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 $n\ge 2$?


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