Test the student's ability to apply Sobolev embedding theorems and compactness results (e.g., Rellich–Kondrachov, Arzelà–Ascoli) to establish existence and regularity of weak solutions for nonlinear elliptic boundary value problems. Scope includes formulation of weak solutions in H^1_0 or W^{1,p}, deriving a priori energy estimates, using embeddings to obtain bounds, applying compactness to pass to limits in nonlinear terms, proving existence via variational methods, monotone operator theory or fixed-point theorems, and performing regularity bootstrap to obtain higher integrability or Hölder continuity under standard structure conditions.