A constrained minimization problem involving the p(x)p(x)-Laplacian in RNRN ☆

We consider a constrained minimization problem for the functional View the MathML sourceI(u)=∫RN1p(x)(|∇u|p(x)+|u|p(x))dx on the space W1,p(x)(RN)W1,p(x)(RN), restricted to the set View the MathML sourceMt={u∈W1,p(x)(RN):∫RNQ(x)q(x)|u|q(x)dx=t} with t>0t>0, where 10t>0, the problem has a nonnegative solution utut, and λ(ut)→0λ(ut)→0 as t→+∞t→+∞, and λ(ut)→+∞λ(ut)→+∞ as t→0t→0, where λ(ut)λ(ut) is the Lagrangian multiplier associated with utut.