Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-Laplacian and its critical exponent

Abstract

Abstract. We give a sufficient condition for the compact embedding from Wwhere Ω is a bounded open set in R N . As an application, we find a nontrivial nonnegative weak solution of the nonlinear elliptic equationWe also consider the existence of a weak solution to the problem above even if the embedding is not compact.

On the Sobolev trace Theorem for variable exponent spaces in the critical range

“…(Ω) ֒→ L q(x) (Ω), this result was obtained in [18]. Following the same ideas we can prove a similar result for the trace immersion.…”

“…First, employing the same ideas as in [18] we obtain some restricted conditions on the exponents p and r guarantying that the immersion W 1,p(x) (Ω) ֒→ L r(x) (∂Ω) remains compact and so the existence of an extremal for T (p(·), r(·), Ω, Γ) holds true. As in the Sobolev immersion Theorem more general conditions for the existence of extremals are needed and these are the contents of our main results.…”

Existence of solution to a critical trace equation with variable exponent

“…In fact, in [18] the authors find conditions on the exponents p and r such that A T = ∅ but the immersion remains compact. This type of conditions were first discovered in [27] where the embedding W 1,p(x) 0…”

Existence of solution to a critical equation with variable exponent

“…In [17] the authors prove that if A is small and there exists a control on the rate of how q reaches the critical value p * , then the immersion W 1,p(·) 0 (U ) ֒→ L q(·) (U ) remains compact, and so the usual techniques can be applied. When the immersion fails to be compact they prove that if the subcriticality set U \ A contains a sufficiently large ball, then (1.1) with h = 0 has a nonnegative solution.…”