A constrained minimization problem involving the p(x)-Laplacian in R-N

作者：Fan Xianling^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2008, 69(10): 3661-3670.

DOI：10.1016/j.na.2007.10.002

• p(x)-Laplacian
• Variable exponent Sobolev spaced
• Concentration-compactness principle
• Constrained minimization

We consider a constrained minimization problem for the functional I(u) = integral(RN)1/p(x) (vertical bar del u vertical bar(p(x)) + vertical bar u vertical bar(p(x)))dx on the space W-1.p(x)(R-N), restricted to the set M-t = {u is an element of W-1.p(x)(R-N) : integral(RN)Q(x)/q(x)v vertical bar u vertical bar(q(x)) dx = t} with t > 0, where 1 < p < q < p*. Under the assumptions that p, q and Q are periodic, or p and q are periodic and 0 < inf(RN) Q(x) = lim(vertical bar x vertical bar ->infinity) Q(x), we prove that for every t > 0, the problem has a nonnegative solution u(t), and lambda(u(t)) -> 0 as t -> +infinity, and lambda(u(t)) -> +infinity as t -> 0, where lambda(u(t)) is the Lagrangian multiplier associated with u(t).

• 出版日期2008-11-15
• 单位兰州大学