I want to prove that if the constant $ k>0$ is sufficiently large, then for every $ f\in L^2(0,1)$ , there exists a unigue $ u\in H^2(0,1) $ satisfying $ $ -u”+ku=f \; on \;(0,1) $ $ $ $ u'(0)=0,\;\; u'(1)=u(1)$ $

for this problem, i want to find appropriate Bilinear form B to satisfy the condition for Lax-Milgram thorem (Similar as Energy estimates so that there is such $ k$ ).

so for $ B[u,v]=\int -u”vdx=-u(1)v(1)+\int u’v’dx$ it cannot be bounded by $ \alpha ||u||||v||$ for some $ \alpha$ since the constant must depend on u,v

and also not sure about the coercivity.

Is there proper bilinear form to satisfy the condition for Lax-Milgram theorem to prove the existence of unique solution u?