Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave-convex terms (Q2341666)

In this paper the authors consider the following nonlinear parametric Dirichlet problem: \[ \begin{cases} -\mathrm{div}(a(Du(z))=\lambda \left| u(z) \right|^{q-2} \;u(z)+f(z,u(z)) \text{ in } \Omega, \\ u |_{ \partial \Omega }=0, \quad \lambda >0. \end{cases}\tag{1} \] where the map \(a:\mathbb R^{n}\rightarrow\mathbb R^{n}\) involved in the differential operator of (1) is strictly monotone and satisfies certain regularity conditions. They suppose that the differential operator in (1) need not be homogeneous, that \(q\in(1,p)\) and so the first term in the right-hand side of (1) is \((p-1)\) sublinear; moreover that \(f(z,x)\) is a Carathéodory function. The authors, combining variational methods based on critical point theory, with suitable truncation and comparison techniques and with Morse Theory, obtain five non trivial smooth solutions and deduce precise sign information for all of them, that is that two of them are positive, two negative and the fifth is nodal and change sign. We observe that all previous results concerning the existence of nodal solutions deal with equations driven by the Laplacian or \(p\)-Laplacian and the reaction satisfies Ambrosetti-Rabinowitz condition. Moreover, because the differential operator in the problem is not homogeneous, it does not allow the use of techniques employed in the previous papers; then this result is the first one on the existence of nodal solutions for nonlinear equations driven by nonhomogeneous differential operators.