Flores-Bazán, Fabián On radial solutions to non-convex variational problems. (English) Zbl 0857.49009 Houston J. Math. 22, No. 1, 161-181 (1996). The paper provides some existence results for non-convex variational problems for integral functionals depending on the second derivatives of the state functions.One of the results proved concerns the existence of a radial symmetric nonnegative solution for the problem \[ \min\Biggl\{\int_\Omega g(|u(x)|)dx+\int_\Omega h(|\Delta u(x)|)dx: u\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)\Biggr\}, \] where \(\Omega\) is the unit ball of \(\mathbb{R}^n\), \((p-1)(n-1)\geq1\), \(g\) is continuous, nonincreasing, verifying \(g(u)\geq-\alpha_1 u^p-\beta_1\) for every \(u\geq0\) with \(\alpha_1\geq0\), \(h\) is continuous verifying \(h(z)\geq\alpha_2 z^p-\beta_2\) for every \(z\geq0\) with \(\alpha_1>0\) and it is assumed that \(\alpha_1|u|^p_{L^p(\Omega)}\leq \alpha_2|\Delta u|^p_{L^p(\Omega)}\) for every \(u\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)\backslash\{0\}\).Similar existence results for problems with homogeneous Neumann boundary conditions are also proved. Reviewer: R.De Arcangelis (Napoli) Cited in 1 Document MSC: 49J45 Methods involving semicontinuity and convergence; relaxation Keywords:non-convex variational problems; integral functionals; radial symmetric nonnegative solution PDFBibTeX XMLCite \textit{F. Flores-Bazán}, Houston J. Math. 22, No. 1, 161--181 (1996; Zbl 0857.49009)