Normalized solutions of nonautonomous Kirchhoff equations: sub- and super-critical cases
The authors study a nonautonomous Kirchhoff-type equation involving a non-constant potential \(K \in C(\mathbb{R}^3,\mathbb{R}^+)\) with some regularities. Precisely, they consider the equation \[(P_\lambda) \quad -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 dx\right)\Delta u-\lambda u=K(x)f(u), \, x \in \mathbb{R}^3, \, u \in H^1(\mathbb{R}^3).\] Here \(a,b>0\), \(\lambda \in \mathbb{R}\), \(0< \lim_{|y|\to +\infty}K(y)\leq \inf_{\mathbb{R}^3}K\) and \(f \in C(\mathbb{R},\mathbb{R})\) satisfies certain \(L^2\)-supercritical or \(L^2\)-subcritical assumptions. The authors look at the critical points of a suitable energy functional associated to \((P_\lambda)\), on the manifold \(S_c=\{u \in H^1(\mathbb{R}^3) : \|u\|^2_2=c \}\), that is, the set of solutions \(u \in H^1(\mathbb{R}^3)\) having a prescribed \(L^2\)-norm, as \(c>0\) is given. The parameter \(\lambda \in \mathbb{R}\) is unknown and appears as a Lagrange multiplier. Thus it is shown that each critical point \(u_c \in S_c\) of the energy functional corresponds to a Lagrange multiplier \(\lambda_c \in \mathbb{R}\) so that \((u_c,\lambda_c)\) solves \((P_\lambda)\).
scientific article; zbMATH DE number 7381020
scientific article