and V (t, x) is γ ’ subadditive with γ > 0 for a.e. t ∈ [0, T ]. All of these authors

studied only the existence of solutions. Here, we studied the problem under much

weaker assumptions and showed the existence of nonconstant solutions.

Little was done concerning nonconstant solutions of problem (17.1). For the

homogeneous case, Ben Naoum“Troestler“Willem [18] proved the existence of a non-

constant solution. For the case T = 2π, Theorem 17.5, with substantially stronger

hypotheses, was proved by Nirenberg (cf. Ekeland and Ghoussoub [60]). In place of

hypothesis 2, they assumed

3

|x| ¤ 1, t ∈ R, x ∈ Rn .

V (t, x) ¤ ,

2π 2

In place of hypotheses 3 and 4, they assumed the superquadraticity condition

V (t, x) > 0, Hμ (t, x) ¤ 0, |x| ≥ C, t ∈ R, x ∈ Rn ,

for some μ > 2, which implies these hypotheses and

V (t, x) ≥ C|x|μ ’ C , x ∈ Rn , C > 0,

among other things.

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