Some remarks on an eigenvalue problem for an anisotropic elliptic equation with indefinite weight

Abstract

In this paper, we consider an eigenvalue problem for an anisotropic elliptic equation with indefinite weight, in which the differential operator involves partial derivatives with different variable exponents. Under some suitable conditions on the growth rates of the anisotropic coefficients involved in the problem, we prove some results on the existence and non-existence of a continuous family of eigenvalues by using variational methods.

Solutions for a quasilinear elliptic p⃗(x)${\vec{p}(x)}$‐Kirchhoff type problem with weight and nonlinear Robin boundary conditions

“…, with homogeneous Dirichlet boundary condition . It is well known (see [11,25] and [12], where intersting results on eigenvalue anisotropic elliptic problems with variable exponents have been given ) that 𝜆 + 1 and 𝜆 − 1 are defined by…”

On some $\overrightarrow{p(x)}$ anisotropic elliptic equations in unbounded domain

“…Lemma 5 Let Ω ⊂ IR N be an unbounded domain, suppose that the assumptions ( 8), ( 10)- (13), there exists at least one weak solution of the problem (P n ).…”

“…For the anisotropic operator with polynomial growth i.e. − → p = (p 1 , p 2 , ..., p N ) p i ∈ IR we mention the reference works of A. G. Korolev ([22]) and N. T. Chung et al ( [12]), for more works in the classical anisotropic spaces W 1, − → p (Ω) we refer the reader to ( [2], [7], [8], [10], [13], [15], [19]). Now for the operators governed by non-standard growth, namely, − → p (.)…”