Our main result (compare with [2, 15]) is the following theorem.

#### Theorem 2.1.

*Let $N\mathrm{\u2a7e}\mathrm{3}$, $\alpha \mathrm{>}\mathrm{0}$ and $p\mathrm{\in}\mathrm{]}\mathrm{1}\mathrm{,}\frac{N\mathrm{+}\alpha}{N\mathrm{-}\mathrm{2}}\mathrm{[}$. Then there exists no radial solution to problem (P).
*

Radial solutions of (P) can be found looking for global positive solutions of the Cauchy problem defined in ${\mathbb{R}}_{+}$ for some $\xi >0$:

$\{\begin{array}{cc}& {\left({\displaystyle \frac{{u}^{\prime}}{\sqrt{1\pm {({u}^{\prime})}^{2}}}}\right)}^{\prime}+{\displaystyle \frac{N-1}{r}}{\displaystyle \frac{{u}^{\prime}}{\sqrt{1\pm {({u}^{\prime})}^{2}}}}+{r}^{\alpha}{|u|}^{p-1}u=0,\hfill \\ & {u}^{\prime}(0)=0,\hfill \\ & u(0)=\xi \hfill \end{array}$(C-xi)

such that ${lim}_{r\to +\mathrm{\infty}}u(r)=0$.

By standard arguments, it is easy to prove that problem (C-xi) has a unique local solution in a right neighborhood of 0. In the sequel, every time we consider any Cauchy problem whose local solution ${u}_{\xi}$ can be continued until it touches the line $u=0$, we will denote by ${R}_{0}({u}_{\xi})>0$ the point such that $u({R}_{0}({u}_{\xi}))=0$ and ${u}_{|[0,{R}_{0}({u}_{\xi})[}>0$.

Assume the following notations:
We set ${\rho}_{u}(r)=|{u}^{\prime}(r)|$ and define the following functions:

${A}^{\pm}(q)={\displaystyle \frac{1}{\sqrt{1\pm {q}^{2}}}},$${\mathrm{\Omega}}^{\pm}(q)=q{A}^{\pm}(q)={\displaystyle \frac{q}{\sqrt{1\pm {q}^{2}}}},$${E}^{\pm}(q)={\displaystyle \frac{d}{dq}}({\mathrm{\Omega}}^{\pm}(q))=1/{(1\pm {q}^{2})}^{\frac{3}{2}},$${H}^{\pm}(q)={\displaystyle {\int}_{0}^{q}}t{E}^{\pm}(t)\mathit{d}t=\pm 1\mp {(1\pm {q}^{2})}^{-\frac{1}{2}},$

which are of class ${C}^{\mathrm{\infty}}$ (respectively in $\mathbb{R}$ and in $]-1,1[$ according to wether the sign is $+$ or $-$). We will use the notations $A,\mathrm{\Omega},E$ and *H* every time our arguments work fine regardless of the sign.
Define also the following functions in $\mathbb{R}$:

$f(t)={|t|}^{p-1}t\hspace{1em}\text{and}\hspace{1em}F(t)={\displaystyle {\int}_{0}^{t}}f(s)\mathit{d}s={\displaystyle \frac{1}{p+1}}{|t|}^{p+1}.$

We are going to present some preliminary lemmas.

#### Lemma 2.2.

*Let $\mathrm{1}\mathrm{<}p\mathrm{<}\frac{N\mathrm{+}\mathrm{2}\mathrm{+}\mathrm{2}\mathit{}\alpha}{N\mathrm{-}\mathrm{2}}$. Then there exists $\overline{\xi}\mathrm{>}\mathrm{0}$ such that for all $\delta \mathrm{>}\mathrm{0}$ there exists $\overline{\epsilon}\mathrm{>}\mathrm{0}$ for which the local solution ${w}_{\epsilon}^{\delta}$ of the problem*

$\{\begin{array}{cc}& {\left(\frac{{u}^{\prime}}{\sqrt{1\pm \epsilon {({u}^{\prime})}^{2}}}\right)}^{\prime}+\frac{N-1}{r}\frac{{u}^{\prime}}{\sqrt{1\pm \epsilon {({u}^{\prime})}^{2}}}+{r}^{\alpha}{|u|}^{p-1}u=0,\hfill \\ & {u}^{\prime}(0)=0,\hfill \\ & u(0)=\overline{\xi}\hfill \end{array}$(2.1)

*corresponding to $\epsilon \mathrm{\in}\mathrm{]}\mathrm{0}\mathrm{,}\overline{\epsilon}\mathrm{]}$ is sign-changing and ${R}_{\mathrm{0}}\mathit{}\mathrm{(}{w}_{\epsilon}^{\delta}\mathrm{)}\mathrm{\in}\mathrm{[}\mathrm{1}\mathrm{-}\delta \mathrm{,}\mathrm{1}\mathrm{+}\delta \mathrm{]}$.*

#### Proof.

Consider the following boundary value problem set in the unit ball centered in 0, denoted by ${B}_{1}$:

$\{\begin{array}{cccc}& \mathrm{\Delta}u+{|x|}^{\alpha}{u}^{p}=0\hfill & & \hfill \text{in}{B}_{1},\\ & u0\hfill & & \hfill \text{in}{B}_{1},\\ & u=0\hfill & & \hfill \text{on}\partial {B}_{1}.\end{array}$(2.2)

By [14], we know that (2.2) possesses at least a radial solution *w*.
Now pick $\delta >0$ and consider the Cauchy problem (2.1) for $\overline{\xi}=w(0)$.
Since $w(r)$ solves

$\{\begin{array}{cc}& {u}^{\prime \prime}+\frac{N-1}{r}{u}^{\prime}+{r}^{\alpha}{|u|}^{p-1}u=0,\hfill \\ & {u}^{\prime}(0)=0,\hfill \\ & u(0)=\overline{\xi},\hfill \end{array}$(2.3)

and the equation in (2.1) is a regular perturbation of
the equation in (2.3), we can find $\overline{\epsilon}>0$ sufficiently small such that, for any $\epsilon \in ]0,\overline{\epsilon}]$, the local solution ${w}_{\epsilon}$ of (2.1) touches the axis somewhere in $[1-\delta ,1+\delta ]$.
∎

#### Lemma 2.3.

*Let $\mathrm{1}\mathrm{<}p\mathrm{<}\frac{N\mathrm{+}\mathrm{2}\mathrm{+}\mathrm{2}\mathit{}\alpha}{N\mathrm{-}\mathrm{2}}$ and $\overline{\delta}\mathrm{>}\mathrm{0}$, and consider $\overline{\xi}\mathrm{>}\mathrm{0}$ as in Lemma 2.2.
If ${\mathrm{(}{w}_{\epsilon}^{\overline{\delta}}\mathrm{)}}_{\epsilon}$ is the family of sign-changing solutions of (2.1) built by means of Lemma 2.2,
the family ${\mathrm{(}{u}_{\epsilon}^{\overline{\delta}}\mathrm{)}}_{\epsilon}$ whose elements are defined for any $r\mathrm{>}\mathrm{0}$ by the relation*

${u}_{\epsilon}^{\overline{\delta}}(r)={\epsilon}^{\frac{2+\alpha}{2(p+\alpha +1)}}{w}_{\epsilon}^{\overline{\delta}}\left({\epsilon}^{\frac{p-1}{2(p+\alpha +1)}}r\right)$

*is such that, for any $\epsilon \mathrm{\in}\mathrm{]}\mathrm{0}\mathrm{,}\overline{\epsilon}\mathrm{]}$, the function ${u}_{\epsilon}^{\overline{\delta}}$ is a local sign-changing solution of (C-xi) with*

$\xi ={\epsilon}^{\frac{2+\alpha}{2(p+\alpha +1)}}\overline{\xi}\mathit{\hspace{1em}}\text{\mathit{a}\mathit{n}\mathit{d}}\mathit{\hspace{1em}}{R}_{0}({u}_{\epsilon}^{\overline{\delta}})\in [{\epsilon}^{-\frac{p-1}{2(p+\alpha +1)}}(1-\overline{\delta}),{\epsilon}^{-\frac{p-1}{2(p+\alpha +1)}}(1+\overline{\delta})].$

#### Proof.

The proof follows from Lemma 2.2 by straightforward computations.
∎

#### Lemma 2.4.

*If **v* is a radial solution of (P), then the following assertions hold:

#### Proof.

(i) By contradiction, let η be a positive number such that for some $R>0$ and for any $r>R$ we have ${\rho}_{v}(r)\u2a7e\eta $.
Then, since ${v}^{\prime}(r)=-{\rho}_{v}(r)$, for any $r>R$ we have

$v(r)=v(R)-{\int}_{R}^{r}{\rho}_{v}(s)\mathit{d}s\u2a7dv(R)+\eta (R-r).$

Of course the inequality is false for *r* large enough.

(ii) We know that, for every $r>0$,

$\frac{d}{dr}\left[{r}^{N-1}\mathrm{\Omega}({\rho}_{v}(r))\right]={r}^{\alpha +N-1}f(v(r))>0.$

Then there exist $C>0$ and $\stackrel{~}{R}>0$ such that, for all $r>\stackrel{~}{R}$,

${r}^{N-1}\mathrm{\Omega}({\rho}_{v}(r))>C.$

Now we distinguish two cases:

•

If we are considering ${\mathrm{\Omega}}^{+}$, then we trivially deduce ${\rho}_{v}(r)>C{r}^{1-N}$.

•

If we are considering ${\mathrm{\Omega}}^{-}$, then either ${\rho}_{v}(r)\u2a7d1/\sqrt{2}$, and then, since ${A}^{-}({\rho}_{v}(r))\u2a7d\sqrt{2}$, we have

${\rho}_{v}(r)>\frac{C}{\sqrt{2}}{r}^{1-N},$

or ${\rho}_{v}(r)>1/\sqrt{2}\u2a7eC{r}^{1-N}$ just, if needed, redefining $\stackrel{~}{R}$.

In any case we can assume that for suitable positive constants *C* and $\stackrel{~}{R}$ we have

${\rho}_{v}(r)>C{r}^{1-N}\mathit{\hspace{1em}}\text{for}\stackrel{~}{R}r.$

Integrating in $]r,+\mathrm{\infty}[$, we get the conclusion.
∎

#### Lemma 2.5.

*Assume **v* is a ground state solution of (C-xi). Then, if *w* is a sign-changing solution of (C-xi) such that $w\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{<}v\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}$, the graphs of *w* and *v* must intersect somewhere in $\mathrm{[}\mathrm{0}\mathrm{,}{R}_{\mathrm{0}}\mathrm{(}w\mathrm{)}\mathrm{]}\mathrm{\times}\mathrm{]}\mathrm{0}\mathrm{,}\mathrm{+}\mathrm{\infty}\mathrm{[}$.

#### Proof.

Assume by contradiction that *w* is a solution as in the statement of the lemma, and the set of points in $[0,{R}_{0}(w)]\times ]0,+\mathrm{\infty}[$ where the graphs of *w* and *v* intersect is empty.

Set ${\xi}_{v}=v(0)$ and ${\xi}_{w}=w(0)$. Since *v* and *w* are decreasing in $[0,{R}_{0}(u)]$ and in ${\mathbb{R}}^{+}$, respectively, we can define the functions $s(u)$ and $t(u)$, the inverses of $v(r)$ and $w(r)$, respectively, the first defined into $]0,{\xi}_{v}]$, the second into $[0,{\xi}_{w}]$.

Observe that, since ${w}^{\prime}({R}_{0}(w))<0$ and ${lim\; sup}_{r\to +\mathrm{\infty}}{v}^{\prime}(r)={0}^{-}$ by Lemma 2.4, we obtain

$\underset{u\to {0}^{+}}{lim}{t}^{\prime}(u)=\underset{u\to {0}^{+}}{lim}{\displaystyle \frac{1}{{w}^{\prime}(t(u))}}={\displaystyle \frac{1}{{w}^{\prime}({R}_{0}(w))}}>-\mathrm{\infty},$$\underset{u\to {0}^{+}}{lim\; inf}{s}^{\prime}(u)=\underset{u\to {0}^{+}}{lim\; inf}{\displaystyle \frac{1}{{v}^{\prime}(s(u))}}=-\mathrm{\infty}.$

Then, for every $\eta >0$, there exist ${u}_{\eta}\in ]0,\eta ]$ and ${\mu}_{\eta}\in ]0,u{}_{\eta}[$ such that we have ${(s-t)}^{\prime}(u)<0$ for any $u\in ]u{}_{\eta}-\mu {}_{\eta},u{}_{\eta}+\mu {}_{\eta}[$.

On the other hand, since ${v}^{\prime}(s({\xi}_{w}))<0$ and ${lim}_{u\to {\xi}_{w}}{w}^{\prime}(t(u))={0}^{-}$, we have

$\underset{u\to {\xi}_{w}^{-}}{lim}{s}^{\prime}(u)=\underset{u\to {\xi}_{w}^{-}}{lim}{\displaystyle \frac{1}{{v}^{\prime}(s(u))}}={\displaystyle \frac{1}{{v}^{\prime}(s({\xi}_{w}))}}>-\mathrm{\infty},$$\underset{u\to {\xi}_{w}^{-}}{lim}{t}^{\prime}(u)=\underset{u\to {\xi}_{w}^{-}}{lim}{\displaystyle \frac{1}{{w}^{\prime}(t(u))}}=-\mathrm{\infty},$

and then, if we have taken $\eta >0$ sufficiently small, we have ${(s-t)}^{\prime}(u)>0$ for $u\in ]\xi {}_{w}-\eta ,\xi {}_{w}[$.

We deduce that, necessarily, the function $s-t$ has a local minimum in the interval $]u{}_{\eta}-\mu {}_{\eta},\xi {}_{w}[$.

On the other hand, by arguments analogous to those in the proof of
[13, Lemma 3.3.1], we have that the following equations hold in the interval $]0,\xi {}_{w}[$:

$E\left(\left|{\displaystyle \frac{1}{{s}_{u}}}\right|\right){s}_{uu}-{\displaystyle \frac{N-1}{s}}A\left(\left|{\displaystyle \frac{1}{{s}_{u}}}\right|\right){s}_{u}^{2}-{s}^{\alpha}{s}_{u}^{3}{u}^{p}=0,$$E\left(\left|{\displaystyle \frac{1}{{t}_{u}}}\right|\right){t}_{uu}-{\displaystyle \frac{N-1}{t}}A\left(\left|{\displaystyle \frac{1}{{t}_{u}}}\right|\right){t}_{u}^{2}-{t}^{\alpha}{t}_{u}^{3}{u}^{p}=0.$

If $\stackrel{~}{u}\in ]0,\xi {}_{w}[$ is a critical point of $s-t$, computing the previous two equations in $\stackrel{~}{u}$ and subtracting one from the other, we should obtain the following relation (observe that ${s}_{u}(\stackrel{~}{u})={t}_{u}(\stackrel{~}{u})$):

${(s-t)}_{uu}(\stackrel{~}{u})=(N-1)Q\left(\left|\frac{1}{{s}_{u}(\stackrel{~}{u})}\right|\right)\left(\frac{1}{s(\stackrel{~}{u})}-\frac{1}{t(\stackrel{~}{u})}\right){s}_{u}^{2}(\stackrel{~}{u})+\frac{{\stackrel{~}{u}}^{p}}{E({|{t}_{u}(\stackrel{~}{u})|}^{-1})}({s}^{\alpha}(\stackrel{~}{u})-{t}^{\alpha}(\stackrel{~}{u})){s}_{u}^{3}(\stackrel{~}{u})$(2.4)

where we have assumed the notation $Q=\frac{A}{E}$.
From (2.4) we deduce that, since ${s}_{u}(\stackrel{~}{u})<0$ and by our contradiction assumption we know that $s(\stackrel{~}{u})>t(\stackrel{~}{u})$, the critical point must be a maximum.
∎

#### Proof of Theorem 2.1.

Assume *v* is a radial solution of (P) and consider the family ${({u}_{\epsilon}^{\overline{\delta}})}_{\epsilon}$ from Lemma 2.3 (which exists since $p<\frac{N+\alpha}{N-2}$).

Set

${R}_{\epsilon}:={R}_{0}({u}_{\epsilon}^{\overline{\delta}}).$

By Lemma 2.3, Lemma 2.4 and since *v* is decreasing, we have that, for $\epsilon >0$ sufficiently small and all $r\in ]0,{R}_{\epsilon}]$,

$v(r)>v({R}_{\epsilon})>v\left({\epsilon}^{-\frac{p-1}{2(p+\alpha +1)}}(1+\overline{\delta})\right)\u2a7eC{\epsilon}^{\frac{(N-2)(p-1)}{2(p+\alpha +1)}}.$(2.5)

On the other hand, for any $\epsilon \in ]0,\overline{\epsilon}]$ and every $r\in ]0,{R}_{\epsilon}]$,

${u}_{\epsilon}^{\overline{\delta}}(r)<{u}_{\epsilon}^{\overline{\delta}}(0)=\overline{\xi}{\epsilon}^{\frac{2+\alpha}{2(p+\alpha +1)}}.$(2.6)

By our assumption on α and taking ε small enough, from (2.5) and (2.6) we find that in $]0,{R}_{\epsilon}]$,

${u}_{\epsilon}^{\overline{\delta}}(r)<\overline{\xi}{\epsilon}^{\frac{2+\alpha}{2(p+\alpha +1)}}<C{\epsilon}^{\frac{(N-2)(p-1)}{2(p+\alpha +1)}}<v(r),$

contradicting Lemma 2.5.
∎

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