Twin positive solutions for Schrödinger-Kirchhoff-type problem with singularity and critical exponents
Summary: We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent \(-(a + b \int_{\Omega}|\nabla u|^2 dx)\Delta u + u = Q(x) u^5 + \mu|x|^{\alpha - 2} u + f(x)(\lambda / u^\gamma)\) in \(\Omega\), \(u = 0\) on \(\partial\Omega\), where \(a, b > 0\) are constants, \(\Omega \subset \mathbb R^3\) is a smooth bounded domain, \(0 < \alpha < 1\), \(\lambda > 0\) is a real parameter, \(\gamma \in(0,1)\) is a constant, and \(0 < \mu < a \mu_1\) (\(\mu_1\) is the first eigenvalue of \(-\Delta u = \mu|x|^{\alpha - 2} u\), under Dirichlet boundary condition). Under appropriate assumptions on \(Q\) and \(f\), we obtain two positive solutions via the variational and perturbation methods.
scientific article; zbMATH DE number 6868708