$\Omega$ is a bounded domain in $\mathbb R^n$ with smooth boundary. Consider the Dirichlet problem:$$-\Delta u = (\lambda - \log u)u ~~~{\rm on}~~~ \Omega~~~ {\rm and}~~~u=0 ~~~{\rm on}~~\partial {\Omega}$$ where the solution $u\in {C^\infty }(\bar \Omega )$ and $u>0$ on $\Omega$ and $\lambda$ is a positive constant.
My question is: what conditions do we need for the existence and uniqueness of the solution $u$ ?
Define $$ f(t) = \left\{ \begin{array}{rl} -t\log{t} &\mbox{ if $t\in[0,1]$} \\ 0 &\mbox{ otherwise} \end{array} \right. $$
Note that $f$ is continuous. COnsider the problem
$$\tag{1} \left\{ \begin{array}{rl} -\Delta u-\lambda u=f(u) &\mbox{ in $\Omega$} \\ u=0 &\mbox{ in $\partial\Omega$} \end{array} \right. $$
1 - Existence
I - There is some methods avaliable to find a solution to $(1)$. For example, we can use the fact that $f$ is continuous with compact support, to prove that the energy functional $I:H_0^1\to\mathbb{R}$ associated with $(1)$ is (under some additional restriction) weakly lower semi continuous and coercive. Remember that if $F(t)=\int_0^t f(s)ds$, then $$Iu=\frac{1}{2}\int_\Omega |\nabla u|^2-\frac{\lambda}{2}\int_\Omega u^2-\int_\Omega F(u)$$
The additional restriction necessary here is $\lambda<\lambda_1$ (to guarantee coercivity).
II - (This is just a guess - I dont have verified the calculations): Also, you can prove existence, by noting that if $\lambda$ is not a eigenvalue of $-\Delta$, then the operator $-\Delta-\lambda I$ is invertible, and you can use arguments similar to this one.
Remark 1: The original function $t\log{t}$ is not monotone, so I dont know if this problem has uniqueness.
Remark 2: The function $f$ being continuous, we can conclude that $u\in C^1(\overline{\Omega})$.
Now, let's find conditions such that the solution $u$ of $(1)$ satisfies $0<u\leq 1$. Because $f\geq 0$ and there is a open set where $f>0$, you can conclude by the maximum principle that $u>0$. On the other hand, by regularity theory, you have that the solution $u$ of $(1)$ (See Breziz chapter 9) satisfies $(1)$ almost everywhere, i.e. $u$ is a Strong solution. If we suppose that the set where $u>1$ is non0empty, then we would have $-\Delta u-\lambda u=0$ almost everywhere in this set. If $\lambda$ is not a eingevalue, then this is a absurd, which implies that $u\geq 1$.
From the last paragraph, we can conclude that in fact $u$ is a solution to the original problem and satisfies $0<u\leq 1$. Hence, by using the first method proposed, we can conclude that if $\lambda<\lambda_1$, then you problem has a solution $u\in C^{2,\alpha}(\overline{\Omega})$ satisfying $0<u\leq 1$. I dont know if we can get more regularity, because the functions $-t\log{t}$ is only Holder continuous. A argument of bootstrap can be applied in the interior of $\Omega$, but in the boudary I dont know, because $-t\log{t}$ is not even Lipschitz for $t=0$.
