On a question about non-uniqueness of global minima
Let X be a topological space, I and J be sequentially l.s.c. real functions on X such that J is non-negative with sequentially compact sub-levels, and I/J is not bounded from below outside the sublevel J−1([0,c[) of J for some c>0. Is it true that for any large enough λ and any increasing l.s.c. real function φ the function x → I(x)+λJ(x)+μφ◦J(x) has at least two global minima for some positive μ? We give a positive answer to this question assuming that I + λ J + μ ◦φ has sequentially compact sublevels for some λ and all μ > 0.
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