The Keller-Osserman condition for quasilinear elliptic equations in Sobolev spaces with variable exponent

Liouville-Type Result for Quasilinear Elliptic Problems With Variable Exponent

“…The study of singular quasilinear elliptic equation, and specifically boundary blow-up problems, has attracted considerable attention starting with the original work of Bieberbach (1916) [6]. This approach was introduced in the last decades of the need to provide answers to important questions of the nonlinear problems with variable exponent for example see [18], [5].…”

“…For the existence and uniqueness of p(. )-solutions u ∈ W 1,p(x) (Ω) where 1 < p(x) < d for all x ∈ Ω, of the variational Dirichlet problem associated with the quasilinear elliptic equation (1) see [3], these p(. )-solutions are obtained by the p(.…”

“…In [15], [19] Keller and Osserman prove that a necessary and sufficient condition for the considered problem to have an entire solution is that B. In the case where p(x) = p is constant the validity of the Keller-Osserman property with variable exponent see [5].…”

Variable exponent Sobolev trace spaces and Dirichlet problem in axiomatic nonlinear potential theory

“…Note that the comparison principle given in [7] can be extended immediately to functions in C(∂Ω) in the following way: Now, letting ε −→ 0, we get L p(.) u = u.…”

Harnack inequality and continuity of solutions for quasilinear elliptic equations in Sobolev spaces with variable exponent

“…For the existence and uniqueness of solutions u ∈ W 1,p(x) (Ω) where 1 < p(x) < d for all x ∈ Ω, of the variational Dirichlet problem associated with the quasilinear elliptic equation (1) see [4], these solutions are obtained by the p(. )-obstacle problem.…”