We consider positive solutions to the singular semilinear elliptic equation −∆ u = 1 u γ + f (u), in bounded smooth domains, with zero Dirichlet boundary conditions. We provide some weak and strong maximum principles for the H 1 0 (Ω) part of the solution (the solution u does generally not belong to H 1 0 (Ω)), that allow to deduce symmetry and monotonicity properties of the solutions, via the Moving Plane Method.

We consider a jumping problem for singular semilinear elliptic equations. Existence of two distinct solutions is proved via a truncation argument and exploiting minimax methods to prove the existence of solutions to the truncated problem. In the final step, to pass to the limit, a refined analysis about the behavior of the solutions is required, because of the singular nature of the problem.

Several studies have showed the subsistence, even in students enrolled in scientific degree courses, of spontaneous ideas regarding the motion of bodies that conflict with Newton’s laws. One of the causes is related to the intuitive preconceptions that students have about the role of friction as a force. In fact, in real world novices do not recognise friction as a force, and for this reason they may believe that a motion with a constant speed needs as a necessary condition the presence of a constant force in the same direction of the motion. In order to face these ‘intuitive ways of reasoning’, in this paper we propose two sequential experiments that can allow undergraduate students to clarify the role of friction forces through the use of the work–energy theorem. This is a necessary first step on the way to a deeper understanding of Newton’s second law. We have planned our experiments in order to strongly reduce quantitative difficult calculations and to facilitate qualitative comprehension of observed phenomena. Moreover, the proposed activities represent two examples of the recurring methodology used in experimental practices, since they offer the possibility to measure very small physical quantities in an indirect way with a higher accuracy than the direct measurements of the same quantities.

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