Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Teaching Philosophy and Enactivism

"The paper presents a concise history of enactivism in education, especially in mathematics education. Cases described by Davis’s, Proulx and Simmt’s work showcase the idea that enactivism is a viable alternative to constructivism or to classical views both in terms of practical teaching and theoretical models related to the process of learning. The idea that the student should solve a fixed problem, discover the universally correct solution, and eventually store that correct solution to find many other universally correct solutions to other fixed problems reduces the student to a very simple mechanism aimed at informational efficiency. This problem is met by the enactivistic tradition that began with Varela and Maturana’s work, now updated to the aforementioned researchers. Contra the classical perspective, enactivism proposes the idea that the student collaboratively produces the problem, being able to see multiple solutions, and eventually becoming a performer of knowledge. The article takes these ideas developed in mathematics education and finds their use in philosophical education. The article especially focuses on the student’s problem of being unable to link a new philosophical text discussed in class with their intuition. The last part of the article offers a lesson design example. The philosophical design focuses on making the students explore their own thinking regarding the topic about to be discussed by using a philosophy text before introducing the text. Keywords: enactivism, phenomenology, philosophy of education, classroom design "