On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category

.In this paper we study the existence of multiple normalized solutions to the class of elliptic problems \(\left\{\!\!\!\!\!\begin{array}{ll} &-\Delta u+V(\epsilon x)u=\lambda u+f(u) \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{array} \right.\) where \(a,\epsilon \gt 0\) , \(\lambda \in \mathbb{R}\) is an unknown parameter that appears as a Lagrange multiplier, \(V:\mathbb{R}^N \to [0,\infty )\) is a continuous function, and \(f\) is a differentiable function with \(L^2\) -subcritical growth. It is proved that the numbers of normalized solutions are related to the topology of the set where the potential \(V\) attains its minimum value. In the proof our main result, we apply minimization techniques and the Lusternik–Schnirelmann category.KeywordsLusternik–Schnirelmann categorynormalized solutionsmultiplicitynonlinear Schrödinger equationvariational methodsMSC codes35A1535J1035B0935B33