Nonlinear, nohonogeneous parametric Neumann problems
The authors consider a parametric nonlinear Neumann problem \[ \begin{cases} -\text{div} \, a(Du(z))+\lambda|u(z)|^{p-2}u(z)=f(z, u(z)) \;\text{ in } \;\Omega\\ \frac{\partial u}{\partial n}=0 \qquad\qquad\qquad\qquad\qquad\qquad \quad \text{ on } \;\partial\Omega, \;\lambda>0 \end{cases} \] driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction \(f\) which is \(p\)-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. A bifurcation type result is obtained describing the dependence of positive solutions on the parameter \(\lambda>0\), the existence of a smallest positive solution \(\bar u_\lambda\) is shown, and the properties of the map \(\bar\lambda\to \bar u_\lambda\) are investigated. In addition the existence of nodal solutions is shown.
nonlinear nonhomogeneous differential operator
scientific article; zbMATH DE number 6700548