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edited Jun 14, 2016 at 21:44

It is possible to do this. Here is a sketch of the construction:

Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary to the interior.
Let $v_0 = \max\{w, \, c_0(|x|^2 - 1) - c_0\}$. For $c_0$ very$c_0 > 0$ small and positive, $v_0 = w$ near the boundary and $v_0$ is the quadratic on a set $E$ whose boundary is a radial graph just inside of $\partial B_1$. Furthermore, $\Delta v_0 > c_1 > 0$ in the distributional sense.
Now let $\eta$ be a smooth cutoff function that is $1$ near $\partial B_1$ and $0$ in a neighborhood of $E$. Take $$v = \eta v_0 + (1-\eta) v_{\epsilon},$$ where $v_{\epsilon}$ is a mollification of $v_0$ (so $\Delta v_{\epsilon} > c_1$). Then $v = w$ near $\partial B_1$. Since $v_0 = w$ is smooth where the derivatives of $\eta$ are supported, it is straightforward to check that $\Delta v$ is smooth and positive in $B_1$ for $\epsilon$ small, completing the construction.

It is possible to do this. Here is a sketch of the construction:

Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary to the interior.
Let $v_0 = \max\{w, \, c_0(|x|^2 - 1) - c_0\}$. For $c_0$ very small and positive, $v_0 = w$ near the boundary and $v_0$ is the quadratic on a set $E$ whose boundary is a radial graph just inside of $\partial B_1$. Furthermore, $\Delta v_0 > c_1 > 0$ in the distributional sense.
Now let $\eta$ be a smooth cutoff function that is $1$ near $\partial B_1$ and $0$ in a neighborhood of $E$. Take $$v = \eta v_0 + (1-\eta) v_{\epsilon},$$ where $v_{\epsilon}$ is a mollification of $v_0$ (so $\Delta v_{\epsilon} > c_1$). Then $v = w$ near $\partial B_1$. Since $v_0 = w$ is smooth where the derivatives of $\eta$ are supported, it is straightforward to check that $\Delta v$ is smooth and positive in $B_1$ for $\epsilon$ small, completing the construction.

It is possible to do this. Here is a sketch of the construction:

Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary to the interior.
Let $v_0 = \max\{w, \, c_0(|x|^2 - 1) - c_0\}$. For $c_0 > 0$ small, $v_0 = w$ near the boundary and $v_0$ is the quadratic on a set $E$ whose boundary is a radial graph just inside of $\partial B_1$. Furthermore, $\Delta v_0 > c_1 > 0$ in the distributional sense.
Now let $\eta$ be a smooth cutoff function that is $1$ near $\partial B_1$ and $0$ in a neighborhood of $E$. Take $$v = \eta v_0 + (1-\eta) v_{\epsilon},$$ where $v_{\epsilon}$ is a mollification of $v_0$ (so $\Delta v_{\epsilon} > c_1$). Then $v = w$ near $\partial B_1$. Since $v_0 = w$ is smooth where the derivatives of $\eta$ are supported, it is straightforward to check that $\Delta v$ is smooth and positive in $B_1$ for $\epsilon$ small, completing the construction.

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created Jun 14, 2016 at 21:24

It is possible to do this. Here is a sketch of the construction:

Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary to the interior.
Let $v_0 = \max\{w, \, c_0(|x|^2 - 1) - c_0\}$. For $c_0$ very small and positive, $v_0 = w$ near the boundary and $v_0$ is the quadratic on a set $E$ whose boundary is a radial graph just inside of $\partial B_1$. Furthermore, $\Delta v_0 > c_1 > 0$ in the distributional sense.
Now let $\eta$ be a smooth cutoff function that is $1$ near $\partial B_1$ and $0$ in a neighborhood of $E$. Take $$v = \eta v_0 + (1-\eta) v_{\epsilon},$$ where $v_{\epsilon}$ is a mollification of $v_0$ (so $\Delta v_{\epsilon} > c_1$). Then $v = w$ near $\partial B_1$. Since $v_0 = w$ is smooth where the derivatives of $\eta$ are supported, it is straightforward to check that $\Delta v$ is smooth and positive in $B_1$ for $\epsilon$ small, completing the construction.