Band 15, Heft 4

In this paper, we consider the ground state solutions for a class of nonlinear Schrödinger equations where 2 < p < 2*. We investigate the case λ > inf σ (−Δ + V), i.e., an indefinite problem. We characterize the ground states as minimizers of the variational functional on a modified Nehari manifold.

In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝ n (n ≥ 3): There exists a constant C(n) > 0 such that for all with , , where . This extends the main result in [27] when n = 4 to all dimensions n ≥ 3. A crucial technical lemma we need is Lemma 4.2 for all p > 1 (corresponding to the Adams inequality for all n ≥ 3) whose proof is quite involved. As an application, we obtain the best constant for Ozawa’s inequality of Adams type in the Sobolev space : For any α < β n , there exists a constant C(α, n) > 0 such that for all satisfying , we have . Moreover, if α ≥ β n then the inequality cannot hold with a uniform constant C(α, n).

In this work, we study the existence and multiplicity of positive solutions for nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator , where k > 0 is a constant, λ > 0 is a parameter and f : [0,∞) → ℝ is continuous. We apply the quadrature arguments to prove how changes in the sign of f (u) lead to multiple positive solutions of the above problem for sufficiently large λ.

This paper studies the Hardy-type inequalities on the discrete intervals. Firstly, two variational formulas for the optimal constants are introduced. Based on these formulas, an approximating procedure and the known basic estimates of the optimal constants are deduced. Thirdly, as the main innovation of this paper, an improved factor for the upper estimates is presented, which is smaller than the known one and is the best possible. Finally, some comparison results are included for comparing the optimal constants on different intervals.

We study the existence and non-existence of nontrivial weak solution of , where N ≥ 5, and . Using Pohozaev type of identity, we prove the non-existence result when 1 < q < 2**. On the other hand when the equation has multiple critical nonlinearities i.e. q = 2** and −(N − 2) 2 ≤ μ < μ 1 , we establish the existence of nontrivial solution, using the Mountain-Pass theorem by Ambrosetti and Rabinowitz.

Consider in hyperbolic space ℍ 3 the one parameter family of immersed (non embedded) constant mean curvature surfaces of revolution Dπ with constant mean curvature H > 1. The parameter π ∈ (−∞, 0) is the analogue of the “necksize” of the Delaunay surfaces in Euclidean space. It is proved that when π → -∞, there exists a branch of surfaces with constant mean curvature H which bifurcate from D π . Furthermore, we prove that these new surfaces have only a discrete group of symmetries. The proof consists in a detailed study of the behaviour of the eigenvalues of the Jacobi operator when π tends to − ∞, together the bifurcation theorem of Crandall-Rabinowitz.

In this paper we give a new Adams type inequality for the Sobolev space W m,2 (ℝ 2m ). By using this inequality in combination with mini-max method we establish the existence of nontrivial solutions to the following class of polyharmonic equations (−Δ) m u(x) + u(x) = f (|x|, u), in ℝ 2m , where the nonlinearity is “superlinear” and has critical exponential growth at infinite.

In this article we study the initial trace problem for the nonnegative solutions of equation u t − Δu + |∇u| q = 0 in Q Ω,T = Ω ×(0, T) , where q > 0, and Ω = ℝ N , or Ω is a bounded domain of ℝ N and u = 0 on ∂Ω × (0, T) . We define the trace at t = 0 as a Borel measure (S, u 0 ), infinite on a closed set S, where u 0 is a Radon measure on Ω\S. We prove that the trace is a Radon measure when q ≦1. We study the existence for q ∈ (1, (N +2)/(N +1) and any given (S, u 0 ). When S = ω̅ Ⴖ Ω(ω ⊂ ΩΩ, ω open) existence is valid for q ≦ 2 when u 0 ∈ L loc 1 loc (Ω) , for q > 1 when u 0 = 0. In particular there exists a self-similar nonradial solution with trace (ℝ N+ , 0), with a growth rate of order |x| q/(q−1) as |x| → ∞ for fixed t. Moreover the solutions with trace (ω̅, 0) in , may present a growth rate of order t −1/(q−1) in ω and of order t −(2−q)/(q−1) on ∂ω.

We consider the semilinear elliptic problem where a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..

In this paper the existence of infinitely many solutions for nonlinear elliptic problems involving the p-Laplacian operator with mixed Dirichlet-Neumann boundary conditions is established. The key assumption is a suitable oscillating behavior, either at infinity or at zero, of the nonlinearity. The approach is based on the critical point theory.

We study the existence and non-existence of Liouvillian first integrals for the generalized Riccati polynomial differential systems of the form xʹ = y, yʹ = a(x)y 2 +b(x)y+c(x), where a(x), b(x) and c(x) are polynomials.

In this paper, the Maslov (P, ω)-index theory for a symplectic path is developed and the Bott-type iteration formula is proved.

In this paper, by using a perturbation method together with the method of invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions of quasilinear elliptic systems in ℝ N .