# How to treat an equation of the form $-\Delta u=G\cdot \nabla u+f(u) ?$

There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear elliptic equation of the form $$-\Delta u=f(u)$$ in the Sobolev space $$H^1_0(\Omega)$$, where $$\Omega$$ is a non-empty open subset of $$\mathbb{R}^n$$, under suitable hypothesis on $$f:\mathbb{R}\to\mathbb{R}$$, thanks to the fact that we can see weak solutions of this problem as the stationary points of the functional: $$I:H^1_0(\Omega)\to\mathbb{R}, u\mapsto\frac{1}{2}\|u\|_{H^1_0}^2-\int_\Omega\int_0^{u(x)}f(s)\operatorname{d}s\operatorname{d}x.$$

If $$G:\Omega\rightarrow\mathbb{R}^n$$, how can we treat the equation: $$-\Delta u=G\cdot\nabla u+f(u),$$ or, more generally, if $$g:\mathbb{R}^n\to\mathbb{R}$$, the equation: $$-\Delta u=g\left(\nabla u\right)+f(u)?$$

If $$n=1$$, I saw in the $$G$$-case that we can transform the equation into another semilinear elliptic equation that hasn't the dissipative term $$u'$$, with the same trick used in Sturm-Liouville theory, and so we can bring back this problem into the realm of the previous variational problem. However, what about the $$g$$-case if $$n=1$$? What about the $$G$$-case if $$n\ge2$$? Can we say anything about the $$g$$-case if $$n\ge 2$$?