Volume 9, Issue 3

We prove some maximum principle results for weak solutions of elliptic inequalities, possibly inhomogeneous, on complete Riemannian manifolds.

In this paper we consider nonlinear Dirichlet problems driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequalities). We assume that the problem is resonant at infinity with respect to λ1 > 0 (the principal eigenvalue of the Dirichlet p-Lapalcian) from the right. Using minimax methods based on the nonsmooth critical point theory we prove an existence and a multiplicity theorem.

We study the quasilinear Schrödinger equation iz t = −∆z +W(x)z − η(|z| 2 )z − k[∆p(|z| 2 )]p′(|z| 2 )z in ℝ 2 , where W : ℝ 2 →ℝ is a positive potential and the nonlinearity η : ℝ 2 × ℝ → ℝ has critical or sub-critical exponential growth. Quasilinear Schrödinger equations of this type have been studied as models of several physical phenomena such as superfluid film equation, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics and in condensed matter theory. In a suitable Orlicz space together with Trudinger-Moser inequality we establish an existence of standing wave solutions for this problem. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.

We take the example of the standard geodesics problem on S 2 to show how the point to circle Morse relations in the periodic orbits variational problem for contact vector-fields lead directly to the existence of critical points at infinity. This is sharply in contrast with the phenomena encountered in symplectic geometry where the variational problems are not necessarily S 1 -invariant and where the Hamiltonians are usually bounded. The related tools defined in this area do not exhibit therefore all the phenomena encountered in S 1 -invariant settings. We then recall our definition of Contact Form Homology which includes these essential Morse relations and we define related tools for its computation.

We study the evolution of spacelike hypersurface moving by mean curvature minus an external force field in Minkowski space. It is shown that the flow will exist for all time if the initial spacelike surface has two compact spacelike barrier surfaces and the external force field satisfies a weak convexity condition.

In this paper minimax methods are employed to establish the existence of a bounded positive solution for semilinear elliptic equation of the form −∆u + V (x)u = P(x)|u| p−1 u + Q(x)|u| q−1 u in ℝ N , where the nonlinearity has supercritical growth and the potential can change sign. The solutions of the problem above are obtained by proving a priori estimates for solutions of a suitable auxiliary problem.

This paper proves the existence of nontrivial solution for a class of quasilinear systems on bounded domains in ℝ N , N ≥ 2, whose nonlinearity has a double criticality. The proof is based on a linking theorem without the Palais-Smale condition.

Let (M, g) be a smooth compact Riemannian N−manifold, N ≥ 2. We show that if the scalar curvature of g is not constant, the problem −ε 2 ∆ g u + u = u p−1 in M has a positive solution with two positive peaks ξ 1 ε and ξ 2 ε , and a sign changing solution with one positive peak η 1 ε and one negative peak η 2 ε , such that as ε goes to zero Scal g (ξ 1 ε ), Scal g (η 1 ε ) → Scal g (ξ) and Scal g (ξ 2 ε ), Scal g (η 2 ε ) → Scal g (ξ). Here p > 2 if N = 2 and

We prove the existence of infinitely many radial solutions for elliptic systems in ℝ n with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest.