For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" and bounded it is clear that we have the Hölder continuity by some Sobolev embedding.

But I want to show the Hölder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

What is the next step?

For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" and bounded it is clear that we have the Hölder continuity by some Sobolev embedding.

But I want to show the Hölder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radius $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_{2r}(0)}\phi^2|\nabla u|^2 +\int_{B_{2r}(0)}2\phi u\nabla u\nabla \phi= \int_{B_{2r}(0)}f \phi^2$$.

What is the next step?

For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" and bounded it is clear that we have the Holder continuity by some Sobolev embedding.

But I want to show the Holder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

What is the next step?

For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" and bounded it is clear that we have the Hölder continuity by some Sobolev embedding.

But I want to show the Hölder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

What is the next step?

For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" it is clear that we have the Holder continuity by some Sobolev embedding.

But I want to show the Holder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

What is the next step?

For all $x$ in the ball $B_R(0) \subset \mathbb{R}^2$, I have the equation $$\Delta u=f(u)$$ with the boundary condition $u(|x|=R)=1$. Suppose $0\leq u \leq 1$ and $u \in W^{1,2}$. If $f$ is "nice enough" and bounded it is clear that we have the Holder continuity by some Sobolev embedding.

But I want to show the Holder continuity by using the Morrey space, that is to show that $r^{-2\alpha}\int_{B_r(0)} |\nabla u|^2$ is finite.

I am trying to choose $\phi^2u$ to be the test function $\phi=0$ outside the ball of radiu $2r$ and equal to $1$ inside the ball of radiu $r$ and integrate both side to get $$\int_{B_2r(0)}\phi^2|\nabla u|^2 +\int_{B_2r(0)}2\phi u\nabla u\nabla \phi= \int_{B_2r(0)}f \phi^2$$.

What is the next step?