A concentration-compactness lemma with applications to singular eigenvalue problems
For any open set \(\Omega\subset {\mathbb R}^n\), the author studies the nonlinear eigenvalue problem \[ \begin{cases} -\text{div}(\nabla u|\nabla u|^{p-2})=\lambda V(x)|u|^{p-2}u, \\ u \in {\mathcal D}_0^{1,p}(\Omega), \end{cases}\tag{1} \] where \(V\in L_{\text{loc}}^1(\Omega)\), \(1<p<n\), and \({\mathcal D}_0^{1,p}(\Omega)\) is the completion of \({\mathcal D}(\Omega)\) for the norm \(\|u\|=(\int_{\Omega}|\nabla u|^p)^{1/p}\). The author uses the condition (\(H_1\)) Suppose \(V=V^+-V^-\) with \(V^+\not=0\), and there exists \(c>0\) such that \[ c\int_{\Omega}V^+|u|^p \leq \int_{\Omega}|\nabla u|^p. \] Let \[ \begin{aligned} S_V &= \inf \left\{\int_{\Omega}|\nabla u|^p, |u\in {\mathcal D}_0^{1,p}(\Omega), \int_{\Omega}V|u|^p=1\right\}, \\ S_{r,V} &= \inf \left\{\int_{\Omega}|\nabla u|^p, |u\in {\mathcal D}_0^{1,p}(\Omega\setminus B_r(0)), \int_{\Omega}V^+|u|^p=1\right\}, \\ S_{\infty,V} &= \sup_{r>0}S_{r,V} = \lim_{r\to 0}S_{r,V}, \\ S^x_{r,V} &= \inf \left\{\int_{\Omega}|\nabla u|^p, |u\in {\mathcal D}_0^{1,p}(\Omega)\cap B_r(x)), \int_{\Omega}V^+|u|^p=1\right\}, \\ S^x_V &= \sup_{r>0}S^x_{r,V} = \lim_{r\to 0}S^x_{r,V}, \\ S_{\ast,V} &= \inf_{x\in\overline{\Omega}}S^x_V, \end{aligned} \] and \[ \Sigma_V = \{x\in\overline{\Omega} |S^x_V<+\infty\}. \] Here \(B_r(x)\) is the ball of radius \(r\) centered at \(x\). The author further uses the assumption: (\(H_2\)) The closure of \(\Sigma_V\) is at most countable. The main results in this paper are: Theorem 1. Let \(V\in L^1_{\text{loc}}(\Omega)\) satisfying (\(H_1\)) and (\(H_2\)). Assume that \(S_V < S_{\infty,V}\) and \(S_V < S_{\ast,V}\). Then there exists a principal eigenvalue for problem (1). Theorem 2. Let \(V\in L^1_{loc}(\Omega)\) satisfying (\(H_1\)) and (\(H_2\)). Assume \(S_{\infty,V}=S_{\ast,V}=+\infty\). Then there exists an increasing sequence of eigenvalues \(\lambda_k\) for problem (1) with \(\lambda_k\to +\infty\). Proofs of these theorems depend on a concentration-compactness lemma related to \(V\)-dependent limiting problem.
scientific article; zbMATH DE number 1367854
scientific article