Qualitative properties of solutions to the Dirichlet problem for a Laplace equation involving the Hardy potential with possibly boundary singularity

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1.   Introduction

The aim of the paper is to investigate symmetry and monotonicity properties of weak solutions to semilinear elliptic equations concerning the Hardy term and a locally Lipschitz continuous from above (see below) nonlinearity in a bounded smooth domain $\Omega\subset \mathbb R^N$, $N\geq 3$ with $0\in\overline \Omega$. More precisely let us consider the problem

where $p > 0$, $1\leq q\leq2$ and $k, \vartheta\geq 0$. Our results will be obtained by means of the moving plane technique, see ^[1,11,26,34]. Such a technique can be performed in general domains providing partial monotonicity results near the boundary and symmetry when the domain is convex and symmetric with respect some direction. In particular, along this paper, we say that a domain is strictly convex with respect a direction, say for example the $x_1$-direction, if and only if

every point on the line segment connecting $P_a, P_b$ other than the end points $P_a$ and $P_b$ is contained in the interior of $\overline\Omega$.

For simplicity of exposition we assume directly in all the paper that $\Omega$ is a strictly convex with respect the $x_1$-direction (or convex when it will be specified, see Theorem 1.5 below) domain which is symmetric with respect to the hyperplane $\{x_1 = 0\}$.

Moreover in all the paper the nonlinearity $f$ will be assumed to be locally Lipschitz continuous from above. More precisely we assume that $f$ satisfies the following condition denoted from now on by $(h_f)$, namely

$(h_f)$ $f: (0, +\infty) \rightarrow {\mathbb{R}}^+_0$ is a continuous function such that for $0 \, < t \leq s\leq M$ it holds

where $C(M)$ is a positive constant depending on $M$.

A typical example is provided by positive solutions to equations involving nonlinearities given by $f(u) = g(u)+{1}/{u^\alpha}$, where $g$ is a locally Lipschitz continuous function and $\alpha > 0$. We recall here the following

Proposition 1.1. (Hardy-Sobolev Inequality) Suppose $N\geq 3$ and $u\in H^1(\mathbb{R}^N)$. Then we have

with ${ C_{N} = ({2}/{N-2}))^2}$ optimal and not achieved constant.

Singular semilinear elliptic equations with Hardy potential have been intensely studied. The problem of the existence of solutions to $({\cal{P}})$ exhibits a different behavior depending on the position of the pole on the domain. This acutually is strongly related to the Hardy-Sobolev inequality stated in Proposition 1.1. Let us consider, as a particular case of the problem $({\cal{P}})$ the following

We analyze the two different cases:

● Case 1: $0\in \partial\Omega$. The problem of the existence of solutions of problem shows a different behaviour depending on the exponent $p$. In the case $0\in \partial \Omega$, the existence of a solutions to (1.2) generally depends on the geometry of the domain.

$(i)$ If $0 < p < 1$ the existence of a solution to (1.2) is independent of the location of the origin. Indeed using Hardy inequality (Proposition 1.1) the functional satisfies

for some positive constant $C$ and therefore the existence of a solution in $H^1_0(\Omega)$ follows by minimization, see ^[3].

$(ii)$ In the linear case $p = 1$ the problem of the existence of (1.2) is related to the attainability of some constant less then $C_N$ in (1.1). In ^[25], the authors give sufficient conditions to get the existence of solutions to (1.2). In this case the geometry of $\Omega$ at the origin plays a fundamental role. See also ^{[16,17,20,27,28]} for related problems.

$(iii)$ The situation for $p > 1$ is also involved. Of course, if $0\not\in \overline \Omega$ the solution follows using the mountain pass theorem ^[9]. On the contrary, if $0\in \Omega$ there is no solutions to (1.2). Actually in ^[13] was shoved that (1.2) has no weak supersolutions since this would be a contradiction with the Hardy inequality (Proposition 1.1). In the case $0\in \partial \Omega$ the existence of solutions to (1.2) depends strongly on the geometry of the domain $\Omega$. For example in starshaped (with respect to the origin) domains there are no solutions since a Pohozaev's identity is in force in this case. On the other hand in some suitable non-starshaped domains, e.g., dumbell domains, there exits a weak solution to (1.2) in the range $1 < p < (N+2)/(N-2)$, see ^[18,29]. Moreover we point out the if we perturb the problem (1.2) adding some sublinear term $u^r$ with $0 < r < 1$, we get the existence of a weak solution without any restriction on the shape of the domain $\Omega$ and on the size of the exponent $p > 1$.

● Case 2: $0\in \Omega$. As in the previous case, the existence of solutions to problem (1.2) is related to the exponent $p$ and to the Hardy inequality (1.1). In particular

$(i)$ If $0 < p < 1$, (as in the case $0\in \partial \Omega$) the existence of a solution to (1.2) is independent of the location of the origin and follows by using a minimization procedure, see ^[3].

$(ii)$ In the linear and superlinear cases $p\geq1$ the problem (1.2) does not admit solutions (even in the weakest possible sense) because Proposition 1.1, see ^{[2,4,5,6,7,8,12,18,29]}. On the contrary, if the problem (1.2) is perturbed adding a first order term in the right hand side (that is, adding a first order term as an absorption term), then the existence of positive solutions of (1.2) can be proved by means of approximation and variational methods, see ^{[2,4,5,6,7,8,12,18,29]}. The absorption term $k |\nabla u|^q$ in $({\cal{P}})$, despite of Proposition 1.1, is sufficient to break down the effect of the obstruction to the existence of solutions due to the presence of the Hardy potential in the problem $({\cal{P}})$.

Therefore, taking into account these considerations on the existence of solutions to $({\cal{P}})$, taking into account the presence of the Hardy potential and the presence of a Hölder nonlinearity in the right hand side and standard elliptic regularity theory results, in all the paper we assume that

Thus the equation is understood in the following sense

Definition 1.2. $u \in H_{0}^1(\Omega) \cap C^1(\overline{\Omega} \setminus \{0\})$ is a weak solution to $({\cal{P}})$ if

and

Let us now state our main results.

Theorem 1.3. Let $\Omega$ be a strictly convex domain with respect to the $x_1$-direction, which is symmetric with respect to the hyperplane $\{x_1 = 0\}$ and let

be a solution to $({\cal{P}})$. Assume that

Then, it follows that $u$ is symmetric with respect to the hyperplane $\{x_1 = 0\}$ and increasing in the $x_1$-direction in $\Omega\cap\{x_1 < 0\}$. Furthermore

As an immediate consequence of the previous result we get the following

Corollary 1.4. Let $\Omega = B_R(0)$, $R > 0$ and let $u\in H_{0}^1(\Omega) \cap C^1(\overline{\Omega} \setminus \{0\})$ be a solution to $({\cal{P}})$. Assume that

Then, it follows that $u$ is radially symmetric with

If we assume more regularity on the data of problem $({\cal{P}})$, we can only assume that $\Omega$ is convex (not strictly) in the $x_1$-direction. In this case our result holds also for domains with a flat part on the boundary, as for example the case of a $N$-dimensional cube. We have the following

Theorem 1.5. Let $\Omega$ be a convex domain with respect to the $x_1$-direction, which is symmetric with respect to the hyperplane $\{x_1 = 0\}$ and let

be a solution to $({\cal{P}})$. Assume that $p\geq 1$ and $f: [0, +\infty) \rightarrow {\mathbb{R}}$ is locally Lipschitz continuous in $[0, \infty)$.

Then, it follows that $u$ is symmetric with respect to the hyperplane $\{x_1 = 0\}$ and increasing in the $x_1$-direction in $\Omega\cap\{x_1 < 0\}$. Furthermore

Symmetry and monotonicity properties of solutions to quasilinear and semilinear elliptic problems involving the Hardy potential (and $0\in \Omega$) or more general singular critical sets where the solution may be not regular, have been studied in ^{[14,15,21,22,23,24,30,31,33]} for the local case and in ^[10,19,32] for the nonlocal case. In this direction our result is new and more general. Indeed in this paper we also deal with the case $0\in \partial \Omega$ and we consider nonlinearities that are sum of a Hölder continuous term (the case $0 < p < 1$) and of a term $f$ that is locally Lipschitz continuous (only) from above in $(0, +\infty)$.

Actually, all the non negative nonlinearities of the form

where $f_1$ is a decreasing continuous function in $[0, \infty)$, $f_2(\cdot),$ is locally Lipschitz continuous in $[0, \infty)$, satisfy our assumptions $(h_f)$.

The remaining part of the paper is devoted to the proofs of our results.

2.   Proof of Theorem 1.3 and Theorem 1.5

Notation. Generic fixed numerical constants will be denoted by $C$ (with subscript in some case) and will be allowed to vary within a single line or formula. Moreover $f^+$ and $f^-$ will stand for the positive and negative part of a function, i.e., $f^+ = \max\{f, 0\}$ and $f^- = \min\{f, 0\}.$ We also denote $|A|$ the Lebesgue measure of the set $A$.

For a real number $\lambda$ we set

which is the reflection through the hyperplane $T_\lambda : = \{ x_1 = \lambda\}$and

Also let

In the following we will exploit the fact that $u_\lambda$ is a solution to:

for all $\varphi\in C^{1}_c(R_\lambda(\Omega))$ and we also observe that, for any $a < \lambda < 0$, the function $w_\lambda\, : = \, u-u_\lambda$ satisfies

and so $w_\lambda^+ \in L^2(\Omega_{\lambda})$, since $u \in C^0(\overline{\Omega_{\lambda}})$. Since in the range $0 < p < 1$ the right hand side of $({\cal{P}})$

is the sum of a Hölder continuous term (with respect to the variable $u$) and of a Lipschitz continuous term from above in $(0, +\infty)$, first we need to prove the following weak comparison principle that holds in subdomain of $\Omega$ that lies far from the boundary of $\Omega$ where the right hand side (2.4) is more regular. Then we have to take into account this fact in the proof of Theorem 1.3, by exploiting the Hopf's boundary lemma and the strictly convexity (in the $x_1$-direction) of $\Omega$. We have the following

Proposition 2.1 (Weak Comparison Principle 1). Assume that

Let $\lambda \leq \hat \lambda < 0$ and $\tilde \Omega$ be a bounded domain such that $\tilde \Omega \subset \subset \Omega_\lambda$. Assume that $u$ is a solution to $({\cal{P}})$ such that $u\leq u_\lambda$ on $\partial \tilde \Omega$. Then there exists a positive constant

such that if we assume $|\tilde \Omega| \leq \hat\delta,$ then it holds

Proof. We have (in the weak sense, see (1.4))

By contradiction, we assume the (2.5) is false.

First of all we start proving that

It is immediate to show that $(u-u_\lambda)^+\in L^{\infty}(\tilde \Omega)$ because $0\leq(u-u_\lambda)^+\leq u\in C^0(\overline{\Omega_{\hat\lambda}})$. On the other hand, the fact that $(u-u_\lambda)^+\in H^{1}_0(\tilde \Omega)$ is not a priori obvious since it can happen that $\partial \tilde \Omega\cap 0_{\lambda}\neq \emptyset$ and there the reflected function $u_{\lambda}$ is not defined. For the reader's convenience we give some details.

Let us define $\varphi_\varepsilon(x)\in C^{\infty}_c (\Omega)$, $\varphi_\varepsilon \geq 0$ such that

where $B_{\varepsilon} = B_{\varepsilon}(0)$ denotes the open ball with center $0$ and radius $\varepsilon > 0$. For $x\in \Omega_{\lambda}$, we consider

with $\varphi_\varepsilon$ defined in (2.9). Let us set

Since by hypothesis $u\in H^{1}_0(\Omega)$ and $u\leq u_\lambda$ on $\partial \tilde \Omega$, we readily have that $\phi_\varepsilon\in H^{1}_0(\Omega_{\lambda})$, for all $\varepsilon > 0$ and that

if $\varepsilon\rightarrow 0$. Setting $w_\lambda = (u-u_{\lambda})$, we deduce

where we used (2.9). Therefore (recall that $N\geq 3$)

Finally, by Sobolev embedding, from (2.10) and (2.11), we deduce (2.8).

Therefore, because (2.8), by a density argument we consider $(u-u_\lambda)^+\in H^{1}_0(\tilde \Omega)\cap L^{\infty}(\tilde \Omega)$ as a test function in both (2.6) and (2.7). We first consider the

Case: $1\leq q < 2$. Subtracting in the weak formulation of (2.6) and (2.7), we get

We note that in the set $\tilde \Omega\subset \subset \Omega_\lambda$ we have that $|x|\geq|x_\lambda|$. Therefore (2.12) becomes

where in the last inequality of (2.13) we used the assumption $(h_f)$ (recall that we are working where $u\geq u_\lambda$). Since $q\geq1$, for every $0_{\hat\lambda}\neq x\in\tilde \Omega_{\hat\lambda}$ by the mean value's theorem we get

Hence from (2.13) we deduce that

In the set $\{x\in\tilde \Omega\, :\, |\nabla u_\lambda| > 2 |\nabla u| \}$ using standard triangular inequalities we can deduce that

Note that, here below, we shall exploit (2.15) in the support of $(u-u_\lambda)^+$ since otherwise the functions involved vanish. By (2.15) we therefore obtain

where we have used weighted Young's inequality and the fact that $2/(2-q)\geq 2$ for $1\leq q < 2$. Therefore from (2.14) we deduce

where in the last inequality we used (2.15). Applying one more time weighted Young's inequality in the r.h.s of (2.16) we obtain

and for $\varepsilon$ small we deduce

where $C = C(k, f, q, \vartheta, \|u\|_{L^{\infty}(\Omega_{\hat\lambda})}, \|\nabla u\|_{L^{\infty}(\Omega_{\hat\lambda})})$ is a positive constant. Taking into account that for $\lambda \leq \hat\lambda < 0$ one has $|x|\geq C$ in $\Omega_\lambda$ for some positive constant $C$ depending only on $\hat \lambda$ (but not on $\lambda$), from (2.17) we obtain

where $C = C(k, f, p, q, \hat \lambda, \vartheta, {{\rm{dist}}}\, (\tilde \Omega, \partial \Omega), \|u\|_{L^{\infty}(\Omega_{\hat\lambda})}, \|\nabla u\|_{L^{\infty}(\Omega_{\hat\lambda})})$ is a positive constant. We note that, in the last inequality, we have used the fact that the term $u^p-u_\lambda^p$ is locally Liptschitz continuous in $(0, +\infty)$ and that the solution $u$ of $({\cal{P}})$ is strictly positive in $\Omega$.

Case: $q = 2$. In this case we consider we consider $e^{-ku}(u-u_\lambda)^+ \in H^{1}_0(\tilde \Omega)\cap L^{\infty}(\tilde \Omega),$ as a test function in (2.6) and $e^{-ku_\lambda}(u-u_\lambda)^+ \in H^{1}_0(\tilde \Omega)\cap L^{\infty}(\tilde \Omega),$ as a test function in (2.7). Subtracting in the weak formulation of (2.6) and (2.7), we get

Notice that we are considering the set $\tilde \Omega \cap \{u\geq u_\lambda\}$ and there $|x|\geq|x_\lambda|$. Then (2.19) becomes

As in the previous case, taking into account that for $\lambda < 0$ one has $|x|\geq C$ in $\Omega_{\hat\lambda}$ for some positive constant $C$, that the term $u^p-u_\lambda^p$ is locally Liptschitz continuous in $(0, +\infty)$ and that $u$ is positive in $\Omega$, we obtain

where $C = C(k, f, p, \hat \lambda, \vartheta, {{\rm{dist}}}\, (\tilde \Omega, \partial \Omega), \|u\|_{L^{\infty}(\Omega_{\hat\lambda})}, \|\nabla u\|_{L^{\infty}(\Omega_{\hat\lambda})})$ is a positive constant. Using weighted Young inequality finally we obtain

We get a similar estimate as the one in (2.18). The conclusion follows using classical Poincaré inequality in (2.18) and in (2.20). Indeed we deduce

where $C_P$ we denotes the Poincaré constant. By choosing

small such that $CC^2_p(\tilde \Omega) < 1$, we get $(u-u_\lambda)^+ = 0$ in $\tilde \Omega$ since by hypothesis we have $u\leq u_\lambda$ on $\partial \tilde \Omega$. This concludes the proof.

In the case $p\geq 1$, adapting straightforwardly the proof of Proposition 2.1 we are able to get the next

Proposition 2.2 (Weak Comparison Principle 2). Assume that

Let $\lambda \leq \hat \lambda < 0$ and $\tilde \Omega$ be a bounded domain such that $\tilde \Omega \subseteq \Omega_\lambda$. Assume that $u$ is a solution to $({\cal{P}})$ such that $u\leq u_\lambda$ on $\partial \tilde \Omega$. Then there exists a positive constant

such that if we assume $|\tilde \Omega| \leq \hat\delta,$ then it holds

Now we are ready to prove our main results. We start with the

Proof of Theorem 1.3. To prove the theorem, we exploit the moving plane method. To start with the procedure, we take advantage of the application of Hopf's boundary lemma. We recall that in the case $0 < p < 1$ we do have to use Proposition 2.1 far from the boundary $\partial \Omega$, since the loss of regularity of the right hand side of $({\cal{P}})$. Thus let $a < \lambda < 0$ with $\lambda$ sufficiently close to $a$, see (2.2). By Hopf's boundary lemma, it follows that

We define

and

Notice that by the continuity of the solution $u$ we obtain $u\leq u_{\lambda_0}$ in $\Omega_{\lambda_0}$. To prove our theorem, we have to show that

Assume by contradiction $\lambda_0 < 0$. We can exploit the strong maximum (or comparison) principle for the laplacian operator, to get that

in $\Omega_{\lambda_{0}}$. It follows now that the case $u\equiv u_{\lambda_0}$ in $\Omega_{\lambda_{0}}$ is not possible, since the Dirichlet condition would imply the existence of some point $x\in \Omega$ such that $u(x) = 0$. This is a contradiction with $({\cal{P}})$, in particular with the assumption $u > 0$. Thus $u < u_{\lambda_0}$ in $\Omega_{\lambda_0}\setminus\{0_{\lambda_0}\}$ (let us observe that $u_{\lambda_0}$ may not defined in $\Omega_{\lambda_0}$).

We point out that, because $0\in \overline\Omega$, in general we have that the reflected point $0_{\lambda_{0}}$ may belong to $\Omega_{\lambda_{0}}$. There, $u_{\lambda_0}$ is not smooth. Since the domain is strictly convex in the $x_1$-direction, by Hopf's boundary lemma and the Dirichlet condition, we get that there exists a neighborhood $\mathcal{I}_{\lambda_0}$ of

such that

for some positive $\delta$. In particular: for $x\in \mathcal{I}_{\lambda_0} \setminus B_\delta(0_{\lambda_0})$ far from $\partial \Omega_{\lambda_{0}} \cap T_{\lambda_0}$ we exploit the uniform continuity of the solution and the zero Dirichlet boundary condition; on the other hand, in a neighborhood of $\partial \Omega_{\lambda_{0}} \cap T_{\lambda_0}$ we exploit the Hopf's boundary lemma since by our assumption, the domain is smooth and strictly convex.

Therefore we deduce that there exists a compact set $K$ in $\Omega_{\lambda_{0}}$ such that $K\cap B_\delta(0_{\lambda_0}) = \emptyset$ and

is sufficiently small (eventually reducing $\delta$) so that $u_{\lambda_{0}}-u$ is positive in $K \cup (\mathcal{I}_{\lambda_0}\setminus B_\delta(0_{\lambda_0}))\big)$ and Proposition 2.1 applies in the set $\Omega_{\lambda_{0}}\setminus \big (K\cup (\mathcal{I}_{\lambda_0}\setminus B_\delta(0_{\lambda_0}))\big)$. We point out that, without loss of generality, we can suppose $0_{\lambda_0}\in \partial \Omega$. Actually in the case $0_{\lambda_0}\in \Omega$ we can choose the neighborhood $\mathcal{I}_{\lambda_0}$ and $\delta$ such that $\mathcal{I}_{\lambda_0}\cap B_\delta(0_{\lambda_0}) = \emptyset$ and (2.23) reduces to $|\Omega_{\lambda_{0}}\setminus \big (K\cup \mathcal{I}_{\lambda_0}\big)|$.

Arguing by continuity, we also have $u_{\lambda_0}-u > 0$ on $\partial \big(K \cup (\mathcal{I}_{\lambda_0}\setminus B_\delta(0_{\lambda_0}))\big)$. Hence it follows $u \leq u_{\lambda_0+\varepsilon}$ on $\partial \big(K \cup (\mathcal{I}_{\lambda_0}\setminus B_\delta(0_{\lambda_0}))\big)$, for sufficiently small $\varepsilon$. Using Proposition 2.1 we obtain that

To get the desired contradiction, it remains to show that

To do this let us consider the ball $B_r = B_r(0)$ for $r$ small such that $B_r\subset \subset \Omega$. Since $u$ is positive in $\Omega$ and (1.3), we infer that $m = m(r): = \min_{x\in \partial B_r} u(x) > 0$. We claim that there exist $r$ such that

Arguing by contradiction, let us define $\varphi = (u-m)^-$ if $x\in B_r$ and $\varphi = 0$ elsewhere. Clearly $\varphi\in H^{1}_0(\Omega)\cap L^{\infty}(\Omega)$. Therefore using the weak formulation of $({\cal{P}})$ we have

For $1\leq q < 2$ (by weighted Young inequality) we obtain

where $C(k, q, \varepsilon)$ is some constant that does not depend on $r$ since $0 < \min\limits_{0 < \varrho\leq r} \leq m(r)$ (recall also that ${(2q-2)}/{(2-q)}\geq0$ since $1\leq q < 2$). For $\varepsilon$ small enough we get

with ${(2q-2)}/{(2-q)}\geq0$ since $1\leq q < 2$. Using Poincaré inequality in the right hand side we obtain

Therefore for $r$ small we have that actually $(u-m)^- = 0$ in $B_r$. This proves (2.26) for the case $1\leq q < 2$.

For $q = 2$ we take $\varphi\in H^{1}_0(\Omega)\cap L^{\infty}(\Omega)$ such that $\varphi = e^{-ku}(u-m)^-$ if $x\in B_r$ and $\varphi = 0$ elsewhere, as test function in the weak formulation of $({\cal{P}})$. Then

that is (2.26) for the case $q = 2$.

Thanks to (2.26) (see also (2.1)), we deduce as well that $u_{\lambda_{0}}(x)\geq m > 0$ in $B_\delta(0_{\lambda_{0}})$, for $0 < \delta\leq r/2$. Hence using the boundary Dirichlet condition and the continuity of $u$ in $B_r(0_{\lambda_{0}})$, reducing $\delta$ if it is necessary, we deduce (2.25) for $\varepsilon$ small.

Consequently from (2.24) and (2.25) we have that $u \leq u_{\lambda_0+\varepsilon}$ in $\Omega_{\lambda_0+\varepsilon}$. This contradicts the assumption $\lambda_0 < 0$. Therefore, $\lambda_0 = 0$. We point out that we are exploiting Proposition 2.1 in the set $\Omega_{\lambda_{0}+\varepsilon}\setminus \big (K\cup (\mathcal{I}_{\lambda_0+\varepsilon}\setminus B_\delta(0_{\lambda_0}))$ which is bounded away from the boundary $\partial \Omega$ and then the constant $\hat\delta$ in the statement is uniformly bounded.

In the same way, performing the moving plane method in the opposite direction, namely $-x_1$, we obtain

that is, $u$ is symmetric. Moreover, it is implicit in the moving plane procedure the fact that the solution is increasing in the $x_1$-direction in $\{x_1 < 0\}$. Since (see (1.3)) $u$ is $C^1$ far away the origin $0\in \partial \Omega$, using the monotonicity of the solution $u$ wen readily get (1.6).

Proof of Corollary 1.4. If $\Omega$ is a ball, applying Theorem 1.5 along any direction, it follows that $u$ s radially symmetric. The fact that $u_r < 0$ for $r\neq 0$, follows by the Hopf's boundary lemma which works in this case since the level sets are balls and therefore fulfill the interior sphere condition.

Proof of Theorem 1.5. Since the origin $0\in \overline \Omega$ is contained in the hyperplane $\{x_1 = 0\}$, then the moving plane procedure can be started in the standard way and, for $a < \lambda < a + \sigma$ with $\sigma > 0$ small, we have that $u-u_{\lambda} \leq 0$ in $\Omega_\lambda$, by Proposition 2.2. In this case the standard weak comparison principle holds since the right hand side (far away to zero) is locally Lipschitz continuous. Moreover note that $u, u_{\lambda}$ are smooth far from zero. Therefore for $\lambda$ close to $a$ the singularity at zero coming from the Hardy potential does not play a role. To proceed further we define as we did above

that is not empty for $\lambda$ close to $a$, and $\lambda_0 = \sup\, \Lambda_0.$ Assuming by contradiction that $\lambda_0 < 0$, we have by the strong maximum principle that $u < u_{\lambda_0}$ in $\Omega_{\lambda_0}$. Therefore exploiting the fact that $u < u_{\lambda_0}$ in $\Omega_{\lambda_0}\setminus \{0_{\lambda_0}\}$ and the fact that the solution $u$ is continuous in $\overline \Omega\setminus\{0\}$ (resp. $u_{\lambda_0}$ is continuous in $\overline \Omega\setminus\{0_{\lambda_0}\}$), we deduce that there exist a compact set $K$ such that $\Omega_{\lambda_0}\setminus K$ is sufficiently small in order to apply the weak comparison principle, Proposition 2.2. The rest of the proof is standard (follow the proof of Theorem 1.3). Moreover, it is implicit in the moving plane procedure the fact that the solution is increasing in the $x_1$-direction in $\{x_1 < 0\}$. Since (see (1.3)) $u$ is $C^1$ far from the origin, using the monotonicity of the solution $u$ we get that $u_{x_1}\geq 0$ in $\Omega\cap\{x_1 < 0\}$. The fact that $u_{x_1}$ is positive for $x_1 < 0$ (see (1.6)), follows by the maximum principle for $u_{x_1}$ that applies in this case and by the Hopf's boundary lemma.

Aknowledgments

The authors would like to thank the anonymous referee for his/her useful suggestions and comments. L. Montoro and B. Sciunzi are partially supported by PRIN project 2017JPCAPN (Italy): Qualitative and quantitative aspects of nonlinear PDEs, and L. Montoro by Agencia Estatal de Investigación (Spain), project PDI2019-110712GB-100.

Conflict of interest

The authors declare no conflict of interest.