1 Introduction
Let
$\Omega $
be a smooth bounded domain in
$\mathbb {R}^{N}$
,
$N\geq 1$
, let
$q\in ( 0,1) $
and let
$m\in C( \overline {\Omega }) $
be a function that changes sign in
$\Omega $
. Let us consider the problem
where
$\partial _{\nu }$
is the exterior normal derivative. To simplify the presentation of Theorem 1.1, we assume that the set
has a smooth boundary and a finite number of connected components.
Let us also introduce the problems
An overview of the main results and the existing literature concerning
$( \mathcal {P}_{D}) $
and
$( \mathcal {P}_{N}) $
can be found in the review paper [Reference Kaufmann, Ramos Quoirin and Umezu15] and its references. Of course,
$( \mathcal {P}) $
is closely related to these problems. In fact, to the best of our knowledge, the only known results for
$(\mathcal {P}) $
come from
$( \mathcal {P}_{D}) $
and
$( \mathcal {P}_{N}) $
. We compile these results in the next theorem. As usual, we set
$m^{\pm }:=\max (\pm m,0)$
.
Theorem 1.1. The following assertions hold.
-
(i) If $( \mathcal {P}) $
has a positive solution, then
$\int _{\Omega }m<0$
; and if
$\Omega ^{+}$
is connected and
$(\mathcal {P}) $
has a solution, then
$\int _{\Omega }m<0$
[Reference Bandle, Pozio and Tesei2, Corollaries 2.1 and 2.2]. -
(ii) The problem $( \mathcal {P}) $
has at most one solution u such that
$u>0$
in
$\Omega ^{+}$
; and if
$\Omega ^{+}$
is connected, then
$(\mathcal {P}) $
has at most one solution [Reference Bandle, Pozio and Tesei1, Theorem 2.1]. -
(iii) There exists some $q_{0}\in [ 0,1) $
such that
$(\mathcal {P}) $
does not have a solution for
$q\in ( q_{0},1) $
[Reference Kaufmann, Ramos Quoirin and Umezu13, Theorem 1.9] -
(iv) Under further conditions on $\Omega ^{+}$
(which, in particular, imply
$m\leq 0$
in the set
${\{ x\in \Omega : \mathrm {dist}( x,\partial \Omega ) <\delta \}}$
for some
$\delta>0$
), there exists
$\mu _{0}>0$
such that for all
${\mu \geq \mu _{0}}$
,
$( \mathcal {P}) $
has a solution with
$\mu m^{-}$
in place of
$m^{-}$
[Reference Bobkov and Ramos Quoirin3, Proposition 4.2].
Let us also mention that, for any fixed
$q\in ( 0,1) $
, one can construct a weight
$m=m_{q}$
such that
$( \mathcal {P}) $
has a positive solution (see [Reference Kaufmann, Ramos Quoirin and Umezu13, Proposition 2.1]). In particular,
$q_{0}$
in Theorem 1.1(iii) can be arbitrarily close to
$1$
.
Let us add that the existence and qualitative properties of nonnegative (and nontrivial) solutions of the problem
where
$\lambda $
is a real parameter, have been recently studied in [Reference Díaz, Hernández and Ilyasov10]. When the solution is strictly positive in the interior while simultaneously satisfying both Dirichlet and Neumann boundary conditions, it is referred to as a flat solution, while if the solution is nonnegative in the interior, it is called a compact support solution, since its support does not necessarily coincide with
$\overline {\Omega }$
. For further articles considering flat and compact support solutions for similar problems, we refer to [Reference Díaz5–Reference Díaz, Hernández and Ilyasov9] and the references therein.
Our aim in this paper is to study the existence and nonexistence of radial solutions for
$( \mathcal {P}) $
when
$\Omega $
is a ball in
$\mathbb {R}^{N}$
,
$N\geq 1$
. Concerning this problem, employing energy methods, in Theorem 2.1, we provide a necessary condition for the existence of solutions for
$( \mathcal {P}) $
, while in Theorem 2.3, we present a necessary condition for the existence of positive solutions for
$( \mathcal {P}) $
. Also, in Theorem 2.5 (see also Remark 2.6 and Corollary 2.7), we give sufficient conditions for the existence of solutions for
$( \mathcal {P}) $
, relying on a sub- and supersolution approach.
2 Main results
By a solution of
$( \mathcal {P}) $
, we mean a strong solution, that is, a function
$u\in W^{2,r}( \Omega ) $
with
$r>N$
(and, hence,
$u\in C^{1}(\overline {\Omega })$
) that satisfies the equation for the weak derivatives and the boundary conditions in the usual sense. If
$u>0$
in
$\Omega $
, we call it a positive solution. Given
$0<r_{0}<R$
, we write
If f is a radial function, we write (with a slight abuse of notation)
$f( x) =f( \vert x\vert ) :=f(r) $
.
The first result is a necessary condition for the existence of solutions for
$(\mathcal {P})$
, which strengthens Theorem 1.1(i) and (iii). The idea for the proof is inspired by some of the computations in the proofs of [Reference Godoy and Kaufmann12, Theorems 2.7 and 3.4].
Theorem 2.1. Let
$\Omega :=B_{R}$
and let
$m\in C( \overline {\Omega }) $
be a radial function. Assume that there exists
$r_{0}>0$
such that:
-
• $m\geq 0$
in
$B_{r_{0}}$
; -
• $m\leq 0$
in
$A_{r_{0},R}$
; -
• $r\to m(r)$
is differentiable and nonincreasing in
$( r_{0},R) $
.
If
$( \mathcal {P}) $
has a solution, then
Remark 2.2.
-
(i) Note that (2.1) says that if $(\mathcal {P})$
has a solution, then (2.2) $$ \begin{align} q<\frac{-\int_{B_{R}}m}{\int_{B_{R}}\vert m\vert }. \end{align} $$
-
(ii) If $m\leq 0$
in
$B_{r_{0}}$
and
$m\geq 0$
in
$A_{r_{0},R}$
, then
$( \mathcal {P}) $
does not have a solution. Indeed, if u is solution, from Hopf’s lemma for strong solutions [Reference López-Gómez16, Theorem 7.11] applied in
$A_{r_{0},R}$
, we deduce that
$u\equiv 0$
in
$A_{r_{0},R}$
. Hence, by the strong maximum principle for strong solutions [Reference López-Gómez16, Theorem 7.10],
$u\equiv 0$
in
$B_{R}$
. -
(iii) In [Reference Kaufmann, Ramos Quoirin and Umezu14, Theorem 1.6], it is proved that if the first three conditions of Theorem 2.1 hold but the inequality in (2.1) is the opposite, then there exists a solution u of $(\mathcal {P}_{N})$
with
$u>0$
in
$\Omega $
. Theorem 2.1 says that u cannot be a solution of
$( \mathcal {P}) $
.
Proof. Suppose
$( \mathcal {P}) $
has a solution u. From the strong maximum principle,
$u>0$
in
$B_{r_{0}}$
and so, in particular,
$u>0$
in
$\Omega ^{+}$
. Since
$u( Sx) $
is also a solution if S is an isometry of
$\mathbb {R}^{N}$
, we infer from the first assertion in Theorem 1.1(ii) that u is radial. Let us first assume that
$u>0$
in
$B_{R}$
. Now, in
$( 0,r_{0}) $
, the equation reads as
$-(r^{N-1}u^{\prime })^{\prime }=r^{N-1}m^{+}(r)u^{q}$
. Thus,
$(r^{N-1} u^{\prime })^{\prime }\leq 0$
for all such r. Since
$u^{\prime }(0)=0$
, integrating in
$(0,r) $
gives
$r^{N-1}u^{\prime }(r)\leq 0$
and so
$u^{\prime }(r)\leq 0$
(and
$u'$
is not identically
$0$
). In other words,
$u(r)$
is nonincreasing. Thus,
and so
On the other hand, in
$( r_{0},R) $
, the equation can be written as
$r^{N-1}m^{-}(r)=(r^{N-1}u^{\prime }(r))^{\prime }/u(r)^{q}$
. Integrating by parts in
$( r_{0},R) $
gives
We claim that
$\lim _{r\rightarrow R}{r^{N-1}u^{\prime }(r)}/{u(r)^{q}}=0$
. Indeed, integrating in
$( r,R) $
with
$r>r_{0}$
, since
$u^{\prime }( R) =0$
,
Hence,
$u^{\prime }(r)\leq 0$
and
$u(r)$
is also nonincreasing in
$( r_{0},R) $
. Therefore,
$u(s)\leq u(r)$
for
$s\in ( r,R) $
, and then
Dividing by
$u(r)^{q}$
, the claim follows. Moreover, from (2.4),
For
$r\in [ r_{0},R] $
, we now define the energy functional
We have
$E(R)=0$
(because
$u( R) =u^{\prime }( R) =0$
) and
Since
$u^{\prime \prime }=m^{-}(r)u^{q}-({(N-1)}/{r})u^{\prime }$
in
$( r_{0},R) $
and
$r\rightarrow m(r)$
is nonincreasing in
$( r_{0},R) $
,
Hence,
$E(r)\geq 0$
for all
$r\in \lbrack r_{0},R]$
. Thus,
and so multiplying by
$r^{N-1}$
and integrating in
$( r_{0},R) $
,
Now, applying this bound to (2.5),
Finally, recalling (2.3), we see that
and this part of the theorem is proved.
To conclude the proof, suppose there exists a solution u of
$( \mathcal {P}) $
that vanishes for some r. Let
$r^{\ast }$
be the smallest r such that
$u( r) =0$
. From the strong maximum principle,
$r^{\ast }>r_{0}$
. In addition, u is a positive solution of
$( \mathcal {P}) $
in
$B_{r^{\ast }}$
. Therefore, by the first part of the proof,
When
$N=1$
, we can also give a lower bound on q and
$\int _{B_{r_{0}}}m^{+}$
for the existence of positive solutions for
$( \mathcal {P}) $
.
Theorem 2.3. Let
$\Omega :=( -R,R) $
and
$m\in C^{1}( \overline {\Omega }) $
be an even function. Assume that there exists
$r_{0}>0$
such that:
-
• $m\geq 0$
in
$B_{r_{0}}$
; -
• $m\leq 0$
in
$A_{r_{0},R}$
; -
• $x\to m(x)$
is nonincreasing in
$( 0,R) $
.
Let
If
$( \mathcal {P}) $
has a positive solution, then
Remark 2.4.
-
(i) Let us note that $C_{m^{-}}<R\Vert m^{-}\Vert _{\infty }$
and that also
$\int _{0}^{R}m+2C_{m^{-}}>0$
by the final paragraph of the following proof. In addition, note that (2.8) can be written as (2.9) $$ \begin{align} \int_{0}^{r_{0}}m^{+}>\int_{r_{0}}^{R}m^{-}-\frac{2q}{1-q}C_{m^{-}}. \end{align} $$
-
(ii) In view of Theorem 2.5, we cannot expect a similar result for solutions of $( \mathcal {P}) $
that may vanish in
$\Omega $
. -
(iii) Under the assumptions of the above theorem, recalling (2.2), we see that if $( \mathcal {P}) $
has a positive solution, then $$ \begin{align*} \frac{-\int_{0}^{R}m}{\int_{0}^{R}m+2C_{m^{-}}}<q<\frac{-\int_{0}^{R}m} {\int_{0}^{R}\vert m\vert }, \end{align*} $$or equivalently,
$$ \begin{align*} \int_{r_{0}}^{R}m^{-}-\frac{2q}{1-q}C_{m^{-}}<\int_{0}^{r_{0}}m^{+}<\frac {1-q}{1+q}\int_{r_{0}}^{R}m^{-}. \end{align*} $$
Proof. Let u be a positive solution of
$( \mathcal {P}) $
. Then, u is even, and multiplying by
$u^{-q}$
and integrating by parts,
where we have used
$u^{\prime }( 0) =0$
(because u is even) and
$\lim _{x\rightarrow R}{u^{\prime }( x) }/{u(x) ^{q}}=0$
(this follows exactly as in the proof of Theorem 2.1). We now define
$E( x) $
also as in the proof of Theorem 2.1, but for all
$x\in ( 0,R) $
, that is,
As in (2.6) (with
$N=1$
), we have
$E^{\prime }(x)=({m^{\prime }(x)}/{(1+q)})u(x)^{1+q}$
for all x and so, since
$E( R) =0$
, integrating gives
$E(x)=-\int _{x}^{R}({m^{\prime }(s)}/{(1+q)})u(s)^{1+q}\,ds$
and then
Therefore, since u and m are nonincreasing,
We claim that
Indeed, note first that since
$u^{\prime }( R) =0$
,
$u^{\prime }( z) =-\int _{z}^{R}u^{\prime \prime }( t) \,dt$
; and since
$u( R) =0$
,
In addition, for
$s\leq t\leq R$
, let
$f( t) ={(t-s)}/{(t-x)}$
. Then, f is increasing and so
$f( t) <f( R) $
if
$t\in [ s,R) $
, that is,
Now, using (2.13) and (2.14), since
$u^{\prime \prime }\geq 0$
in
$( r_{0},R) $
, for
$r_{0}\leq x<s<R$
,
and the claim is proved. We now divide our argument into two cases.
Case 1:
$x\in [ r_{0},R) $
. Recalling that m is nonincreasing, from (2.11) and (2.12), and integrating by parts,
Case 2:
$x\in ( 0,r_{0}) $
. Using (2.11), from the computation made above and arguing as in Case 1,
Finally, from (2.10), (2.15) and (2.16),
and so
$q( 2C_{m^{-}}+\int _{0}^{R}m)>-\int _{0}^{R}m>0$
. Thus, (2.8) is proved.
We now give a sufficient condition for the existence of a solution for
$( \mathcal {P}) $
.
Theorem 2.5. Let
$\Omega :=B_{R}$
and let
$m\in C( \overline {\Omega }) $
be a radial function that changes sign in
$\Omega $
. Assume that there exists
$r_{0}>0$
such that
$0\not \equiv m\leq 0$
in
$A_{r_{0},R}$
. For
$r\in ( 0,R) $
and
$\rho \in ( r_{0},R) $
, let
If there exists some
$\overline {r}\in ( r_{0},R) $
such that
then
$( \mathcal {P}) $
has a solution.
Remark 2.6. Since
$I_{1}(\overline {r})\leq \Vert m^{+}\Vert _{\infty }\overline {r}^{N}/N$
and
$I_{2}(\overline {r})\leq \Vert m^{+}\Vert _{\infty }\overline {r}^{2}/( 2N) $
, one can see that if
then (2.17) holds. Note that clearly the supremum is attained at some
$\overline {r}\in ( r_{0},R) $
.
Proof. Let
$\overline {r}$
be given by (2.17), and define
and
We note that
$\overline {u}( R) =\overline {u}^{\prime }( R) =\overline {u}^{\prime }(0)=0$
. In addition, by the choice of C,
and hence
$\overline {u}\in C^{1}( \overline {B}_{R}) $
. Furthermore, since
$\gamma -2=\gamma q$
and recalling the definition of k, for
$r\in (\overline {r},R)$
,
On the other hand, for
$r\in ( 0,\overline {r}) $
, we claim that
Indeed, we only have to check that
$C\geq v( 0) ^{q}$
, since the other inequalities are clearly true. Moreover, recalling (2.18),
Now, since
$\gamma =2/( 1-q) $
and
$\gamma -1=( 1+q) /( 1-q) $
,
and so (again using
$( \gamma -1) ( 1-q) =1+q $
),
Next, observe that
and thus, from (2.19), (2.20) and (2.21), it follows that
$C\geq v( 0) ^{q}$
if
and a quick look at (2.17) shows that (2.22) holds. So,
$\overline {u}$
is a positive weak supersolution of
$( \mathcal {P}) $
.
To provide an arbitrarily small weak subsolution of
$( \mathcal {P}) $
, we proceed as follows. Let
$A_{r_{1},r_{2}}\Subset \Omega ^{+}$
and
$0<\underline {m}:=\min _{\overline {A}_{r_{1},r_{2}}}m^{+}$
, and take
$\underline {u}:=\varepsilon \phi $
, with
$\varepsilon>0$
small and
$\phi>0$
in
$A_{r_{1},r_{2}}$
, where
$\phi $
is defined such that
$-\Delta \phi =\lambda _{1}( \underline {m}) \underline {m}\phi =\lambda _{1}\phi $
in
$A_{r_{1},r_{2}}$
and
$\phi =0$
on
$\partial A_{r_{1},r_{2}}$
, and extend
$\underline {u}$
by 0 to
$\overline {\Omega }$
. It is easy to check that if
$\varepsilon $
is small enough, then
$-\Delta \underline {u}\leq m( r) \underline {u}^{q}$
in
$A_{r_{1},r_{2}}$
(and
$-\Delta \underline {u}=0=m( r) \underline {u}^{q}$
in
$\Omega \setminus \overline {A}_{r_{1},r_{2}}$
), and choosing
$\varepsilon $
smaller if necessary, we also have
$\varepsilon \phi \leq \overline {u}$
in
$\overline {\Omega }$
. Moreover,
$\underline {u} {\vert _{A_{r_{1},r_{2}}} }\in W^{2,2}( A_{r_{1},r_{2}}) $
,
$\underline {u}{\vert _{\Omega \setminus \overline {A}_{r_{1},r_{2}}} }\in W^{2,2}( \Omega \setminus \overline {A}_{r_{1},r_{2}}) $
and
$\underline {u}\in C( \overline {\Omega }) \cap W_{0}^{1,2}( \Omega ) $
. Taking these facts into account, using the definition of
$\underline {u}$
and integrating by parts, we deduce that
$\underline {u}$
is a weak subsolution of
$( \mathcal {P}) $
, that is,
$\int _{\Omega }\langle \nabla \underline {u},\nabla w\rangle \leq \int _{\Omega }m\underline {u} ^{q}w$
for
$0\leq w\in W^{1,2}( \Omega ) $
. Indeed, let
$\nu _{A}$
denote the exterior unit normal to
$A_{r_{1},r_{2}}$
. Then,
$\partial _{\nu _{A}}\underline {u}<0$
on
$\partial A_{r_{1},r_{2}}$
. In addition, by the divergence theorem (see [Reference Cuesta and Takáč4, Lemma A.1]), we see that the usual Green’s formulae hold here. So, recalling that
$w\geq 0$
,
as desired. Hence, the weak sub- and supersolutions method (see [Reference Du11, Theorem 4.9]) gives a weak solution u of
$( \mathcal {P}) $
, and by elliptic regularity,
$u\in W^{2,r}( \Omega ) \cap W_{0} ^{1,r}( \Omega ) $
for some
$r>N$
.
Extending the solution given by the above theorem by 0 gives the following corollary.
Corollary 2.7. Suppose
$0\in \Omega $
and let
$m\in C( \overline {\Omega }) $
. If there exists some
$B_{R}\subset \Omega $
such that m fulfils the hypothesis of Theorem 2.5 in
$B_{R}$
, then
$( \mathcal {P}) $
has a solution.
In addition, as a direct consequence of Theorems 2.3 and 2.5, we get the existence of a solution of
$( \mathcal {P}) $
that must vanish somewhere in
$\Omega $
.
Corollary 2.8. Let
$\Omega :=( -R,R) $
and
$m\in C^{1}( \overline {\Omega }) $
be an even function.Assume that there exists
$r_{0}>0$
such that:
-
• $m\geq 0$
in
$B_{r_{0}}$
; -
• $m\leq 0$
in
$A_{r_{0},R}$
; -
• $x\rightarrow m(x)$
is nonincreasing in
$( 0,R) $
.
Let
$C_{m^{-}}$
be given by (2.7) and let
If m satisfies (2.17) and
then there exists an even solution u of
$( \mathcal {P}) $
such that
$u>0$
in
$( -r^{\ast },r^{\ast }) $
and
$u\equiv 0$
in
$[ r^{\ast },R] \cup [ -R,-r^{\ast }] $
for some
$r^{\ast } \in ( r_{0},R) $
.
Proof. The existence of u follows directly from Theorem 2.5. Also, note that (2.23) implies that the right member of (2.24) is positive. Now, recalling Theorem 2.3 (and 2.9), from (2.24), u must vanish at some
$r^{\ast }\in ( r_{0},R) $
, and it is easy to see that then it vanishes in
$[ r^{\ast },R] \cup [ -R,-r^{\ast }] $
using a convexity argument.
