Try:
v = x^2 + x^4;
pfun = ParametricNDSolveValue[{-u''[x] + v u[x] == \[Epsilon] u[x],
u[0] == 1, u'[0] == 0}, u, {x, -10, 10}, {\[Epsilon]}];
\[Epsilon]Val1 = \[Epsilon] /. FindRoot[pfun[\[Epsilon]][9], {\[Epsilon], 40},
Method -> "AffineCovariantNewton"]
(*40.6904*)
