Xianling Fan

A Knobloch-type result for p(t)-Laplacian systems

We consider the periodic boundary value problem of ordinary differential systems with p(t)-Laplacian of the form |u′|p(t)−2u′′=f(t,u),u(0)−u(T)=u′(0)−u′(T)=0, where p∈C(R,R) is a T-periodic function and p(t)>1 for t∈R, f∈C(R×RN,RN) and f(t,u) is T-periodic with respect to t. We prove that, if there exists some r>0 such that 〈f(t,u),u〉⩾0 for t∈R and u∈RN with |u|=r, then the problem has at least one solution u satisfying |u(t)|⩽r for t∈R. This is a generalization of the results obtained by Knobloch and Mawhin under the case of p(t)≡2 and p(t)≡p∈(1,∞) respectively.

POSITIVE SOLUTIONS TO p ( x )-LAPLACIAN–DIRICHLET PROBLEMS WITH SIGN-CHANGING NON-LINEARITIES

Consider the p(x)-Laplacian-Dirichlet problem with sign-changing non-linearity of the form {-div(vertical bar del mu vertical bar(p(x)-2) del u) + m(x) vertical bar mu vertical bar(p(x)-2) u = lambda a(x) f(u) in Omega u = 0 on partial derivative Omega, where Omega subset of R(N) is a bounded domain, p is an element of C(0)((Omega) over bar) and (infx is an element of(Omega) over bar)p(x) > 1, m is an element of L(infinity)(Omega) is non-negative, f : R -> R is continuous and f (0) > 0, the coefficient a is an element of L(infinity)(Omega) is sign-changing in Omega. We give some sufficient conditions to assure the existence of a positive solution to the problem for sufficiently small lambda > 0. Our results extend the corresponding results established in the p-Laplacian case to the p(x)-Laplacian case.