Let ${\mathrm{\Omega}}_{0}$ and ${\mathrm{\Omega}}_{1}$ be two bounded smooth domains of ${\mathbb{R}}^{N}$, $N\ge 2$, such that ${\mathrm{\Omega}}_{0}\subset {\mathrm{\Omega}}_{1}$.
Moreover, let $f\in {C}^{1}([0,+\mathrm{\infty}))$ and $c\ge 0$.
We consider weak solutions to the problem

$\{\begin{array}{cc}-\mathrm{\Delta}u=f(u)\hfill & \text{in}{\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}\text{,}\hfill \\ uc\hfill & \text{in}{\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}\text{,}\hfill \\ u=c\hfill & \text{on}\partial {\mathrm{\Omega}}_{1},\hfill \\ u=+\mathrm{\infty}\hfill & \text{on}\partial {\mathrm{\Omega}}_{0},\hfill \end{array}$(1.1)

i.e., we consider $u\in {C}^{1}(\overline{{\mathrm{\Omega}}_{1}}\setminus \overline{{\mathrm{\Omega}}_{0}})$ such that

${\int}_{{\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}}\nabla u\nabla \phi ={\int}_{{\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}}f(u)\phi \mathit{\hspace{1em}}\text{for all}\phi \in {C}_{c}^{\mathrm{\infty}}({\mathrm{\Omega}}_{1}\setminus \overline{{\mathrm{\Omega}}_{0}})$

and

$\underset{\begin{array}{c}x\to {x}_{0}\\ x\in {\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}\end{array}}{lim}u(x)=+\mathrm{\infty}\mathit{\hspace{1em}}\text{for all}{x}_{0}\in \partial {\mathrm{\Omega}}_{0}.$

When $c=0$, actually we deal with the case of positive solutions.
Necessary and sufficient conditions for the existence of solutions to (1.1) are provided by the classical results of Keller [18] and Osserman [19], under suitable assumptions on the nonlinearity.
The literature regarding boundary blow-up solutions is really wide (see, for example, [1, 2, 3, 4, 6, 5, 7, 13, 14, 12, 15, 16, 17, 20, 21]).
Here we exploit an adaptation of the celebrated moving plane technique (see [8]) in order to obtain monotonicity properties of the solutions to (1.1). The domain that we consider is not convex and the solutions are not in ${H}_{0}^{1}({\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0})$ as in the classical case.
This is the same difficulty
that occurs when dealing with the study of the uniqueness, symmetry and monotonicity properties of solutions to singular semilinear elliptic equations, see [9, 10, 11].
These problems exhibit in fact some similarities with problem (1.1) although the proofs cannot be adapted to our case.

In the second part of the paper, we prove the radial symmetry of the solutions on annular domains, under suitable assumptions.
In our setting, this cannot be done just using the moving plane method, since the domain is not convex, and we prove that the solution is radially symmetric showing directly that the angular derivative is zero. The technique is based on a refined maximum principle for the linearized equation.

Let us introduce some notations. Let ν be a direction in ${\mathbb{R}}^{N}$, with $|\nu |=1$.
Given a real number λ, we set

${T}_{\lambda}^{\nu}=\{x\in {\mathbb{R}}^{N}:x\cdot \nu =\lambda \}$

and

${x}_{\lambda}^{\nu}={R}_{\lambda}^{\nu}(x)=x+2(\lambda -x\cdot \nu )\nu ,$

that is, the reflection of *x* trough the hyperplane ${T}_{\lambda}^{\nu}$.
We will make the following assumption throughout the paper:

Moreover, we set

${\mathrm{\Omega}}_{\lambda}^{\nu}=\{x\in {\mathrm{\Omega}}_{1}:x\cdot \nu <\lambda \}\setminus {R}_{\lambda}^{\nu}({\mathrm{\Omega}}_{0})\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{({\mathrm{\Omega}}_{\lambda}^{\nu})}^{\prime}={R}_{\lambda}^{\nu}({\mathrm{\Omega}}_{\lambda}^{\nu}).$

Also we set

$a(\nu )=\underset{x\in {\mathrm{\Omega}}_{1}}{inf}x\cdot \nu \mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}b(\nu )=\underset{x\in {\mathrm{\Omega}}_{0}}{inf}x\cdot \nu .$

Observe that, by assumption (A), it follows that ${\mathrm{\Omega}}_{\lambda}^{\nu}$ is nonempty and ${({\mathrm{\Omega}}_{\lambda}^{\nu})}^{\prime}\subset {\mathrm{\Omega}}_{1}\setminus {\mathrm{\Omega}}_{0}$ for any $a(\nu )<\lambda \le 0$.
We define

${u}_{\lambda}^{\nu}(x)=u({x}_{\lambda}^{\nu})\mathit{\hspace{1em}}\text{for all}x\in {\mathrm{\Omega}}_{\lambda}^{\nu}$

and for any $a(\nu )<\lambda \le 0$.

We are now ready to state our main results.

#### Theorem 1.1.

*Let $u\mathrm{\in}{C}^{\mathrm{1}}\mathit{}\mathrm{(}\overline{{\mathrm{\Omega}}_{\mathrm{1}}}\mathrm{\setminus}\overline{{\mathrm{\Omega}}_{\mathrm{0}}}\mathrm{)}$ be a weak solution to (1.1).
Then*

$u<{u}_{\lambda}^{\nu}\mathit{\hspace{1em}}\mathit{\text{in}}{\mathrm{\Omega}}_{\lambda}^{\nu}\mathit{\text{for any}}a(\nu )\lambda b(\nu ).$(1.2)

*Consequently, it follows that **u* is strictly increasing with respect to the ν-direction in the set

$\{x\in {\mathrm{\Omega}}_{1}:a(\nu )<x\cdot \nu <b(\nu )\},\mathit{\text{with}}\frac{\partial u}{\partial \nu}0.$

In order to get symmetry results for the solution to (1.1), we restrict our attention to annular domains.
We denote by ${B}_{R}$ the open ball of center 0 and radius $R>0$ in ${\mathbb{R}}^{N}$.
By Theorem 1.1, we immediately deduce the following.

#### Corollary 1.2.

*Let $\mathrm{0}\mathrm{<}{R}_{\mathrm{0}}\mathrm{<}{R}_{\mathrm{1}}$ and $u\mathrm{\in}{C}^{\mathrm{1}}\mathit{}\mathrm{(}\overline{{B}_{{R}_{\mathrm{1}}}}\mathrm{\setminus}\overline{{B}_{{R}_{\mathrm{0}}}}\mathrm{)}$ be a weak solution to (1.1) in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus}{B}_{{R}_{\mathrm{0}}}$.
Then $\frac{\mathrm{\partial}\mathit{}u}{\mathrm{\partial}\mathit{}r}\mathrm{<}\mathrm{0}$ in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus}{B}_{{R}_{\mathrm{0}}}$.*

In the following we state sufficient conditions in order to deduce the
radial symmetry of the solution, once we prove the monotonicity.
We set

$v=-x\cdot \nabla u,$

and we denote by ${u}_{\theta}$ the angular derivative of *u*.

#### Theorem 1.3.

*Let $\mathrm{0}\mathrm{<}{R}_{\mathrm{0}}\mathrm{<}{R}_{\mathrm{1}}$ and let $u\mathrm{\in}{C}^{\mathrm{1}}\mathit{}\mathrm{(}\overline{{B}_{{R}_{\mathrm{1}}}}\mathrm{\setminus}\overline{{B}_{{R}_{\mathrm{0}}}}\mathrm{)}$ be a weak solution to (1.1) in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus}{B}_{{R}_{\mathrm{0}}}$ that satisfies*

${u}_{\theta}=o(v)\mathit{\hspace{1em}}\mathit{\text{as}}|x|\to {R}_{0}.$(1.3)

*If $f\mathrm{\le}\mathrm{0}$, then **u* is radially symmetric and radially decreasing in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus}{B}_{{R}_{\mathrm{0}}}$.

For the reader’s convenience, let us point out that the condition in (1.3) is inspired by the results in [20, 21].

In Section 2 we give the proof of Theorem 1.1.
We prove Theorem 1.3 in Section 3.

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