Nonlinear Transmission Eigenvalue Problems with Nonhomogeneous Operators of Different p-Growth
Let $\Omega \subset \mathbb{R}^N$, $N \ge 2$, be a bounded domain with Lipschitz boundary, divided by a Lipschitz hypersurface $\Sigma$ into two open, disjoint Lipschitz subdomains $\Omega_1$ and $\Omega_2$. We study a nonlinear transmission eigenvalue problem driven by nonhomogeneous operators with $p_i$- growth in each subdomain $\Omega_i$, $i=1,2$, and subject to continuity and flux transmission conditions across the interface $\Sigma$. The real parameter $\lambda$ appears both in the equations and in the nonlinear boundary conditions. Using variational methods, we prove the existence of an unbounded sequence of eigenvalues. Under additional assumptions, we establish that the set of eigenvalues coincides with the entire interval $(0,\infty)$. As a particular case, we obtain the corresponding eigenvalue results for the associated single-domain problem.