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

Zbl 0738.73025
Full Text:

### References:

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