I am interested in the study of the (semi-linear, I suppose) equation
$$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\
u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$
on an open bounded domain $\Omega\subset\mathbb{R}^2$ with piecewise $C^1$ boundary. Here $k\in\mathbb{R}$ is a constant, and $f$ is a continuous non-linearity that is strictly greater than linear growth, i.e., $|f(x)|\geq |a|+b|x|^\gamma$ with $\gamma> 1$.



> 
> **Question.** What are known results about such equations? Is there a reference for them?
> 
> 
> 


It would be highly appreciated if the answer contains a brief discussion on the results in the literature if possible, although I'd be satisfied with just the references.


**Added.** I am mainly interested in existence results for the weak formulation of the equation. As for boundary conditions, we could take Dirichlet for simplicity but I would be interested in more generality.



Reference request for semilinear PDEs in dimension 2
You are in dimension $2$, which is nice, because $H^{1}(\Omega)\hookrightarrow L^N(\Omega)$ for any $N<\infty$ so you can look for a normal, weak solution. The other good news is that behind your problem is a compact operator.


Let us assume $q\in L^{\infty}\left(\Omega\right)$, $q\geq0$, $g\in H^{1/2}\left(\partial\Omega\right),$
$h\in L^{N}\left(\Omega\right)$, and
$$
\left|f\left(x\right)\right|\leq N+\left|x\right|^{N},
$$
for some $N>1.$ Then, the operator
\begin{align\*}
T\_q: H^{-1}(\Omega) \to &H^1\_0(\Omega) \\ 
\ell \to & u \text{ the solution of } \\
&-\Delta u + q u = \ell
\end{align\*}
is well defined and continuous, thanks to Lax--Milgram (this is where $q\geq0$ is useful) and Sobolev
embeddings (in dimension $2$, $L^N(\Omega)\hookrightarrow H^{-1}(\Omega)$).


Let $u\_{0}\in H^{1}\left(\Omega\right)$ be the weak
solution of
\begin{align\*}
-\Delta u\_{0}+qu\_{0}+h & =0\text{ in }\Omega,\\
u\_{0} & =g\text{ on }\partial\Omega.
\end{align\*}


Writing $\kappa:x\to kx$ your problem writes


\begin{align\*}
-\Delta\left(u-u\_{0}\right) & +q\left(u-u\_{0}\right)=f\left(u - \kappa\right)\text{ in }\Omega\\
u-u\_{0} & =0\text{ on }\partial\Omega.
\end{align\*}


In other words,  

\begin{align\*}
u -u\_0 &= T\_q f\left(u-u\_0+u\_0-\kappa\right)\\
u -u\_0 &= K(u-u\_0) \\
&\text{with } K:=a\to T\_q f(a +u\_0- \kappa)
\end{align\*}
Now $K: H^1\_0\to H^1\_0$ is a compact operator, as a composition of compact and continuous operators:


* The continuous map in $H^1(\Omega)$ $a\to a+ u\_0-\kappa$,
* The compact embedding
$
H^1\_0(\Omega)\hookrightarrow L^{2N}(\Omega),
$
* The continuous map $f:L^{2N}(\Omega)\to L^2(\Omega)$
* The compact embedding $L^{2}(\Omega)\hookrightarrow H^{-1}(\Omega)$
* The continuous map $T\_q: H^{-1}(\Omega) \to H^1\_0(\Omega)$.


So $u-u\_0$ is a fixed point of $K$...[Schauder's Fixed Point Theorem](https://en.wikipedia.org/wiki/Schauder_fixed-point_theorem) comes to mind. Alternatively, because you have positive operators and a maximum principle you can probably get away with a constructive method of iterating sub/super solutions as decribed in Evan's PDE book.


Lecture notes detailing both points of view are [here](https://courses.maths.ox.ac.uk/node/view_material/2037), and many references are included.


