On the extreme value of the Nehari manifold method for a class of Schrödinger equations with indefinite weight functions (Q2187171)

In this paper, the authors consider the following nonlinear problem \[ \begin{cases} -\Delta_p u-\lambda h(x)|u|^{p-2}u=f(x)|u|^{\gamma-2}u\quad\text{in }\mathbb{R}^N,\\ u\in D^{1,p}(\mathbb{R}^N)\cap L^\gamma(\mathbb{R}^N), \end{cases} \tag{1} \] where \(p>1\), \(p<\gamma<p^*\), \(\lambda\) is a real parameter, \(h\in L^{\frac{N}{p}}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\), \(f\in L^\infty(\mathbb{R}^N)\), \(\Delta_p\) is the \(p\)-Laplacian operator and \(D^{1,p}(\mathbb{R}^N)\) is the closure of \(C_0^\infty(\mathbb{R}^N)\) with respect to the norm \(\|u\|_{D^{1,p}(\mathbb{R}^N)}=\int |\nabla u|^p\,dx\). Using an approach based on the Nehari manifold method, the authors establish the existence of two solutions of problem (1) when \(\lambda>\lambda^*\), where \(\lambda^*\) is the extreme value of the Nehari manifold method associated to problem (1).