Normalized solutions for $ p $ -Laplacian equations concentrating at a potential well

In this paper, we study the existence and multiplicity of normalized concentrating solutions for a class of $ p $-Laplacian equations with weighted potential, focusing on the following problem:$ \begin{cases} -\varepsilon^p \mathrm{div}\left(a(x) \vert \nabla u \vert^{p - 2} \nabla u\right) = \lambda \vert u \vert^{p - 2} u + f(u)\; \; \; \mbox{in}\; \; \mathbb{R}^N, \\ \displaystyle {\int }_{\mathbb{R}^N} \vert u \vert^p \, dx = m^p \varepsilon^N, \quad u \in W^{1, p}(\mathbb{R}^N), \end{cases} $where $ m, \varepsilon > 0 $, $ \lambda \in \mathbb{R} $ is an unknown parameter that appears as a Lagrange multiplier, $ a: \mathbb{R}^N \to (0, +\infty) $ is a continuous potential such that $\min_{\partial \Lambda} a(x)> \min_{\Lambda} a(x)$ for some open bounded subset $ \Lambda\subset \mathbb{R}^N $, and $ f $ is a continuous function with $ L^p $-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential $ a $ attains its minimum value. In the proof of the main result, we apply minimization techniques, Ljusternik-Schnirelmann theory and the penalization method. Our studies improve and complement the studies [20]: G.M. Figueiredo and M.F. Furtado, On the number of positive solutions of a quasilinear elliptic problem, Indiana Univ. Math. J. (2006), to the context that we are concerned with the multiple normalized concentrating solutions. As far as we are aware, we did not find any works on the multiplicity of normalized concentrating solutions of $ p $-Laplacian Schrödinger equations with potential $ a(x)$ $\not\equiv 1 $.