An eigenvalue problem involving the \((p, q)\)-Laplacian with a parametric boundary condition. (English) Zbl 1518.35395

Barbu, Luminiţa; Burlacu, Andreea; Moroşanu, Gheorghe

Mediterr. J. Math. 20, No. 4, Paper No. 232, 18 p. (2023).

Summary: Let \(\Omega\subset\mathbb{R}^N\), \(N \geq 2\), be a bounded domain with smooth boundary \(\partial\Omega\). Consider the following nonlinear eigenvalue problem \[ \begin{cases} -\Delta_p u-\Delta_q u + \rho(x)|u|^{q-2}u = \lambda\alpha(x)|u|^{r-2}u \text{ in }\Omega ,\\ \frac{\partial u}{\partial\nu_{pq}} + \gamma(x)|u|^{q-2}u = \lambda\beta (x)|u|^{r-2}u \text{ on }\partial\Omega, \end{cases} \] where \(p, q, r\in(1, \infty)\) with \(p \neq q\); \(\alpha, \rho \in L^\infty(\Omega)\), \(\beta, \gamma \in L^\infty(\partial\Omega)\), \(\Delta_\theta u:= \operatorname{div}(\|\nabla u\|^{\theta - 2}\nabla u)\), \(\theta\in\{p, q\}\), 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.

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

\((p, q)\)-Laplacian; spectrum; eigenvalues; Krasnosel’skiĭ genus; Ljusternik-Schnirelmann theory

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