Existence and uniqueness for the p(x)‐Laplacian‐Dirichlet problems

Abstract

Two results on the existence and uniqueness for the p( u) in Ω, u = 0 on ∂Ω, are obtained. The first one deals with the case that f (x, u) is nonincreasing in u. The second one deals with the radial case in which f (r, u) is nondecreasing in u and satisfies the sub-p− − 1 growth condition.

Existence and multiplicity results for a newp(x)-Kirchhoff problem

“…In recent years, there has been a great deal of work done on problem (1.1), especially concerning the existence, multiplicity, uniqueness and regularity of solutions. Some important and interesting results can be found, for example, in [4,5,3,2,7,10,8,11,12,13,15,18,21,19,22,23,24,25,29,30,31,32,33,40] and references therein.…”

Quasilinear parabolic problem with p(x)-laplacian: existence, uniqueness of weak solutions and stabilization

“…Remark 2.11. [11] establish uniqueness results for quasilinear elliptic equations involving the p(x)-laplacian under different conditions on f (see Theorem 1.2 for instance). Theorem 2.10 is still valid in this case.…”

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

“…on Ω. There have been many studies about p(x)-Laplacian (see [3,5,7,8,13,16] and the references therein). Throughout the paper, we assume that w ∈ P + (Ω) and p ∈ C + (Ω); and for each q ∈ C + (Ω), set q − := min x∈Ω q(x), q + := max x∈Ω q(x), and let q ′ denote the conjugate function of q, i.e., 1 q(x) + 1 q ′ (x) = 1, ∀x ∈ Ω.…”