Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's Equation.
Schrodinger's equation is homogeneous -- so if $\phi_1,\phi_2,\cdots,\phi_n$ are solutions, $c_1\phi_1 + c_2\phi_2 + \cdots +c_n\phi_n$ is a solution.
More importantly, if $\phi$ is a solution, $A\phi$ is a solution as well. If $A$ is the normalization constant, we see that both non-normalized and normalized versions are valid solutions of Schrodinger's equation, making it consistent.
MATHEMATICALLY, if a function $\phi$ is a solution of the SE: $$ H\phi=E\phi $$ then $c\phi$ (c is a constant) is also a solution of it due to its linearity.
PHYSICALLY, however, we cannot choose any mathematical solution of the above SE equation to be the physical solution. Remember that we are working with Physics, not Mathematics. In Quantum Mechanics, wave function is probability amplitude and wave function squared is probability density. It's clear that we can find a particle in whole space with a probability of 1; or,in other work, we will definitely find out the particle in whole space. So, we must choose the constant $c$ so that the integral of the wave function squared over whole space must be 1 (normalization condition). The normalized solutions of the SE equation are really physical solutions, or wave functions.