Volume 13, Issue 3

This paper is concerned with the following H’enon type problem where B 1 (0) is the unit ball in ℝ N centered at the origin, 1 < p, q and if N ≥ 3. We show that there exists α∗ > 0 such that the ground state solution of the problem is non-radial if α > α∗. We also consider the limiting behavior of the ground state solution (u, v) as p + q → 2∗. We prove that the maximum points of two components u and v concentrate at the same point on the boundary ∂B 1 (0).

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution for all λ ≠ 0 and sufficiently small ε > 0. Moreover, for each λ ≠ 0 we show that u ε converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of b(x) in certain sense as ε → 0. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

In this paper we complete our work started in [31], where the present paper was announced; in order to set a unified theory of the irrigation problem. The main result of the paper is the equivalence of the various formulations introduced so far as well as a new one introduced here. To this aim we introduce several geometric and analytical concepts which are essential for reaching our final goal even if they may deserve an intrinsic interest in themselves.

This paper is about prescribing the Webster scalar curvature on a compact CR manifold of dimension 2n+1 ≥ 5, which is locally CR equivalent to the standard CR unit sphere S 2n+1 . The associated variational problem being noncompact we approach this issue with techniques from the critical points at infinity theory, combined with topological tools. Our existence results are based on a generalized Bahri-Coron type criteria. We also give an upper bound on the Morse index of the obtained solutions.

We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of boundary curvature appears.

We give conditions on the variable exponent q that ensure the existence and nonexistence of a positive solution to the elliptic equation Δu = u q(x) on ℝ N (N ≥ 3) which satisfies lim |x|→∞ u(x) = ∞. The nonnegative function q is required to be locally Hölder continuous on ℝ N . We prove existence for q > 1 provided q(x) decays to unity rapidly as |x| → ∞. We treat the case q ≤ 1 as a special case of q − 1 changing signs and show that a solution exists provided q is asymptotically radial. In addition, we give an example to show that our results are nearly optimal.

In this paper we study the generalized Hénon equation in the unit ball, where the coefficient function may or may not change its sign. We prove that the least energy solution is not radial and moreover we show the existence of a group invariant positive solution without radial symmetry.

The aim of this paper is investigating the existence of solutions of the quasilinear elliptic Problem where Ω is an open bounded domain of R N with C 2 boundary ∂Ω, Δ p u = div(|∇u| p−2 ∇u), 1 < p < q < p∗, f ∈ C(Ω̅) and ϕ ∈ C 2 (Ω̅). By means of the so-called Bolle’s method in [3, 4], we extend a result in [10] where the authors consider Ω =]0, 1[ N and u = 0 on ∂Ω.

We consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

We study the following boundary value problem of semi-linear elliptic systems Δu + f(x, u, ∇u) =0 in Ω, u =0 on ∂Ω where Ω ⊂ ℝ n (n ≥ 2) is a connected and bounded smooth domain, and u and f are real vector-valued functions. Under appropriate conditions on the function f, we establish a general existence result for positive solutions. We continue our earlier works [15, 16, 17] and, in particular, remove a requirement of the cooperative structure on elliptic systems. As examples, we consider the stationary Klein-Gordon-Maxwell system and Maxwell- Schrodinger system, both lacking the cooperative structure..