×

On radial solutions to non-convex variational problems. (English) Zbl 0857.49009

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.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
PDFBibTeX XMLCite