SOLUTIONS FOR DOUBLY RESONANT NONLINEAR NON-SMOOTH PERIODIC PROBLEMS

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

THREE SOLUTIONS FOR A QUASILINEAR TWO-POINT BOUNDARY-VALUE PROBLEM INVOLVING THE ONE-DIMENSIONAL p-LAPLACIAN

In this paper we prove the existence of at least three classical solutions for the problem$$ \left\{ \begin{aligned} \amp-(|u'|^{p-2}u')'=\lambda f(t,u)h(u'), \\ \ampu(a)=u(b)=0, \end{aligned} \right. $$when $\lambda$ lies in an explicitly determined open interval.Our main tool is a very recent three-critical-points theorem stated in a paper by D. Averna and G. Bonanno (Topolog. Meth. Nonlin. Analysis22 (2003), 93–103).AMS 2000 Mathematics subject classification: Primary 34B15