A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

Abstract

The main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as theOur present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Anisotropic Robin problems with logistic reaction

“…Let dom j = u ∈ L 1 (Ω) : j(u) < +∞ (the effective domain of j(•)). From Theorem 2.2 of Takáč and Giacomoni [30], we know that j(•) is convex.…”

Anisotropic equations with indefinite potential and competing nonlinearities

“…Since η > 0 is arbitrary, we infer that (26)φ λ (t n u + n ) → +∞ as n → ∞. We know that (27) (26) and (27) it follows that we can find n 2 ∈ N such that (28) t n ∈ (0, 1) for all n ≥ n 2 .…”

Anisotropic (p, q)-equations with gradient dependent reaction

“…Suppose that v n → v in C 1 0 (Ω). According to Proposition 10, we can find u n ∈ S + vn ⊆ int C + n ∈ N such that (44) u n → β(v) =ũ v in C 1 0 (Ω) as n → ∞.…”