If "solving the problems" means finding something that will replace cheap petroleum as a fuel source, then the first thing we need to acknowledge is that nothing short of nuclear fusion is ever going to do that for us. Once you recognize that it isn't about that; that we're in for severe reductions in energy consumption no matter what, then approaches that didn't look like they wanted to work -- because you were trying to get them to sustain a way of life that is just not sustainable -- suddenly start to look more promising. When your daily commute is maybe two or three miles, you don't have the same requirements that you had when it was maybe twenty or thirty.
I don't agree that this is absolutely necessary. I agree that there isn't a single technology short of nuclear fusion that will solve all energy needs; but we can maintain at least our present level, and possibly even a decent amount of future growth, with a combination of technologies, applied to their strengths.
Fuel technology: Hydrogen is not going to fly with current technology. Right now, it's a net cradle-to-grave negative energy economy. It costs more than it produces. Plus, despite what all the eco-woos claim, it's not even remotely "green". It's highly dependent on fossil fuels, being cracked from natural gas, and thus generates just as much carbon dioxide as using the fossil fuels directly. If a good bio-generation source can be developed, then it'll be worthwhile; but until then it's useless.
Ethanol is a decent fuel, and should have a net positive energy economy; but crops need high-quality land, displacing food crops, and are still dependent on fossil fuel-based fertilizers. Because of the land requirements, and the low yeild per acre/kilogram, it's unlikely to be viable in the long term. If cracking cellulose for methanol is ever made commercially viable, then it'll be a hugely useful fuel source; but until then, it's worse than useless.
Biodiesel is an almost ideal fuel, in the right implementation. In the current implementation, far less so. Being currently based on soy as a feedstock, it requires high-quality land and displaces food crops. Because of the nature of the crop, it's not all that dependent on fossil fuel fertilizers. Because of the land requirement and moderate to low yeild per acre/kilogram, it's energy economy is posivite, but not very high; so it's not likely to be more than marginal in the long term. There is a crop which could make it highly viable as a fuel source. Hemp has a much higher yield per acre/kilogram, orders or magnitude higher than soy or rapeseed. It also grows well in marginal land, so it doesn't displace food crops. Biodiesel production based on hemp would be a significant component of an overall replacement fuel source program.
Thermal Depolymer Process petroleum is a relatively new technology that also has the potential to be a significant component of a replacement fuel source porgram. The feedstock is any waste biological or plastic substance; neither of which are in short supply. There is an excellent positive energy economy, particularly as it's powered by waste natural gas ocurring as a by-product in non-commercially-viable quantities. The problem with TDP petrol production facilities is that they need to be localized close to the waste/feedstock stream; and has a signifigant negative community presence (mainly smell and noise); and therefore is highly affected by NIMBY concerns, which must be overcome in order to scale up to commercially significant production levels.
Grid technology:
Current technologies should be adequate to replace existing usage, as well as expected increases in demand over the short term.
Hydro, Geothermal, and Wind are useful for localized uses in areas where they're feasible; but are too limited overall.
Solar still has a negative overall cradle-to-grave energy economy due to costs of producing the solar energy collectors at current technology. And it's also very localized and space-intensive.
Coal reserves will last for quite some time yet, and with new advances in processing technology increasing the efficiency of energy extraction, it'll become more important as a component of grid-energy production. However, it cannot handle much increase, if any.
The big one for future demand increases is nuclear. Using fast-breeder reactor technology, nuclear power can easily handle grid power demands for quite some time. Unfortunately, there's a lot of resistance to building nuclear reactors.