I am given with partial differential equation $$-(au_x)_x+cu= f \qquad u(0) = u(1) = 0$$ where $a \in H^1[0,1]$ sobolev space, $c, f \in L^2[0,1]$, $c \geq 0 $ a.e. and $a \geq p > 0$.
Now for $a$ as defined above we have an operator $$A(a): H^2\cap H^1_0 \to L^2$$ defined by $$A(a)\phi = -(a\phi_x)_x +c\phi$$ Now Clearly $A(a)$ is a linear operator. I have question that is it possible to write the following equation as this involves inverse of operator $A(a)$ $$A(a)u = -(au_x)_x+cu = f \implies u = A(a)^{-1}f$$ Is $A(a)$ invertible or how it is written?
Your problem is solved by Theorem 4.22 in these lecture notes. You should notice that $a \in L^\infty(0,1)$ by the Sobolev Embedding Theorem, and that the assumption $a \geq p>0$ implies the coercivity of the associates bilinear form.
Anyway, you could also minimize $$ I(u) = \frac{1}{2} \int_0^1 a(x) |u'(x)|^2 \, dx + \frac{c}{2} \int_0^1 |u(x)|^2 \, dx - \int_0^1 f(x)u(x)\, dx $$ on $H_0^1(0,1)$ and show that there exists one and only one minimizer $u$. Then you could prove that this minimizer solves the boundary value problem and belongs to $H^2(0,1)$.