Let $n > 2$ and let $f : \mathbb R \rightarrow \mathbb R$ be a continuous function satisfying $$\lvert f(t)\rvert ≤ c \lvert t\rvert^{q−1}\quad \forall t \in \mathbb R,$$ for some constants $c > 0$ and $1 \leq q < 2$. We say that $u \in H^1_0(U)$ is a weak solution of $$ −\Delta u = f(u) \text{ in } U,$$ $$u = 0 \text{ on } \partial U,$$ if $$\int_U \nabla u\cdot\nabla v \, dx = \int_U f(u) v\, dx \quad \forall v \in H^1_0(U).$$ Show that there exists a constant $C > 0$ such that $\lVert u\rVert_{H^1_0(U)} \leq C$ for all weak solutions u.
Can anyone give me an idea about how to solve this? Very much appreciated for any thoughts. Thanks.
In the definition of $u\in H^1_0(U)$ being a weak solution to $-\Delta u =f(u)$, we can use $u$ as a test function (since $u\in H^1_0(U)$) to obtain that $$ \| \nabla u \|^2_{L^2(U)} = \int_U \vert \nabla u \vert^2 \, dx = \int_Uf(u) u \, dx . $$ Using the assumption on $f$, we have that $$ \| \nabla u \|^2_{L^2(U)} \leqslant C \int_U\vert u\vert^q \, dx \tag{$1$} .$$ Next, assuming that $\Omega$ is bounded, by Hölder's inequality with $p=2/q>1$, it follows that $$ \int_U\vert u\vert^q \, dx \leqslant \vert U\vert^{1-\frac q2} \bigg ( \int_U\vert u\vert^2 \, dx \bigg)^{q/2} \tag{$2$}. $$
Finally, using again that $\Omega$ is bounded, the Poincaré inequality implies that $$\|u\|_{H^1(U)}^2 \leqslant C \| \nabla u \|^2_{L^2(U)} . \tag{$3$}$$ with $C=C(U)$.
Combining ($1$), ($2$), and ($3$) gives $$ \| u\|_{H^1(U)}^2 \leqslant C(\Omega,q) \bigg ( \int_U\vert u\vert^2 \, dx \bigg)^{q/2} \leqslant C(\Omega,q)\|u\|_{H^1(U)}^q. $$ To obtain the final result, divide both sides by $\|u\|_{H^1(U)}^q$ then raise everything to the power of $\frac 1 {2-q}$.