Abstract. This paper concerns semilinear elliptic equations whose nonlinear term has the form W(x)f(u) where W changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part of W is contained in a condition which is shown to be necessary for homogeneous /. More general existence questions are also discussed.

We study entire solutions on R 2 of the elliptic system −∆U + ∇W (u) = 0 where W : R 2 → R 2 is a multiple-well potential. We seek solutions U (x 1 , x 2 ) which are "heteroclinic," in two senses: for each fixed x 2 ∈ R they connect (at x 1 = ±∞) a pair of constant global minima of W , and they connect a pair of distinct one dimensional stationary wave solutions when x 2 → ±∞. These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen-Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a "saddle" geometry, which describe the structure of solutions near a crossing point of smooth interfaces.

Let 0/R N , N 3, be a bounded open set with smooth boundary and * # R. We study the Dirichlet problem,with 1

We consider a two-dimensional model for a rotating Bose-Einstein condensate (BEC) in an anharmonic trap. The special shape of the trapping potential, negative in a central hole and positive in an annulus, favors an annular shape for the support of the wave function u. We study the minimizers of the energy in the Thomas-Fermi limit, where a small parameter tends to 0, for two different regimes of the rotational speed Ω. When Ω is independent of , we observe that the energy minimizers acquire vorticity beyond a critical Ω, but the vortices are strongly pinned in the central hole where the potential is negative. In this regime, minimizers exhibit no vortices in the annular bulk of the condensate. There is a critical rotational speed Ω = O(| ln |) for which this strong pinning effect breaks down and vortices begin to appear in the annular bulk. We derive an asymptotic formula for the critical Ω, and determine precisely the location of nucleation of the vortices at the critical value. These results are related to very recent experimental and numerical observations on BEC.

A thorough study of domain wall solutions in coupled Gross-Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that the domain wall solutions are energy minimizing within a class of vector-valued functions with nontrivial conditions at infinity. The admissible energy functionals include those corresponding to coupled Gross-Pitaevskii equations, arising in modeling of Bose-Einstein condensates. The results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall. The methods apply to many systems of interest and integrability is not germane to our analysis. Finally, sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential, thus giving rigorous justification to earlier results in the physics literature.