I ahev some problem with Laplace equation: I found a function $ u \in W_0^{1,p}( \Omega) $ where $ p < 2 $ and $ \Omega $ is an open set of $ \mathbb{R}^2 $ such that $ \Delta u = 0 $ on $ \Omega. $ Can we deduce that $ u = 0? $ The problem is that the function $ u $ is not necessarily in $ H_0^1( \Omega). $ It is in $ L^2( \Omega) $ but its gradient is not in $ L^2( \Omega). $
I tried with variational formulation but I'm blocked with the fact that $ p < 2. $ Any help please.