An Eigenvalue Problem Involving the (p, q)-Laplacian With a Parametric Boundary Condition

Abstract

Let $\Omega \subset \mathbb {R}^N$, $N\ge 2$, be a bounded domain with smooth boundary $\partial \Omega$. Consider the following nonlinear eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u-\Delta _q u+\rho (x) \mid u\mid ^{q-2}u=\lambda \alpha (x) \mid u\mid ^{r-2}u\ \ \text{ in } ~ \Omega ,\\ \frac{\partial u}{\partial \nu _{pq}}+\gamma (x)\mid u\mid ^{q-2}u=\lambda \beta (x) \mid u\mid ^{r-2}u ~ \text{ on } ~ \partial \Omega , \end{array}\right. \end{aligned}$$

where $p,q,r\in (1,\infty )$ with $p\ne q;$ $\alpha , \rho \in L^{\infty }(\Omega )$, $\beta , \gamma \in L^{\infty }(\partial \Omega )$, $\Delta _{\theta }u:= \text{ div }~ (\Vert \nabla u\Vert ^{\theta -2}\nabla u)$, $\theta \in \{p,q\}$, and $\frac{\partial u}{\partial \nu _{pq}}$ denotes the conormal derivative corresponding to the differential operator $-\Delta _p -\Delta _q$. Under suitable assumptions, we provide the full description of the spectrum of the above problem in eight cases out of ten, and for the other two complementary cases, we obtain subsets of the corresponding spectra. Notice that when some of the potentials $\alpha , \beta , \rho , \gamma$ are null functions, the above eigenvalue problem reduces to Neumann-, Robin- or Steklov-type problems, and so we obtain the spectra of these particular eigenvalue problems.