Volume 12, Issue 2

In this paper we prove existence and summability results for solutions of equations whose model is with a > 0 and f in some Marcinkiewicz space, thus extending the results of [4]. We the use these results to prove similar theorems for equations whose model is with 0 < θ < 1 and p in some Marcinkiewicz space, thus extending the results of [3].

We study the relationship between the first eigenvalue of the p-Laplacian (p > 1) and Cheeger’s isoperimetric constant for families of compact manifolds and graphs with the Cheeger constant converging to zero.

In this paper we prove some comparison results for Neumann elliptic problems whose model involves the anisotropic Laplacian Δ H u = div (H(Du)H ξ (Du)), where H is a positively homogenous convex function. Finally, we find a Poincaré inequality in the anisotropic setting.

In this paper, we show that the following system −Δu + λV(x)u = g(x, v), −Δv + λV(x)v = f (x, u), x ∈ ℝ N (0.1) possesses at least one non-trivial solution pair (u, v) for λ > 0 large enough, where f (x, t), g(x, t) are continuous functions on ℝ N × ℝ and super-linear at t = 0 as well as at t = +∞, V(x) is allowed to be sign-changing. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition.

We are concerned with the asymptotic behavior of positive radial solutions for a class of quasilinear elliptic equation arising from plasma physics. By the variational argument and dual approach, we show the asymptotic uniqueness and non-degeneracy of the ground state.

We consider the stationary semilinear Schrödinger equation −Δu + a(x)u = f (x, u), u ∈ H 1 (ℝ N ), where a and f are continuous functions converging to some limits a ∞ > 0 and f ∞ = f ∞ (u) as |x| → ∞. In the indefinite setting where the Schrödinger operator −Δ + a has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on a and f . We stress that for some of our results we also allow zero to lie in the spectrum of −Δ+a.

Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type u t + Δ 2 u = γu ± Δ(|u| p−1 u) in Ω × ℝ + , (0.1) are considered in a smooth bounded domain Ω ⊂ ℝ N with Navier-type boundary conditions on ∂Ω, or Ω = ℝ N , where p > 1 and γ are given real parameters. The sign “+” in the “diffusion term” on the right-hand side means the stable case, while “−” reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for γ = 0, u t + Δ 2 u = ±Δ(|u| p−1 u) in ℝ N × R + . (0.2) The following three main problems are studied: (i) for the unstable model (0.1), with the −Δ(|u| p−1 u), existence and multiplicity of classic steady states in Ω ⊂ ℝ N and their global behaviour for large γ > 0; (ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝ N × R + for bounded initial data u 0 (x) in the subcritical case ; and (iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.

This is a continuation of our previous paper Chang, Wang, Zhang [6]. We investigate the multiple non semi-trivial solutions for nonlinear elliptic systems by two kinds of index theory. In particular the pseudo index theory for the Z 2 × Z 2 index theory is developed

This work is devoted to the Dirichlet problem for the equation −Δu = λu + |x|α|u| 2*−2 u in the unit ball of ℝ N . We assume that λ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided α is small enough. This solution has a variational characterization as a ground state.

By the use of a generalized version of the Poincaré-Birkhoff fixed point theorem, we prove the existence of at least two periodic solutions for a class of Hamiltonian systems in the plane, having in mind the forced pendulum equation as a particular case. Our approach is closely related to the one used by Franks in [15], but the proof remains at a more elementary level.

In this paper we study the persistence properties of decay for the solutions to the two component b-family system. Using the method of characteristics, we establish that certain decay properties of the initial data persist as long as the solution exists. We also examine the propagation behavior of compactly supported solutions. We show that solutions have an infinite speed of propagation, that is, a non-trivial strong solution with compact initial value can not be compactly supported at any later time.