Existence and multiplicity results for some eigenvalue problems involving variable exponents

On p(x)-Kirchhoff equations with concave-convex terms in unbounded domains

“…Tacking account that M is continue on [0, +∞), by assumptions (H 0 ) − (H 2 ) and standard arguments, we see that I λ ∈ C 1 W 1,p(x) (Ω), R (see for example, [20,21,22]. Moreover, the critical points of I λ are weak solutions of problem (P λ ).…”

Hölder continuity of quasiminimizers with nonstandard growth

“…Clearly By discussion in [3] we have that exponent P (x) = max{α(x) − 1, β(x) − 1, p(x) − 1} is bounded continuous and P − > 0. One immediately checks that the differential operator A satisfies (28) Proof of Theorem 3.…”

“…Example 2. In Aouaoui [3], the following eigenvalue problem is studied in the context of existence and multiplicity of solutions:…”

“…where p, α and β are variable exponents, and h, g and K are positive functions. Moreover, [3,Theorem 1.1] shows the existence of a solution for some λ > 0 provided that 2 < α(x) < p * (x) in the set of points x such that α(x) ≥ β(x). Let us find conditions implying that u is a quasiminimizer of an energy subject to Theorem 3 and, thus, u is Hölder continuous.…”