On minima of radially symmetric functionals of the gradient. (English) Zbl 0819.49013

The authors study the problems of existence, uniqueness and qualitative properties (symmetry) of the minima to the problem \[ \min_{u\in W_ 0^{1,1} (B)} \int_ B [g(| x|,\;|\nabla u(x)|)+ h(u(x)) ]dx, \] where \(B\) is the unit ball of \(\mathbb{R}^ n\) and the map \(v\to g(r,v)\) is lower semicontinuous but not necessarily convex. Such a problem was considered by R. Tahraoui [SIAM J. Math. Anal. 21, No. 1, 37-52 (1990; Zbl 0738.73025)].
From the text: “Our results present the following features: (a) no smoothness on \(g\) or \(h\) is required: \(g\) is either a normal integrand or a lower semicontinuous function; (b) the case \(h\equiv 0\) is allowed; in this case the assumption on \(g\) reduces, for the existence of solutions, to \(g\) being lower semicontinuous and growing at infinity, as is to be expected; for the uniqueness, in addition, to \(g^{**}\) being strictly increasing, as is shown to be expected; (c) the case \(h= au\) is allowed: for \(a\neq 0\) our theorems yield at once existence and uniqueness of solutions with no further assumptions on \(g\) besides lower semicontinuity and growth at infinity”.


49J45 Methods involving semicontinuity and convergence; relaxation
49J35 Existence of solutions for minimax problems


Zbl 0738.73025
Full Text: DOI


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