Existence and some properties of solutions for degenerate elliptic equations with exponent variable

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

“…In this section, we only review the weighted variable exponent Lebesgue-Sobolev spaces L p(x) (w, Ω) and W 1,p(x) (w, Ω), which were studied in [9,12] and for the variable exponent Lebesgue-Sobolev spaces L p(x) (Ω) and W 1,p(x) (Ω), we refer to [5,13] and the references therein.…”

“…Proof. By modifying the proof of Lemma 3.1 in [9], we can easily obtain the differentiability of Φ and Ψ and their derivative formulas.…”

A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractionalp(⋅)-Laplacian

“…To prove Theorem 4.2 we employ the De Giorgi iteration argument used in [9] for which the following result is essential. , n = 0, 1, 2, · · · , (4.6)…”

“…As compared with elliptic equations involving the p(·)-Laplacian, the value of (−∆) s p(x) u(x) at any point x ∈ Ω relies not only on the values of u and p(·) on the whole Ω, but actually on the entire space R N . In this regard, more complicated analysis than the papers [5,9,22] has to be carefully carried out. To the best of the authors' knowledge, the present paper seems to be the first to study the regularity of weak solutions to the fractional p(·)-Laplacian problems.…”

Properties of eigenvalues and some regularities on fractionalp-Laplacian with singular weights

“…where f satisfies the W p -Carathéodory condition (see Definition 2.2 and condition (F) in Section 2). Under the W p -Carathéodory condition, we obtain a-priori bounds for solutions of (1.7) by the De Giorgi iteration argument, which was used in [18]. Moreover under an additional condition, we can obtain the continuity of solutions.…”