In 2019, I posted an answer to a relevant question. See: Can a multivariate function only have local minimum?, and Can a smooth function with compact sublevel sets only admit local minimizers?

In [1], some examples are given.

The function $f(x, y) = (x^2-1)^2 + (x^2y-x-1)^2$ has exactly two stationary points $(-1, 0), \ (1, 2)$ which are both strictly local minima (also are both global minima). There is no another stationary points.

The function $f(x,y) = -\mathrm{e}^{-x} (x\mathrm{e}^{-x} + \cos y)$ has infinitely many strictly local minima. Thre is no another stationary points.

Reference

[1] Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy and Ina Westby, “Counting Critical Points of Real Polynomials in Two Variables,”, The American Mathematical Monthly, Vol. 100, No. 3 (Mar., 1993), pp. 255-271.

In 2019, I posted an answer to a relevant question. See: Can a multivariate function only have local minimum?, and Can a smooth function with compact sublevel sets only admit local minimizers?

In [1], some examples are given.

The function $f(x, y) = (x^2-1)^2 + (x^2y-x-1)^2$ has exactly two stationary points $(-1, 0), \ (1, 2)$ which are both strictly local minima (also are both global minima). There is no another stationary point.

The function $f(x,y) = -\mathrm{e}^{-x} (x\mathrm{e}^{-x} + \cos y)$ has infinitely many strictly local minima. There is no another stationary point.

Reference

[1] Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy and Ina Westby, “Counting Critical Points of Real Polynomials in Two Variables,”, The American Mathematical Monthly, Vol. 100, No. 3 (Mar., 1993), pp. 255-271.