We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ ∈ R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ 2 = the second eigenvalue of (− p , W 1,p 0 (Z)), then the problem has at least three nontrivial solutions. Our approach combines the method of upper-lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p = 2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].

No abstract

No abstract

No abstract

In this paper we study second order non-linear periodic systems driven by the ordinary vector p-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the p-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.