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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.

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    $\begingroup$ $x\in\mathbb{R}$ so that $h,q$ are functions of one variable only. I'll edit the question. $\endgroup$
    – UserA
    May 4 at 7:16
  • $\begingroup$ This covers so, so many equations, even when $h = 0 = q$ and $k = 0$. You'll get a much better answer if you explain what sort of results you're after. $\endgroup$
    – Leo Moos
    May 4 at 8:30
  • $\begingroup$ @LeoMoos Let's say for simplicity that $q\equiv 1$ and and $h$ is linear, and definitely $k\neq 0$. This is good enough for now, but I would like relax these assumptions to $h,q\in L^\infty$. $\endgroup$
    – UserA
    May 4 at 8:44
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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 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, and many references are included.

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  • $\begingroup$ Thank you for the details on Schauder fixed point theorem and the notes. Could you just outline the constructive method? If not can you at least pinpoint exactly where it is in the Evans book/ notes you linked? $\endgroup$
    – UserA
    May 4 at 10:12
  • $\begingroup$ Pages 31 to 37 in the notes for both, and Chapter "non variational methods" in Evans (monotonicity etc.). $\endgroup$
    – username
    May 4 at 10:15
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    $\begingroup$ How do you justify the claim that there is a maximum principle when no sign is specified for $q$ in the question? Without this, the operator $T_q$ seems ill-defined. $\endgroup$
    – Leo Moos
    May 4 at 11:55
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    $\begingroup$ @UserA All details explicitely written now. $\endgroup$
    – username
    May 5 at 8:31
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    $\begingroup$ @userA : if $u\in L^{Np}$ then $f(u)\in L^p$. Luckily if $u\in H^1$ in dimension two then $u\in L^{Np}$ for any $N>1$ and $p>1$ not infinity. the "N" in the previous comment means "some N not infinity". You use $H^1_0\hookrightarrow L^{Np}$, not $L^2$. $\endgroup$
    – username
    May 6 at 8:43

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