational methods. Results from Morse theory are used to produce nodal solutions for the above problem. The authors deal with the resonant case whereas the earlier existence results considered non-resonant equations. The study of the problem is important since it finds application in quantum physics.