Eigenvalues of the p,q,r-Laplacian with a parametric boundary condition

Barbu, Luminiţa   and  Moroşanu, Gheorghe

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

    \begin{equation*} \left\{\begin{array}{l} -\sum_{\alpha \in \{ p,q,r\}}\rho_{\alpha}\Delta_{\alpha}u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega,\\[1mm] \big(\sum_{\alpha \in \{p,q,r\}}\rho_{\alpha}\mid \nabla u\mid ^{\alpha-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^{r-2}u ~ \mbox{ on} ~ \partial \Omega, \end{array}\right. \end{equation*}

where p, q, r\in (1, +\infty),~q<p,~r\not\in \{p, q\}; \rho_p, \rho_q, \rho_r are positive constants; \Delta_{\alpha} is the usual \alpha-Laplacian, i.e., \Delta_\alpha u=\, \mbox{div} \, (|\nabla u|^{\alpha-2}\nabla u); \nu is the unit outward normal to \partial \Omega; a\in L^{\infty}(\Omega), b\in L^{\infty}(\partial\Omega) {are given nonnegative functions satisfying} \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. Such a triple-phase problem is motivated by some models arising in mathematical physics.

If r \not\in (q, p), we determine a positive number \lambda_r such that the set of eigenvalues of the above problem is precisely \{ 0\} \cup (\lambda_r, +\infty ). On the other hand, in the complementary case r \in (q, p) with r < q(N-1)/(N-q) if q<N, we prove that
there exist two positive constants \lambda_*<\lambda^* such that any \lambda\in \{0\}\cup [\lambda^*, \infty) is an eigenvalue of the above problem, while the set (-\infty, 0)\cup (0, \lambda_*) contains no eigenvalue \lambda of the problem.




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Barbu, Luminiţa, Moroşanu, Gheorghe