The variational analysis of a nonlinear anisotropic problem with no-flux boundary condition

Abstract

We study the nonlinear degenerate anisotropic problem $$\begin{aligned} \left\{ \begin{array}{ll} - \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{p_M(x)-2}u = \lambda |u|^{q(x)-2}u &{}\quad \text {in } \Omega ,\\ u(x)=\text {constant} &{}\quad \text {on } \partial \Omega \\ \sum _{i=1}^N\int _{\partial \Omega }a_i(x,\partial _{x_i}u)\nu _id\sigma =0, \end{array}\right. \end{aligned}$$ where \(\Omega \subset \mathbb R^N\) is a bounded domain with smooth boundary. The constant value of the boundary data is not specified, whereas the zero integral term corresponds to a no-flux boundary condition. In the case when \(|u|^{q(x)-2}u\) “dominates” the left-hand side, we show that a nontrivial solution exists for all positive values of \(\lambda \). If the term \(|u|^{q(x)-2}u\) is dominated by the left-hand side, we prove that a solution exists either for small or for large values of \(\lambda >0\). The proofs combine variational arguments with energy estimates.