This paper is focused on the search for low-amplitude solar gravity modes between 150 and 400 μHz, corresponding to low-degree, low-order modes. It presents results based on an original strategy that looks for multiplets instead of single peaks, taking into consideration our knowledge of the solar interior from acoustic modes. Five years of quasi-continuous measurements collected with the helioseismic GOLF experiment aboard the SOHO spacecraft are analyzed. We use different power spectrum estimators and calculate confidence levels for the most significant peaks. This approach allows us to look for signals with velocities down to 2 mm s-1, not far from the limit of existing instruments aboard SOHO, amplitudes that have never been investigated up to now. We apply the method to series of 1290 days, beginning in 1996 April, near the solar cycle minimum. An automatic detection algorithm lists those peaks and multiplets that have a probability of more than 90% of not being pure noise. The detected patterns are then followed in time, considering also series of 1768 and 2034 days, partly covering the solar cycle maximum. In the analyzed frequency range, the probability of detection of the multiplets does not increase with time as for very long lifetime modes. This is partly due to the observational conditions after 1998 October and the degradation of these observational conditions near the solar maximum, since these modes have a ``mixed'' character and probably behave as acoustic modes. Several structures retain our attention because of the presence of persistent peaks along the whole time span. These features may support the idea of an increase of the rotation in the inner core. There are good arguments for thinking that complementary observations up to the solar activity minimum in 2007 will be decisive for drawing conclusions on the presence or absence of gravity modes detected aboard the SOHO satellite.