Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition

The fractional p(.,.)-Neumann boundary conditions for the nonlocal p(.,.)-Laplacian operator

“…Proposition 1 . (See [14]) If (u n ), u ∈ L q(x) (Ω) and q + < ∞, then the following relations hold true:…”

EXISTENCE OF WEAK SOLUTIONS FOR $p(x)$-KIRCHHOFF-TYPE EQUATION

“…By using the conditions (A2) and (A1) ,the embedding from X to L r(x) m(x) (Ω) is compact (see Theorem 2.8 [19]). Then, we can write…”

Existence and nonexistence of positive solutions of some elliptic problems with variable exponents

“…Proposition 2.1 (Mashiyev et al 31 ). For all u ∈ L p(x) (Ω) and all sequence (u k ) in L p(x) (Ω), we have the following results:…”