Volume 13, Issue 4

We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.

We study the Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = -g(x) - f(x)y, where g(x) and f(x) are arbitrary polynomials such that 2 ≤ deg g ≤ deg f.

In this paper, we study the Paneitz curvature problem Δ 2 u = (1 + εK(x))u p , x ∈ ℝ N , where . If K(x) has two critical points satisfying certain conditions, we prove the existence of two peak solutions for the above problem, provided ε is small enough.

This paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691] introduced a new kind of Lax-Oleinik type operator associated with any time-periodic Tonelli Lagrangian. Firstly, using the new operator we give an equivalent definition of the backward weak KAM solution. Then we prove a result on the asymptotic behavior of the new operators with an arbitrary continuous function as initial condition, by taking advantage of the definition mentioned above. Finally, for a specific class of time-periodic Tonelli Lagrangians, we discuss the rate of convergence of the new operators.

In this paper we use a modified test function method to derive nonexistence results for the semilinear wave equation with time-dependent speed and damping. The obtained critical exponent is the same exponent of some recent results on global existence of small data solutions.

In this paper, we consider the following system: where Ω ⊂ ℝ N (N ≥ 3) is a smooth bounded domain and α, β, λ, μ > 1 are parameters with . By using the perturbation method, we prove the existence of positive solutions for the problem (P). Indeed, the method we are using in this paper can be applied to the more general quasilinear system (see (1.7)).

We study the nonlinear elliptic problem -Δu = Fʹ (u) in ℝ n where F(u) is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist α ∈ ℝ n and C > 0 such that (*) |u - α · x| ≤ C. He also showed the existence of solutions with any prescribed α ∈ ℝ n . In this note, we first prove that an solution satisfying (*) with nonzero vector α must be one dimensional. Then we show that in ℝ 2 , for any positive integer d ≥ 1 there exists a solution with polynomial growth |x| d .

We study the global existence and regularity of weak solutions to strongly coupled parabolic systems whose diffusion matrices are almost triangular.

In this paper, some existence theorems are obtained for non-constant periodic solutions of second order Hamiltonian system involving the p-Laplacian by using the linking method. Our results generalize and improve several known results in the literature.

We consider the existence of infinitely many solutions of a class of quasilinear problems, for which the corresponding functionals are non-symmetric

The main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

In this work we consider the following Dirichlet problem corresponding to the anisotropic (p(·), q(·))−Laplacian operator with variable exponents and study the behaviour of the solutions as the exponents p(x, y), q(x, y) → ∞ uniformly in Ω ⊂ ℝ N+K . We denote by and similarly the gradients of u with respect to the first N variables (x variables) and with respect to the last K variables (y variables), respectively. We consider sequences of exponents p n (x, y), q n (x, y) going to infinity uniformly in Ω and verifying suitable assumptions. We prove that u n , the solutions with p(x, y) = p n (x, y) q(x, y) = qn(x, y), converge uniformly in Ω̅ to some nontrivial limit, u ∞ , as an → ∞. We determine the limit equation verified by u ∞ as well as some properties of this limit. Finally, we focus on the case f = 0 (with nontrivial Dirichlet boundary conditions, u = g ≢ 0 on ∂Ω) and show uniqueness for the limit problem.

Though there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ n (n ≥ 2). These include sharp constants for the Moser-Trudinger inequalities on both bounded and unbounded domains of the hyperbolic space ℍ n (see Theorems 1.1 and 1.2), sharp constants for the singular Moser-Trudinger inequality on unbounded domains when we impose restrictions only on the gradient norms (Theorem 1.3) or on the full hyperbolic Sobolev norms (Theorem 1.4). Our results are surprisingly general and extend most results in Euclidean spaces to hyperbolic spaces of any dimension. In particular, we have used a rearrangement-free argument in the hyperbolic spaces to establish Theorems 1.3 and 1.4 where symmetrization argument does not work to prove such sharp singular Moser-Trudinger inequalities on the entire hyperbolic space.