Volume 10, Issue 4

We study here finite Morse index solutions of -∆u = f(u) on the entire space or half space and their application to smooth bounded domain problems when the growth of the non-linearity is faster than the usual Sobolev critical exponent.

We deal with existence of large solutions of ∆ p u = a(x)f(u)+b(x)g(u) in ℝ N . It is shown that if a, b, f, g are non-negative real valued functions with a, b ∈ C(ℝ N ), f, g ∈ C([0,∞)) and f + g ≥ h where h is a continuous, non-negative, non- decreasing function satisfying the Keller-Osserman condition then the equation above admits a large solution if the equation -∆ p v = a(x) + b(x) in ℝ N has a positive upper solution decaying to zero at infinity. No monotonicity condition is required from either f or g. Our proof is based on the method of lower and upper-solutions. We extend recent results by A. V. Lair and A. Mohammed.

In this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator -Δ + αI, where α is a positive constant.

We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

We consider semilinear periodic ordinary differential equations with the nonlinearity exhibiting double resonance at infinity in the spectral interval [λ k , λ k+1 ], k ≥ 1. Using minimax methods based on the critical point theory together with Morse theory, we show that the problem has at least four nontrivial solutions.

We are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a suficiently high order nonlinearity, the solution of the Boussinesq system exists for all time and tends to zero as the time tends to infinity.

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime ℳ. The proof is based on both variational and geometric arguments involving the causal structure of ℳ, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non-spacelike submanifolds can be handled.

We refine the analysis, initiated in [3], [4] of the blow up phenomenon for the following two dimensional uniformly elliptic Liouville type problem in divergence form: We provide a partial generalization of a result of Y.Y. Li [18] to the case A ≠ I. To this end, in the same spirit of [2], we obtain a sharp pointwise estimate for simple blow up sequences. Moreover, we prove that if {p 1 , ...,p N } is the blow up set corresponding to a given simple blow up sequence, then, (∆detA)(pj) = 0, ∀ j = 1, ...,N. This characterization of the blow up set yields an improvement of the a priori estimates already established in [3].

We investigate the minimizers of the energy functional under the constraint of the L 2 -norm. We show that for the case L 2 -norm is small, the minimizer is unique and for the case L 2 -norm is large, the minimizer concentrate at the maximum point of b and decays exponentially. By this result, we can show that if V and b are radially symmetric but b does not attain its maxi- mum at the origin, then the symmetry breaking occurs as the L 2 -norm increases. Further, we show that for the case b has several maximum points, the minimizer concentrates at a point which minimizes a function which is defined by b, V and the unique positive radial solution of -∆φ + φ - φ p = 0. For the case when V and b are radially symmetric, we show that if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once but after that it recovers to be symmetric as the L 2 -norm increases.

Motivated by some relevant physical applications, we study the existence and uniqueness of T-periodic solutions for a second order differential equation with a piecewise constant singularity which changes sign. Other questions like the stability and robustness of the periodic solution are considered.

By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.