Volume 5, Issue 1

The famous Lusternik-Schnirelmann theorem claims that on a surface of genus 0 there exist at least three simple closed geodesics without self-intersections. A variational proof of this theorem is given in the book “Riemannian Geometry (2ed Ed.)” of W. Klingenberg. In this paper, firstly we point out that the construction of the three non-trivial relative homology classes in the circle space on S 2 in this proof (the proof of Proposition 3.7.19 of [7]) is incorrect, and give explicit con- structions of these homology classes. Secondly, we construct a counter-example to show that the deformation constructed in this proof in the closure of non- self-intersecting geodesic polygons (on pp.343-344 of [7]) is discontinuous, and therefore this proof is not complete.

Let Ω be a bounded domain in ℝ N , N ≥ 2, λ > 0, q∊(0, N −1) and In this article we show the existence of at least two positive solutions for the following quasilinear elliptic problem with an exponential type nonlinearity: We use Monotonicity and Variational methods to obtain this multiplicity result.

Integrodifferential equations with non-symmetric kernels are considered. The existence of a non-negative solution is stated through an iterative scheme and a mountain-pass technique.

We consider the simple pendulum type equation with two-parameters μ, λ > 0 in a ball B R . For a given μ > 0, a solution triple (μ, λ(μ), u μ ) ∈ R + 2 × C 2 (B̅ R ) is obtained by a variational method, and u μ develops a boundary layer as μ → ∞. We establish the two-term asymptotic formulas for ∥u μ ∥ ∞ , u′ μ (R) 2 and the variational eigencurve λ(μ) as μ → ∞. It is shown that the coefficients of the second terms of ∥u μ ∥ ∞ , u′ μ (R) 2 and λ(μ) depend on 1/R, the mean curvature of ∂B R . Since the second term of the maximum norm of the solution for the corresponding one parameter problems does not depend on the mean curvature of ∂B R , the results obtained here seem to be new.

In this paper we consider n+1 competing species such that n of them are permanent and we study the behavior of the rest of the species. Our study complements a work initiated by S. Ahmad and A.C. Lazer concerning the extinction or persistence of the rest species.

In this paper, we show the existence and multiplicity of solutions for the following class of quasilinear equations −Δ p u + |u| p−2 u = f(u) in Ω λ , u(x) > 0 a.e in Ω λ , u = 0 on ∂ Ω λ where Ω λ = λΩ, Ω is a bounded domain in ℝ N , λ is a positive parameter, 2 ≤ p < N, f is a continuous functions, and Δ p u = div(|▽u| p−2 ▽u) is the p-Laplacian operator. Here, we use variational method to get multiplicity of solution involving the Lusternick- Schnirelman category of B̅ ∥∂ Ω̅ in itself.

We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p-Laplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations.