Volume 12, Issue 1

We establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝ N . We show, among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any C 2 (ℝ N ) positive entire semi-stable solution of the following Lane-Emden system, is necessarily constant, whenever the dimension N < 8 + 3α + , provided p = 1, q ≥ 2 and f (x) = (1 + |x| 2 ). The same also holds for p = q ≥ 2 provided We also consider the case of bounded domains Ω ⊂ ℝ N , where we extend results of Brown et al. [1] and Tertikas [18] about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.

In this paper we deal with the following quasilinear elliptic problems where Ω ⊂ ℝ N is a bounded regular domain, d(x) = dist(x, ∂Ω), p > 1, λ > 0 and 0 < ν ≤ p. Moreover, g, q and f are nonnegative functions verifying suitable hypotheses. The main goal of this work is to analyze the interaction between the functions d, g and the gradient term to get existence and nonexistence results. The singular behavior of g near 0 will provide a new phenomenon that does not appear in the case where the absorption term g(u) q(d(x)) is dropped or substituted by a regular one.

Various properties of radial solutions of the supercritical elliptic equation with Hardy Potential are studied, where Ω = int{x ∈ ℝ N | α ≤ |x| < b}, which is a ball if 0 = α < b < +∞, an annulus if 0 < α < b < +∞, an exterior domain if 0 < α < b = +∞, and the whole space ℝ N if α = 0, b = +∞. We assume p is supercritical, that is, p > 2∗ with 2∗ = being the critical Sobolev exponent, and N ≥ 3.

The aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

In this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

We investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.

We consider the Liouville equation −Δu = λeu in Ω, u =0 on∂Ω, on a smooth bounded domain Ω in ℝ 2 , where λ > 0 is a parameter. Let {u n } be an m-point blowing up solution sequence of the problem for λ = λn ↓ 0, which satisfies for m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of u n for n sufficiently large. As a corollary, we show that if a solution u n of Morse index one has the property that , then the number of blow up points of the sequence is exactly one. Note that in the last result, we do not need any geometrical assumption such as the convexity of the domain.

We address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion. The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

This paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (P ε ) : −Δu = K|u| 4/(n−2)+ε u in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ n , n ≥ 3, K is a C 3 positive function and ε is a positive real parameter. We show first that in dimension 3, for ε small, (P ε ) has no sign-changing solutions with low energy which blow up at two points. For n ≥ 4, we prove that there are no sign-changing solutions which blow up at two nearby points. We also show that (P ε ) has no bubble-tower sign-changing solutions.

We show the variational structure of a multiplicity result of positive solutions u ∈ H 1 (ℝ N ) to the equation −Δu + a(x)u = u p , where N ≥ 2, p > 1 with p < 2∗ − 1 = and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques.