Volume 15, Issue 1

In this paper we study the ground state solutions for a nonlinear elliptic system of three equations which comes from models in Bose-Einstein condensates. Comparing with existing works in the literature which have been on purely attractive or purely repulsive cases, our investigation focuses on the effect of mixed interaction of attractive and repulsive couplings. We establish the existence of least energy positive solutions and study asymptotic profile of the ground state solutions, giving indication of co-existence of synchronization and segregation. In particular we show symmetry breaking for the ground state solutions.

We study the following class of nonhomogeneous Schrödinger equations −Δu + V(|x|)u = Q(|x|) f(u) + h(x) in ℝ 2 , where V and Q are unbounded or decaying radial potentials, the nonlinearity f (s) has exponential critical growth and the nonhomogeneous term h belongs to the dual of an appropriate functional space. By combining minimax methods and a version of the Trudinger-Moser inequality, we establish the existence and multiplicity of weak solutions for this class of equations.

In this paper, we consider the existence of solutions to spinorial Yamabe type equations on S 3 with a prescribed function H. We give a new condition about H under which at least one non-trivial solution exists. It is proved by a method based on Conley index theory applied to a reduced functional defined on a certain finite dimensional non-compact manifold. Our proof is also applicable to similar equations on S m for all m ≥ 2.

This paper studies two related problems. A first one, where Δ p u = div(|∇u| p−2 ∇u) stands for the p-Laplacian operator, Ω is a ball in ℝ N , ν stands for the outer unit normal and λ > 0 is a parameter. Exponents are supposed to satisfy 1 < q < p < r ≤ p*, p* = N p/(N − p) if 1 < p < N, p* = ∞ otherwise. The existence of Λ > 0 is shown so that (P) does not admit positive solutions if λ > Λ, a minimal positive solution exists when 0 < λ ≤ Λ and most importantly, a further second positive solution arises if 0 < λ < Λ. Hence, extensions of results in [2], [3], [12] and [13] in the framework of (P) are provided. The second problem considered is the variant of (P): where Ω is a ball and 1 < q < p < r. Features described above are shown to be also exhibited by (Q) and more importantly, it is proved that minimal solutions to (Q) develop flat patterns in the degenerate regime p > 2. Finally, it should be stressed that some of the properties satisfied by (P) and (Q) hold true when Ω is a general smooth domain.

The p-Laplace equation ∇ · (|∇u| p−2 ∇u) = 0, where p > 2, in a bounded domain Ω ⊂ ℝ 2 , with inhomogeneous Dirichlet conditions on the smooth boundary ∂Ω is considered. In addition, there is a finite collection of curves Γ = Γ 1 ∪ ... ∪ Γ m ⊂ Ω, on which we assume homogeneous Dirichlet conditions u = 0, modeling a multiple crack formation, focusing at the origin 0 ∈ Ω. This makes the above quasilinear elliptic problem overdetermined. Possible types of the behaviour of solution u(x, y) at the tip 0 of such admissible multiple cracks, being a “singularity” point, are described, on the basis of blow-up scaling techniques and a “nonlinear eigenvalue problem” via spectral theory of pencils of non self-adjoint operators. Specially interesting is the application of those techniques to non-linear problems as the one considered here. To do so we introduce a very novel change of variable compared with the classical one introduced by Kondratiev for the analysis of non-smooth domains, such as domains with corner points, edges, etc, studying the behaviour of the solutions at those problematic points. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of nonlinear eigenfunctions, which are obtained via branching from harmonic polynomials that occur for p = 2. Using a combination of analytic and numerical methods, saddle-node bifurcations in p are shown to occur for those nonlinear eigenvalues/ eigenfunctions.

In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝ n + : where for i = 1, 2. First, we show that, under some mild growth conditions, (0.1) is equivalent to the IE system where is the Green’s function in ℝ n + with the same Dirichlet boundary conditions. Then, inspired by the work [12] of Y. Fang and W. Chen on the Dirichlet problem for (−Δ) m u = up in ℝ n + , we use method of moving planes in integral forms to prove the nonexistence of nontrivial nonnegative solutions for IE system (0.2), and as a consequence, we derive the nonexistence of nontrivial nonnegative classical solutions for problem (0.1).

In this paper, we investigate a Kirchhoff type elliptic problem, where Ω ⊂ ℝ 3 is an open ball, λ ∈ ℝ and b ≥ 0. We give an extension of the result by Brezis-Nirenberg in 1983 to the case b > 0. In particular, we can observe several effects of the nonlocal coefficient on the well known results related to the existence, nonexistence and uniqueness.

In this work we establish existence and multiplicity of solutions for resonant-superlinear elliptic problems using appropriate variational methods. The nonlinearity is resonant at −∞ and superlinear at +∞ and the resonance phenomena occurs precisely in the first eigenvalue of the corresponding linear problem. Our main theorems are stated without the well known Ambrosetti-Rabinowitz condition.

We consider the prescribed mean curvature problem of spacelike graphs in Robertson- Walker spacetimes of flat fiber with homogeneous Dirichlet conditions on an Euclidean ball. Under reasonable assumptions, it is shown that every possible solution must be radially symmetric. Besides, an existence result for a singular nonlinear equation is proved by making use of the classical Schauder fixed point Theorem. The results are applied to realistic examples of Robertson-Walker spacetimes.

We find a solution of the Dirichlet problem for the prescribed mean curvature equation in Ω with u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ N , N ≥ 1 and f : Ω × [0,∞) → ℝ is an unbounded continuous function with oscillatory behavior near the origin.

We consider the nonlinear Schrödinger equation −Δu + V(x)u = f (x, u), x ∈ ℝ N , where V is invariant under an orthogonal involution and converges to a positive constant as |x| → +∞ and the nonlinearity f is asymptotically linear at infinity. Existence of a positive solution, as well as some results on the existence of a class of sign-changing solutions is presented. The new idea is to use the projection on the Pohozaev manifold associated with the problem at infinity in order to show that a min-max energy level is in the appropriate range for compactness.

We study the structure of approximate solutions of autonomous variational problems on large finite intervals. In our previous research we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. In the present paper we study the convergence of approximate solutions to solutions of the corresponding infinite horizon problems, in regions close to the endpoints of the time intervals.

In this paper, Levinson’s problem is introduced to affine-periodic systems. It is proved that every affine-dissipative-repulsive system admits an affine-periodic solution, which extends previous well-known results for dissipative systems.