Eigenvalues for anisotropic p−Laplacian under a Steklov-like boundary condition
DOI:
https://doi.org/10.24193/subbmath.2021.1.07Keywords:
Eigenvalues, anisotropic p−Laplacian, Steklov-like boundary condi- tion, Sobolev spaces, variational methods.Abstract
The eigenvalue problem −div (1p∇ξ(Fp(∇u))=λa(x)∣u∣q−2u, with $q\in (1, \infty),~ p\in \Big(\frac{Nq}{N+q-1}, \infty\Big),~ p\neq q,$ subject to Steklov-like boundary condition, Fp−1(∇u)∇ξF(∇u)⋅ν=λb(x)∣u∣q−2u is investigated on a bounded Lipschitz domain $\Omega\subset \mathbb{R}^ N,~N\geq 2$. Here, $F$ stands for a $C^2(\mathbb{R}^N\setminus \{0\})$ norm and $a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega)$ are given nonnegative functions satisfying ∫Ωa dx+∫∂Ωb dσ>0. Using appropriate variational methods, we are able to prove that the set of eigenvalues of this problem is the interval $[0, \infty)$.
Mathematics Subject Classification (2010): 35J60, 35J92, 35P30.
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