We study the behaviour, on a lattice, of a scalar bidimensional theory with quartic and sextic interactions, the theory. We investigate its phase diagram identifying the sectors where first- and second-order phase transitions occur. We build the effective potential for this theory for different sectors of the phase diagram and verify that the effective potential can be undefined for certain values of the fields, both in the symmetric and broken symmetry sectors, when these sectors are separated by first order phase transitions. We also discuss the existence or not of objects like solitonic kinks and polarons and their dependence with the phase transition order.