Volume 11, Issue 2

We investigate symmetry properties of solutions to equations of the form -Δu = + f(|x|, u) in ℝ N for N ≥ 4, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best constants of Sobolev type inequalities with different symmetry constraints.

First, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

In this work we study multiplicity and concentration of nontrivial solutions for the following class of quasilinear problems: -div(a(|▽u| p )|▽u| p-2 ▽u) + V(εx)b(|u| p )|u| p-2 u=f(u), where ε is a positive parameter, a, b, f : ℝ → ℝ are C 1 functions and V : ℝ N →ℝ is a continuous function.

We establish compact embeddings of the radial Sobolev space H rad 1,p (ℝ N ) into weighted Lebesgue spaces L q w (ℝ N ) under various assumptions on the weight function w. We use these, along with a variety of variational techniques, to prove the existence of nontrivial nonnegative solutions for a class of nonlinear Schrödinger type equations.

In this paper, we study relationship between the stability and the growth of the number of closed characteristics on compact convex hypersurfaces in ℝ 2n , and also the stability of closed characteristics on compact convex hypersurfaces in ℝ 2n under some pinching conditions.

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

In this paper we are concerned with the nonlocal elliptic problem where Ω ⊂ℝ N is a bounded smooth domain, f, g : ℝ →ℝ are given functions and p is a fixed real number. We use variational methods to prove multiplicity results.

In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ 3 , when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in ℝ 3 . Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4,...,2(d + 1)}.

We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

This paper is devoted to studying the behavior as ε → 0 of the equations u ε + H(x, x/ε, Du ε , ε γ D 2 u ε ) = 0 with γ > 0. It is known that, under some periodicity and ellipticity or coercivity assumptions, the solution u ε converges to the solution u of an effective equation u + H̅(x, Du) = 0, with an effective Hamiltonian H̅ dependent on the value of γ. The main purpose of this paper is to estimate the rate of convergence of u ε to u. Moreover we discuss some examples and model problems.

For nonautonomous linear equations xʹ = A(t)x with a nonuniform exponential dichotomy, we show that under sufficiently small C 1 parameter-dependent perturbations, namely of the form xʹ = A(t)x + f (t, x, λ), the topological conjugacies in an appropriate version of the Grobman-Hartman theorem are of class C 1 in the parameter. The proof uses the fiber contraction principle.

The existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential function. First we establish an alternative minimax expression for the first nonzero eigenvalue for the negative periodic scalar p-Laplacian and then using it we prove the existence of three nontrivial solutions, two of which have constant sign. Our approach is variational, based on the nonsmooth critical point theory.

Solutions u of the equation -∇ ・ A(x, ∇u) = b(x)|∇u| q are studied in the subcritical case p - 1 ≤ q ≤ p/n + p - 1, q < p < n. Here A(x, h) satisfies A(x, h) ・ h ≈ |h| p and b a bounded function in an open set Ω ⊂ R n . It is shown that the solutions u are local quasiminimizers in Ω. From the quasiminimizing property several conclusions on the behavior of the solutions u can be derived. In particular, the method provides an easy solution to the uniqueness problem in the class W 0 1,p (Ω).