Abstract

We develop the P. Lions concentration-compactness principle for a sequence of Radon measures on RnR^{n}, the P. Lions principle is extended to variable exponent Lebesgue spaces Lp(⋅)(Ω)L^{p\left(\cdot \right)} \left(\Omega \right), Ω⊆Rn\Omega \subseteq R^{n}, n≥3n\ge 3. Employing this Lp(⋅)L^{p\left(\cdot \right)}-extension of the concentration-compactness principle, we establish almost exact conditions under which the Dirichlet problem u∣∂Ω=0\left. u\right|_{\partial \Omega } =0 for variable exponent Laplace equation −div(∣∇u∣p(x)−2∇u)+λ∣u∣p(x)−2u=a(x)∣u∣s(x)−2u+f(x,  u)-div\left(\left|\nabla u\right|^{p\left(x\right)-2} \nabla u\right) + \lambda \left|u\right|^{p\left(x\right)-2} u = a\left(x\right)\left|u\right|^{s\left(x\right)-2} u+f\left(x,\; u\right) has a weak solution in variable exponent Sobolev space W1p(⋅)(Ω)W_{1}^{p\left(\cdot \right)} \left(\Omega \right), with critically grown coefficients.

Author information

Contact details are reproduced from the original publication and may be historical.

Mykola I. Yaremenko

National Technical University, Igor Sikorsky Polytechnic Institute, Kyiv, Ukraine

math.kiev@gmail.com

Keywords

  • Concentration-compactness
  • variable exponent Lebesgue spaces
  • variable exponent Sobolev space
  • mountain pass theorem
  • concentration-compactness principle
  • Radon measure
  • Levy distribution

Mathematics Subject Classification

49A22, 47H15, 35J65, 35J66, 49N99

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Suggested citation

M. I. Yaremenko. “The Lions Concentration-Compactness Principle for the Dirichlet Problem for Partial Differential Equations with Variable Exponent Laplace Operator.” Journal of Convex Analysis 33 (2026), No. 1&2, 155–170. https://doi.org/10.68381/jca33010

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.