We study the existence and nonexistence of positive (super) solutions to a semilinear elliptic equation −∆u − Ax |x| 2 • ∇u − B |x| 2 u = c |x| σ u p in cone-like domains of R N. On the plane R 2 we determine the set of (p, σ) such that the equation has no positive (super) solutions, depending on the parameters A, B ∈ R and the geometry of the domain.

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equationin exterior domains of R N (N 2). Here p ∈ (1, +∞) and μ C H , where C H is the critical Hardy constant. We provide a sharp characterization of the set of (q, σ ) ∈ R 2 such that the equation has no positive (super)solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the p-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the p-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prüfer transformation.

Education in the UK is failing to provide the increases in the numbers of school-leavers with science and mathematics qualifications required by industry, business and the research community to assure the UK's future economic competitiveness" (The Royal Society, 2008, p17). Furthermore, the proportion of students in Wales following mathematics courses post 16 is lower than in England (GSR, 2014). In particular, although the situation has improved, fewer students in Wales choose to study for the Further Mathematics (FM) A-level. This paper explores the reasons behind student choices in studying mathematics between the ages of 16 and 18, with a particular focus on the FM A-level, in order to make recommendations about how to increase participation in FM. Phase One of the study used a questionnaire to access the opinions of students studying mathematically based courses in sixth forms and colleges to explore the reasons behind their choices and the factors influencing their progression in mathematics. In Phase Two, small focus groups of students in selected schools and colleges were interviewed to enrich the questionnaire data and provide further insight into their decisions. The study identified a lack of information from peers, siblings, parents and teachers about FM as a factor restricting choice. Current models of delivery contribute to the false perception that the FM A-level is harder than the Mathematics A-level and only suitable for the most talented mathematicians. We suggest: developing teachers' knowledge and skills so that whenever possible students can be offered FM as a fully timetabled subject; promoting FM to parents; and establishing student champions to encourage participation.

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