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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”.

MSC:

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

Citations:

Zbl 0738.73025
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References:

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