leqn=Laplacian[u[x,y],{x,y}]==0;

bc={u[x,0]==x^2 (1-x),u[x,2]==0,u[0,y]==0,u[1,y]==0};

sol=FullSimplify[u[x,y]/.DSolve[{leqn,bc},u[x,y],{x,y}][[1]]]

asol=sol/.∞->4//Activate

Plot3D[asol,{x,0,1},{y,0,2},PlotRange->All]

leqn=Laplacian[u[x,y],{x,y}]==0;
bc={u[x,0]==0,u[x,2]==0,u[0,y]==0,u[1,y]==y^2 (2-y)};
sol=FullSimplify[u[x,y]/.DSolve[{leqn,bc},u[x,y],{x,y}][[1]]]
asol=sol/.∞->4//Activate

Plot3D[asol,{x,0,1},{y,0,2},PlotRange->All]

leqn=Laplacian[u[x,y],{x,y}]==0;
bc={u[x,0]==x^2 (1-x),u[x,2]==0,u[0,y]==0,u[1,y]==y^2 (2-y)};
sol=FullSimplify[u[x,y]/.DSolve[{leqn,bc},u[x,y],{x,y}][[1]]]
asol=sol/.∞->4//Activate

Plot3D[asol,{x,0,2},{y,0,2},PlotRange->All]

leqn=Laplacian[u[x,y],{x,y}]==0;
bc={u[x,0]==x^2 (2-x),u[x,2]==0,u[0,y]==0,u[2,y]==y^2 (2-y)};
sol=FullSimplify[u[x,y]/.DSolve[{leqn,bc},u[x,y],{x,y}][[1]]]
asol=sol/.∞->4//Activate

Plot3D[asol,{x,0,2},{y,0,2},PlotRange->All]

Plot3D[asol,{x,0,2},{y,0,2},PlotStyle->Directive[RGBColor[0.96`,0.19`,1.`],Opacity[1.`]],PlotRange->All]

Plot3D[asol,{x,0,2},{y,0,2},PlotStyle->RGBColor[0.19`,1.`,0.91`],PlotRange->All]