On a Steklov eigenvalue problem associated with the (p,q)-Laplacian
Luminiţa Barbu, Gheorghe Moroşanu · Carpathian Journal of Mathematics · 2021
"Consider in a bounded domain \Omega \subset \mathbb{R}^N, N\ge 2, with smooth boundary \partial \Omega, the following eigenvalue problem (1) \begin{eqnarray*} &~&\mathcal{A} u:=-\Delta_p u-\Delta_q u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega, onumber \\ &~&\big(\mid abla u\mid ^{p-2}+\mid abla u\mid ^{q-2}\big)\frac{\partial u}{\partial u}=\lambda b(x) \mid u\mid ^ {r-2}u ~ \mbox{ on} ~ \partial \Omega, onumber \end{eqnarray*} where 1 0. \] Under these assumptions we prove that the set of all eigenvalues of the above problem is the interval [0, \infty). Our result complements those previously obtained by Abreu, J. and Madeira, G., [Generalized eigenvalues of the (p, 2)-Laplacian under a parametric boundary condition, Proc. Edinburgh Math. Soc., 63 (2020), No. 1, 287–303], Barbu, L. and Moroşanu, G., [Full description of the eigenvalue set of the (p,q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, in press], Barbu, L. and Moroşanu, G., [Eigenvalues of the negative (p,q)– Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equations, 64 (2019), No. 4, 685–700], Fărcăşeanu, M., Mihăilescu, M. and Stancu-Dumitru, D., [On the set of eigen-values of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal. Theory Methods Appl., 116 (2015), 19–25], Mihăilescu, M., [An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10 (2011), 701–708], Mihăilescu, M. and Moroşanu, G., [Eigenvalues of -\triangle_p-\triangle_q under Neumann boundary condition, Canadian Math. Bull., 59 (2016), No. 3, 606–616]."