The Lions Concentration-Compactness Principle for the Dirichlet Problem for Partial Differential Equations with Variable Exponent Laplace Operator

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.