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On a nonlinear transmission eigenvalue problem with a Neumann-Robin boundary condition
  • Luminita Barbu,
  • Andreea Burlacu,
  • Gheorghe Morosanu
Luminita Barbu
Ovidius University of Constanta

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Andreea Burlacu
Ovidius University of Constanta
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Gheorghe Morosanu
Babes-Bolyai University
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Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N\geq 2,$ with smooth boundary $\Sigma$ and let $\Omega_1$ be a subdomain of $\Omega$ with smooth boundary $\Gamma,$ such that $\overline{\Omega}_1\subset \Omega$. Denote $\Omega_2 = \Omega \setminus \overline{\Omega}_1.$ Consider the transmission eigenvalue problem \begin{equation*} \left\{\begin{array}{l} -\Delta_p u_1+\gamma_1(x)\mid u_1\mid ^{r-2}u_1=\lambda \mid u_1\mid ^{p-2}u_1\ \ \mbox{in} ~ \Omega_1,\\[1mm] -\Delta_q u_2+\gamma_2(x)\mid u_2\mid ^{s-2}u_2=\lambda \mid u_2\mid ^{q-2}u_2\ \ \mbox{in} ~ \Omega_2,\\[1mm] u_1=u_2,~~\frac{\partial u_1}{\partial\nu_{p}}=\frac{\partial u_2}{\partial\nu_{q}} ~~ \mbox{on} ~ \Gamma,\\[1mm] \frac{\partial u_2}{\partial\nu_{q}}+\beta (x) \mid u_2\mid^{\zeta-2} u_2=0 ~~ \mbox{on} ~ \Sigma, \end{array}\right. \end{equation*} where $\lambda$ is a real parameter $p, q, r, s, \zeta \in (1, \infty)$ and $\gamma_i\in L^{\infty}(\Omega_i), ~i=1, 2, \beta\in L^{\infty}(\Sigma),$ $\beta\geq 0$ a.e. on $\Sigma.$ Under additional suitable assumptions on $p, q, r, s, \zeta$ we prove the existence of a sequence of eigenvalues $\big(\lambda_n\big)_n, \lambda_n\rightarrow \infty.$ The proof is based on the Lusternik-Schnirelmann theory on $C^1-$ manifolds.
03 Aug 2022Submitted to Mathematical Methods in the Applied Sciences
03 Aug 2022Assigned to Editor
03 Aug 2022Submission Checks Completed
02 Sep 2022Reviewer(s) Assigned
03 Jul 2023Review(s) Completed, Editorial Evaluation Pending
03 Jul 2023Editorial Decision: Accept