Journal of Convex Analysis 33 (2026), No. 1&2, 155--170
Copyright Heldermann Verlag 2026
The Lions Concentration-Compactness Principle for the Dirichlet Problem for Partial Differential Equations with Variable Exponent Laplace Operator
Mykola I. Yaremenko
National Technical University, Igor Sikorsky Polytechnic Institute, Kyiv, Ukraine
math.kiev@gmail.com
[Abstract-pdf]
We develop the P.\,Lions concentration-compactness principle for a sequence of Radon measures on $R^{n}$, the P.\,Lions principle is extended to variable exponent Lebesgue spaces $L^{p\left(\cdot \right)} \left(\Omega \right)$, $\Omega \subseteq R^{n}$, $n\ge 3$. Employing this $L^{p\left(\cdot \right)}$-extension of the concentration-compactness principle, we establish almost exact conditions under which the Dirichlet problem $\left. u\right|_{\partial \Omega } =0$ for variable exponent Laplace equation \[ -div\left(\left|\nabla u\right|^{p\left(x\right)-2} \nabla u\right) + \lambda \left|u\right|^{p\left(x\right)-2} u = a\left(x\right)\left|u\right|^{s\left(x\right)-2} u+f\left(x,\; u\right) \] has a weak solution in variable exponent Sobolev space $W_{1}^{p\left(\cdot \right)} \left(\Omega \right)$, with critically grown coefficients.
Keywords: Concentration-compactness, variable exponent Lebesgue spaces, variable exponent Sobolev space, mountain pass theorem, concentration-compactness principle, Radon measure, Levy distribution.
MSC: 49A22, 47H15, 35J65, 35J66, 49N99.
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