Generalized Concentration-Compactness Principles for Variable Exponent Lebesgue Spaces with Asymptotic Analysis of Low Energy Extremals

Abstract

In this paper, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the asymptotic behaviour of low energy extremals.

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

“…The first task is to determine the possible Γ-convergence limit, in this regard: for the generalized concentration/compactness principle ( [18], Theorem 3) plays a vital role, but during our work on establishing the Γ-convergence, we realize that there is a need to refine the generalized concentration/compactness principle and find more sharp bounds. Therefore, we prove a refined version (of [18], Theorem 3) before proving the main result. Theorem 3.…”

“…Note that the refinement is done in Inequalities ( 19) and ( 22) in comparison to Ref. [18], Theorem 3. Thus we only need to prove these inequalities.…”

“…Recently, Bashir et al [18] extended the work of Bonder and Silva [12] and proved generalized concentration/compactness principles for variable exponent Lebesgue spaces. In addition, they initiated the work on Problem (1) and proved the existence of low energy extremals.…”

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

“…We refer to some results in variable exponent Sobolev or Orlicz-Sobolev spaces [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”