A nonlinear elliptic eigenvalue–transmission problem with Neumann boundary condition

Abstract

Let $$\varOmega \subset \mathbb {R}^N$$ , $$N\ge 2$$ , be a bounded domain which is divided into two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ . Consider in $$\varOmega $$ an eigenvalue–transmission problem associated with the p-Laplacian acting in $$\varOmega _1$$ and the q-Laplacian acting in $$\varOmega _2$$ , $$1<p<q$$ , with Dirichlet–Neumann conditions on the interface separating the two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ [see (1.1)]. The main result Theorem 2.1 states the existence of a sequence of eigenvalues for this eigenvalue problem. The proof is based on the Lusternik–Schnirelmann principle. Using the method of Lagrange multipliers for constrained minimization problems, we show (see Theorem 2.2) that if $$2\le p<q$$ then there exists an eigenfunction in any set of the form $$\begin{aligned} \left\{ u\in W^{1,p}(\varOmega );\, u|_{\varOmega _2}\in W^{1,q}(\varOmega _2), \, \frac{1}{p}\int _{\varOmega _1}|u|^p + \frac{1}{q}\int _{\varOmega _2}|u|^q = \alpha \right\} , \quad \alpha >0. \end{aligned}$$ The case of Robin conditions on $$\partial \varOmega $$ and the Riemannian setting are also addressed.