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BY 4.0 license Open Access Published by De Gruyter September 23, 2017

Structure Results for Semilinear Elliptic Equations with Hardy Potentials

Matteo Franca ORCID logo and Maurizio Garrione ORCID logo EMAIL logo

Abstract

We prove structure results for the radial solutions of the semilinear problem

Δ u + λ ( | x | ) | x | 2 u + f ( u ( x ) , | x | ) = 0 ,

where λ is a function and f is superlinear in the u-variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearities f having different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in 3, for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.

1 Introduction

In this paper, we discuss structure results for positive solutions of the following semilinear problem with Hardy potential in the whole of n:

(1.1) Δ u + λ ( | x | ) | x | 2 u + f ( u ( x ) , | x | ) = 0 ,

where xn, n>2 and f is a suitable C1 function, satisfying f(0,r)=0, which is superlinear in u. Since we just deal with radial solutions, we set r=|x| and we consider the equivalent singular ODE

(1.2) u ′′ + n - 1 r u + λ ( r ) r 2 u + f ( u , r ) = 0 , r ( 0 , + ) ,

where the symbol denotes the differentiation with respect to r and, with a little abuse in the notation, we have set u(r)=u(x) for |x|=r. Our plan is to perform a classical change of variable, known as Fowler transformation [11], which allows us to apply phase plane analysis.

It has to be mentioned that equation (1.1) has attracted a lot of interest in recent years (see, e.g., [2, 15, 19, 24]), and several results regarding the asymptotic behavior of the solutions have been produced. Far from being complete, we mention as possible examples the papers [6, 9, 10, 23, 24] and the references therein for the subcritical and critical cases, mainly with variational techniques. The recent contributions [15, 19], whose techniques are more in line with the present paper, deal with the supercritical case under the assumption that λ is constant.

We begin our discussion from the very special case where the auxiliary system is autonomous, so that the analysis and the presentation are simpler. As a model case, we deal with the equation

(1.3) u ′′ + n - 1 r u + λ r 2 u + c r δ u | u | q - 2 = 0 ,

where q>2, δ>-2 and λ is a real constant. We introduce the parameter

l = 2 ( q + δ ) 2 + δ

to take into account the interplay between the values of q and δ, which both influence the criticality of the equation. Notice that if δ=0, we get l=q. Our purpose is to give structure results and to study the asymptotic behavior of the positive solutions as l overcomes several critical values to be introduced below. We will also mention some complementary results related to solutions which vanish at some point.

First, we need to define the following values which are relevant for the asymptotic behavior of the solutions, being the “natural” exponents which control their growth:

(1.4) α l = 2 l - 2 , κ = ( n - 2 ) - ( n - 2 ) 2 - 4 λ 2 .

With these definitions, we distinguish the solutions according to their asymptotic behavior: positive solutions may be

  1. regular if limr0u(r)rκ=d>0 (in this case, we set u(r)=uR(r;d));

  2. singular if limr0u(r)rκ=+.

Further, they may be

  1. fast decaying (f.d.) if limr+u(r)rn-2-κ=L>0 (in this case, we set u(r)=u(r;L));

  2. slow decaying (s.d.) if limr+u(r)rn-2-κ=+.

In fact, the leading term of the asymptotic behavior of singular and slow-decay solutions can be computed; see Theorem 1.2 below. We emphasize that if λ=0, then κ=0 and regular solutions are in fact smooth, positive and bounded. If λ<0, then κ<0, so u(r) is equal to 0 for r=0 and it is monotone increasing for small r. If 0<λ<(n-2)2/4, the term regular is abused since limr0u(r)=+; however in this case no bounded solutions may exist.

Moreover, we divide (strictly) positive solutions into ground states (G.S.) and singular ground states (S.G.S.):

  1. by G.S. we mean a regular solution u(r) such that

    (1.5) lim r + u ( r ) = 0 ;

  2. by S.G.S. we mean a singular solution u(r) which satisfies (1.5).

As mentioned, we will also make some considerations about oscillatory solutions, namely solutions u(r) such that there exists R>0 for which u(r)>0 for any 0r<R and u(R)=0. These may be seen as solutions of the Dirichlet problem in the ball of radius R.

In order to give a classification of all the positive solutions of (1.1), we preliminarily observe that equation (1.3) is characterized by the presence of several critical values 2<2*(λ)<2*<I(λ), where

2 * ( λ ) = 2 n + ( n - 2 ) 2 - 4 λ n - 2 + ( n - 2 ) 2 - 4 λ , 2 * = 2 n n - 2 ,
I ( λ ) = { + if  λ 0 , 2 n - ( n - 2 ) 2 - 4 λ n - 2 - ( n - 2 ) 2 - 4 λ if  0 < λ < ( n - 2 ) 2 4 .

The value 2* is the Sobolev critical exponent, and is independent of λ; the value 2*(0) is the Serrin exponent and is related to the continuity of the trace operator. An interesting fact in our approach is that these quantities, which are relevant from a functional point of view, find an easy interpretation from a dynamical system perspective.

Now we are ready to state in a simpler context the results which are the object of this paper; we begin with a statement concerning the regularity of the solutions.

Theorem 1.1.

Consider (1.3), where we assume that λ<(n-2)2/4.

  1. If 2 < l 2 * ( λ ) , then regular and singular solutions are oscillatory. There are uncountably many singular solutions and they are oscillatory. No positive solutions exist.

  2. If 2 * ( λ ) < l < 2 * , then all the regular solutions are oscillatory. There is a unique S.G.S. with slow decay. There are uncountably many S.G.S. with fast decay.

  3. If l = 2 * , then all the regular solutions are G.S. with fast decay. There are uncountably many S.G.S. with slow decay.

  4. If 2 * < l < I ( λ ) , then all the regular solutions are G.S. with slow decay. There is a unique S.G.S. with slow decay.

  5. If lI(λ), then fast and slow-decay solutions are oscillatory and there are no solutions which are positive in a neighborhood of r=0.

If λ0, then all G.S. and S.G.S. are decreasing, while if λ<0, then the G.S. are increasing, reach a positive maximum and then converge to 0 monotonically.

As for the asymptotic behavior, we introduce the notation

(1.6) P x = ( α l ( n - 2 - α l ) - λ c ) 1 / ( q - 2 )

and state the following result. Henceforth, we will denote by uR the regular solutions, and by u the fast-decay ones, while the symbols used for singular and slow-decay solutions might change according to the statement.

Theorem 1.2.

Consider (1.3), where we assume that λ<(n-2)2/4. If 2<l<I(λ), there is a unique regular solution uR(r,d) for any d>0. Further, the following statements hold:

  1. If 2 < l < 2 * ( λ ) , then all the singular solutions u ( r ) are such that

    lim r 0 u ( r ) r n - 2 - κ = A ,

    where A>0 depends on u.

  2. If l = 2 * ( λ ) , then all the singular solutions u ( r ) are such that

    lim r 0 u ( r ) r n - 2 - κ | ln ( r ) | 1 / ( q - 2 ) = B > 0 ,

    where

    B = [ n - 2 - κ c ( q - 2 ) ] 1 / ( q - 2 ) ,

    so the limit is independent of u.

  3. If 2 * ( λ ) < l < I ( λ ) , then all the singular solutions u ( r ) are such that u ( r ) r α l is uniformly positive and bounded as r 0 ; if we also assume l 2 * , then

    lim r 0 u ( r ) r α l = P x ,

    so the limit is independent of u.

We have a dual result for slow-decay solutions.

Theorem 1.3.

Consider (1.3), where we assume that λ<(n-2)2/4. If l>2*(λ), there is a unique fast-decay solution uF(r,L) for any L>0. Further, the following statements hold:

  1. If 2 * ( λ ) < l < I ( λ ) , then all the slow-decay solutions u ( r ) are such that u ( r ) r α l is uniformly positive and bounded as r + ; if we also assume l 2 * , then

    lim r + u ( r ) r α l = P x ,

    so the limit is independent of u.

  2. If l = I ( λ ) , then all the slow-decay solutions u ( r ) are such that

    lim r + u ( r ) r κ | ln ( r ) | 1 / ( q - 2 ) = B > 0 ,

    where

    B = [ n - 2 - κ c ( q - 2 ) ] 1 / ( q - 2 )

    as above, so the limit is independent of u.

  3. If l > I ( λ ) , then all the slow-decay solutions u ( r ) are such that

    lim r + u ( r ) r κ = E ,

    where E>0 depends on u.

Remark 1.4.

We think it is worthwhile to stress that if f(u,r)=u|u|q-2 and q is small enough (namely 2<q<2*(λ)), then singular solutions of (1.3) have the singularity induced by the principal solution of the Laplacian, i.e., u(r)Lr-(n-2) when λ=0 and u(r)Lr-(n-2)+κ when λ0, where L>0 depends on u (so, in the leading term of the asymptotic behavior, the homogeneity is preserved for small q larger than 2).

When q is large enough (namely q>2*(λ)), then the leading term in the asymptotic behavior of singular solutions (1.3) is ruled by the nonlinearity and is given by Pxr-αq, where αq=2/(q-2) does not depend on λ and Px depends on λ but not on u.

The value at which such a change takes place is precisely q=2*(λ), i.e., the value at which we have αq=(n-2-κ).

This is a general fact that is found even in the nonautonomous setting and in more general nonlinearities, as we will see below in Theorems 2.1 and 2.2.

The main purpose of this paper is to extend the results described in Theorems 1.1, 1.2 and 1.3 to a wider class of nonlinearities f and to a nonhomogenous context, as in (1.2). We emphasize that we need to ask λ(r) to be monotone, but it may vary, and f is just required to be subcritical or supercritical with respect to 2*: for the generalization of this notion we rely on the use of the Pohozaev function, which is a standard tool in this context [22]; see Section 3.3 below. We stress that in the critical case l=2*, assuming λ to be monotone, we find a new structure result for positive solutions analogous to the one found for λ=0, f(u,r)=k(r)u|u|2*-2 and k monotone (see Section 2). Namely, by setting f(u)=u|u|2*-2, if λ is monotone increasing, then we are in a subcritical situation, while if λ is monotone decreasing, we are in a supercritical situation.

The paper is divided into two main sections: In Section 2 we state our structure results for the general nonautonomous equation (1.1), and we list a series of corollaries, focusing in particular on the so-called Matukuma potentials and on nonlinearities of the kind f(u)=uq1-1+uq2-1. Section 3 is instead completely devoted to the proofs of our results, giving first a brief overview of the autonomous case and then moving to the general equation (1.2). In particular, we explicitly recall the Fowler transformation and the Pohozaev identity, and we show how they can be used to deal with the subcritical, critical and supercritical cases.

2 Statement of the Results

In this section, we illustrate our main results, which will be proved in a slightly more general setting in Section 3. With respect to the statements given in the Introduction, we consider a more general class of nonlinearities f and we replace the constant λ by a monotone function: we limit ourselves to the simpler setting where the system always exhibits either subcritical or supercritical behavior (with respect to 2*). In this context, the generalization of such a notion is related to the sign of the Pohozaev functions introduced below. The more difficult case of mixed Sobolev conditions will be the object of future investigations.

For simplicity, throughout this section we will assume that

(2.1) f ( u , r ) = k ( r ) u | u | q - 2 with  k ( r ) = { c r δ + o ( r δ ) for  r 0 , C r η + o ( r η ) for  r + ,

with c,C>0, δ,η>-2 and q>2. Under this assumption, we set

l u = 2 ( q + δ ) 2 + δ , l s = 2 ( q + η ) 2 + η ,

and define αlu, αls as in (1.4). Moreover, we set λ(0)=limr0λ(r), λ(+)=limr+λ(r) and we denote by κ(0), κ(+) the corresponding values defined in (1.4) (replacing λ with λ(0) and λ(+), respectively). We refer the reader to Remark 2.5 for possible extensions (which will be used to state our corollaries). Further, we consider the following assumption:

  1. Assume that f is as in (2.1) and let λ(0)<(n-2)2/4, λ(+)<(n-2)2/4. Moreover, assume that k(r) is C1 and positive for any r>0, and that there exists ϖ>0 such that

    lim r 0 r 1 - ϖ d d r [ k ( r ) r 2 l u - 2 ( l u - q ) - δ ] = 0 , lim r 0 r 1 - ϖ λ ( r ) = 0 ,
    lim r + r 1 + ϖ d d r [ k ( r ) r 2 l s - 2 ( l s - q ) - η ] = 0 , lim r + r 1 + ϖ λ ( r ) = 0 .

These weak assumptions on the weighted derivative of k are in fact technical and may be dropped paying the cost of some technicalities and lowering the precision of the asymptotic estimates, see Remark 3.6 in Section 3.

Now we are ready to state our results concerning the asymptotic behavior of the solutions both as r0 and as r+, showing that regular/singular and fast/slow-decay solutions follow the statements of Theorems 1.2 and 1.3, with the natural modifications: lu controls how regular the solutions are as r0, while ls is related to their decay properties as r+.

Notation.

We denote by Px->0 and Px+>0 the unique solutions of the equations (in the x-unknown)

c x q - 2 = α l u ( n - 2 - α l u ) - λ ( 0 ) ,
C x q - 2 = α l s ( n - 2 - α l s ) - λ ( + ) ,

respectively, generalizing the constant Px defined in (1.6).

Theorem 2.1.

Consider (1.2) and assume that f satisfies (F0). If 2<lu<I(λ(0)), then for any d>0 there exists a unique regular solution uR(r,d). Further, the following statements hold:

  1. If 2 < l u < 2 * ( λ ( 0 ) ) , then all the singular solutions u ( r ) are such that

    lim r 0 u ( r ) r n - 2 - κ ( λ ( 0 ) ) = A ¯ ,

    where A¯ is a positive constant depending on u.

  2. If l u = 2 * ( λ ( 0 ) ) , then all the singular solutions u ( r ) are such that

    lim r 0 u ( r ) r n - 2 - κ ( λ ( 0 ) ) | ln ( r ) | 1 / ( q - 2 ) = B ( 0 ) > 0 ,

    where

    B ( 0 ) = [ n - 2 - κ ( λ ( 0 ) ) c ( q - 2 ) ] 1 / ( q - 2 ) ,

    so the limit is independent of u.

  3. If 2 * ( λ ( 0 ) ) < l u < I ( λ ( 0 ) ) , then all the singular solutions u ( r ) are such that

    lim sup r 0 u ( r ) r α l u

    is positive and finite; if we also assume l u 2 * , then

    lim r 0 u ( r ) r α l u = P x - ,

    so the limit is independent of u.

Theorem 2.2.

Consider (1.2) and assume that f satisfies (F0). If ls>2*(λ(+)), then for any L>0 there exists a unique fast-decay solution uF(r,L). Further, the following statements hold:

  1. If 2 * ( λ ( + ) ) < l s < I ( λ ( + ) ) , then all the slow-decay solutions u ( r ) are such that

    lim sup r + u ( r ) r α l s

    is positive and finite; if we also assume l s 2 * , then

    lim r + u ( r ) r α l s = P x + ,

    so the limit is independent of u.

  2. If l s = I ( λ ( + ) ) , then all the slow-decay solutions u ( r ) are such that

    lim r + u ( r ) r κ ( λ ( + ) ) | ln ( r ) | 1 / ( q - 2 ) = B ( + ) ,

    where

    B ( + ) = [ n - 2 - κ ( λ ( + ) ) C ( q - 2 ) ] 1 / ( q - 2 ) ,

    so the limit is independent of u.

  3. If l s > I ( λ ( + ) ) , then all the slow-decay solutions u ( r ) are such that

    lim r + u ( r ) r κ ( λ ( + ) ) = E ¯ ,

    where E¯ is a positive constant depending on u.

We now state the analog of Theorem 1.1, splitting the statement into the subcritical and the supercritical case, separately. For this purpose, we introduce the following assumptions, which amount to ask for (1.2) to be respectively subcritical or supercritical with respect to 2*.

  1. The functions λ(r) and J(r)=k(r)r((n-2)/2)(2*-q) are increasing for any r>0, and λ(R)+J(R)>0 for some R>0.

  2. The functions λ(r) and J(r)=k(r)r((n-2)/2)(2*-q) are decreasing for any r>0, and λ(R)+J(R)<0 for some R>0.

Theorem 2.3 (Subcritical Case).

Assume that f satisfies (F0) and (H+), and let 2<lu2*, 2<ls2*. Then there exist infinitely many regular solutions and they are oscillatory, and there exist infinitely many singular solutions. Further,

  1. if 2 < l s 2 * ( λ ( + ) ) , then no positive solutions exist for large r : singular solutions are oscillatory;

  2. if 2*(λ(+))<ls2*, then there are infinitely many S.G.S. with fast decay, and there is at least one S.G.S. with slow decay, which is unique if 2*(λ(+))<ls<2*.

If λ0, then the S.G.S. are strictly decreasing.

Theorem 2.4 (Supercritical Case).

Assume that f satisfies (F0) and (H-), and let lu2*, ls2*. Then there exist infinitely many fast-decay solutions and they are oscillatory. Further,

  1. if 2 * l u < I ( λ ( 0 ) ) , then all the regular solutions are G.S. with slow decay (they are infinitely many). Moreover, there is at least an S.G.S. with slow decay, and it is unique if 2 * < l u < I ( λ ( 0 ) ) ;

  2. if l u I ( λ ( 0 ) ) , then no positive solutions exist for small r and there are infinitely many slow-decay solutions, which are oscillatory.

If λ0, then the G.S. are strictly decreasing.

Remark 2.5.

We can also deal with more general nonlinearities having the form

f ( u , r ) = i = 1 j k i ( r ) u | u | q i - 2 with  k i ( r ) = { c i r δ i + o ( r δ i ) for  r 0 , C i r η i + o ( r η i ) for  r + ,

where ci>0, Ci>0 and δi,ηi>-2, qi>2 for any i. In this case, we set

l u = max { 2 ( q i + δ i ) 2 + δ i | i = 1 , , j } , l s = min { 2 ( q i + η i ) 2 + η i | i = 1 , , j } .

Notice that when δi=ηi=0 for any i, and q1<q2<<qj, we have ls=q1 and lu=qj. Moreover, we assume (H+) and (H-) for every i. The statements then hold with the natural changes.

2.1 Corollaries

In this subsection, we give some applications of our result which we consider possibly relevant. In particular, we give a corollary which is new even in the case when λ(r)0, and a new structure result which appears just in the case when λ(r) is not constant.

As we are going to see, a positive aspect of our approach is that it is particularly useful to deal with singular solutions, and to detect the asymptotic behavior when singular and slow-decay solutions have their leading terms behaving like different powers. In particular, this is the case of the so-called Matukuma potentials (and of their generalizations), when f in (1.2) is given by one of the following functions:

f 1 ( u , r ) = u | u | q - 2 1 + r a , f 2 ( u , r ) = r b u | u | q - 2 1 + r a + b ,

with q>2, 0a<2 and b0. We recall that (1.2) with λ=0 and f=f1 is known as the Matukuma equation and finds application in the study of globular clusters in astrophysics (see, e.g., [8, 18, 20] and the references therein). By taking also into account Remark 2.5, our approach also allows us to include in a unique class the following potentials:

f 3 ( u , r ) = u | u | q 1 - 2 + u | u | q 2 - 2 , f 4 ( u , r ) = ( r a + r b ) u | u | q - 2 ,

with 2<q1<q2, q>2 and a>b>-2.

Remark 2.6.

When f=f1, we have lu=q<ls=2(q-a)2-a, while if f=f2, then lu=2(q+b)2+b and ls=2(q-a)2-a. Similarly, when f=f3, we have lu=q2>ls=q1, while if f=f4, then

l u = 2 ( q + b ) 2 + b > l s = 2 ( q + a ) 2 + a .

For the Matukuma potentials we have the following consequences of Theorem 2.3, which are not found in the literature.

Corollary 2.7.

Consider (1.2) with λ(r)0. Assume either

f = f 1 , l u = q 2 * ( 0 ) , 2 * ( 0 ) < l s = 2 ( q - a ) 2 - a < 2 * ,

or

f = f 2 , l u = 2 ( q + b ) 2 + b 2 * ( 0 ) , 2 * ( 0 ) < l s = 2 ( q - a ) 2 - a < 2 * .

Then there are uncountably many monotone decreasing S.G.S. uF(r) with fast decay, and a unique S.G.S. uS(r) with slow decay. Moreover,

u ( r ) C ¯ r 2 - n , 𝑤ℎ𝑖𝑙𝑒 u S ( r ) P x + r - α l s    as  r + ,

and uF(r),uS(r)c¯r2-n as r0 if q<2*(0), while if q=2*(0), then uF(r),uS(r)Br2-n|ln(r)|-1/(q-2) as r0, where C¯,c¯>0 depend on the solutions (while Px+,B>0 do not).

We also have the following consequences of Theorems 2.1 and 2.3, which go in a different direction and, as far as we are aware, have not appeared previously.

Corollary 2.8.

Consider (1.2) where λ(r)0. Assume either

f = f 3 , 2 < l s = q 1 2 * ( 0 ) < q 2 = l u 2 * ,

or

f = f 4 , l s = 2 q + a 2 + a 2 * ( 0 ) < l u = 2 q + b 2 + b 2 * .

Then there are uncountably many singular solutions u(r) which are oscillatory as r+: no positive solutions exist. Further,

u ( r ) r α l u P x - > 0

for r0 if lu<2*, while if lu=2*, then

u ( r ) r α l u

is uniformly positive and bounded as r0.

Clearly, Theorem 2.3 can also be used to give new contributions when λ(r)0. The dual result obtained via Kelvin inversion (see, e.g., [8]) appears just if λ>0; we state it in the easier case when λ(r)λ(0,(n-2)2/4).

Corollary 2.9.

Consider (1.2) where λ(r)λ(0,(n-2)2/4). Assume either

f = f 1 , 2 * l u = q < I ( λ ) l s = 2 q - a 2 - a ,

or

f = f 2 , 2 * l u = 2 q + b 2 + b < I ( λ ) l s = 2 q - a 2 - a .

Then all the regular solutions uR(r,d) are monotone decreasing G.S. with slow decay, and there is at least one singular solution uS(r) which is an S.G.S. with slow decay. Further,

u S ( r ) r α l u P x - > 0

as r0, and if lu>2*, then uS(r) is the unique singular solution, while if lu=2*, uniqueness is not ensured.

Corollary 2.10.

Consider (1.2) where λ(r)λ(0,(n-2)2/4). Assume either

f = f 3 , 2 * l s = q 1 < I ( λ ) l u = q 2 ,

or

f = f 4 , 2 * l s = 2 q + a 2 + a < I ( λ ) l u = 2 q + b 2 + b 2 * .

Then there are uncountably many fast-decay solutions uF(r,L), and at least one slow-decay solution uS(r), but they are all oscillatory as r0: no positive solutions exist. Further,

u S ( r ) r α l s P x + > 0

as r+, and uS(r) is the unique slow-decay solution if ls>2*, while if ls=2*, uniqueness is not ensured.

Corollaries 2.9 and 2.10 are straightforward consequences of Theorems 2.3 and 2.4.

Another interesting consequence of Theorems 2.3, 2.4 and of Theorems 2.1, 2.2 is given by Corollaries 2.13, 2.14 and 2.15 below, displaying a situation for positive solutions which appears just when λ is not a constant, e.g., for the equation

(2.2) u ′′ + n - 1 r u + λ ( r ) r 2 u + u | u | q - 2 = 0 .

To describe such results we need the following simple remarks referring to (1.6) (notice that here c=1).

Remark 2.11.

The expression Px(l)=[αl(n-2-αl)-λ]1/(q-2) behaves like a parabola in αl, so it is strictly increasing in l for l<2*, strictly decreasing for l>2* and reaches its maximum for l=2*, given by

P x ( 2 * ) = [ ( n - 2 ) 2 4 - λ ] 1 / ( q - 2 ) > 0 .

Observing moreover that Px is strictly decreasing in λ, we get the following remark.

Remark 2.12.

The values 2*(λ) and I(λ) are the smallest and the largest solutions of the equation in l given by Px(l)=0. They are respectively strictly increasing and decreasing in λ, and they are respectively smaller and larger than 2* for any λ<(n-2)2/4.

Corollary 2.13.

Consider (2.2) and assume q=2* and λ(r)<(n-2)2/4.

  1. If λ ( r ) is monotone increasing, then we are in a subcritical situation. All the regular solutions are oscillatory, there are uncountably many S.G.S. with fast decay and at least one S.G.S. with slow decay;

  2. If λ ( r ) is monotone decreasing, then we are in a supercritical situation. All the regular solutions are G.S. with slow decay, there are uncountably many fast-decay solutions and they are oscillatory, and there is at least one S.G.S. with slow decay.

Further, in both cases singular and slow-decay solutions are such that u(r)r(n-2)/2 is uniformly positive and bounded, respectively, for r1 if u(r) is singular, and for r1 if u(r) has slow decay.

Corollary 2.14.

Consider (2.2); assume λ(r)0, λ(+)<(n-2)2/4, and 2*(λ(0))<q2*(λ(+)). Then there are uncountably many regular solutions and uncountably many singular solutions, and they are all oscillatory. For large r, no positive solutions exist. Further, singular solutions u(r) are such that

u ( r ) r α q P x - as  r 0 .

We emphasize that if q>2*(λ(+))2*(λ(0)), then there are S.G.S. with fast and slow decay, which do not exist in the assumption of Corollary 2.14; if q2*(λ(0))2*(λ(+)), then regular and singular solutions are oscillatory, but the asymptotic behavior of singular solutions is of a different type. We have also the following dual result.

Corollary 2.15.

Consider (2.2); assume λ(r)0, 0<λ(0)<(n-2)2/4 and I(λ(0))q<I(λ(+)). Then no positive solutions exist for small r. There are uncountably many fast- and slow-decay solutions, and they are all oscillatory. Moreover, slow-decay solutions u(r) are such that

u ( r ) r α q P x + as  r + .

Again, notice that if I(λ(0))I(λ(+))<q, then the asymptotic behavior of slow-decay solutions is of a different type; if q<I(λ(0))I(λ(+)), then there exist regular solutions and they are G.S. with slow decay, and there is at least one S.G.S. with slow decay.

Remark 2.16.

Corollaries 2.13, 2.14 and 2.15 can be applied, e.g., to this possibly interesting example:

u ′′ + n - 1 r u + K 1 + r 2 u + u | u | q - 2 = 0 .

If q=2* and 0<K<(n-2)2/4, then hypothesis (+) of Corollary 2.13 is fulfilled, with λ(0)=0, λ(+)=K. S.G.S. are monotone decreasing. On the other hand, if q=2* and K<0, then hypothesis (-) of Corollary 2.13 is satisfied, with λ(0)=0, λ(+)=K.

Furthermore, if 2*(0)<q2*(K) and 0<K<(n-2)2/4, then we are in the hypotheses of Corollary 2.14; if q>I(0) and K<0 so that I(K)=+>q, then we are in the hypotheses of Corollary 2.15.

3 The Fowler Transformation: Proof of the Main Results

In this section, we introduce the auxiliary nonautonomous planar system which will be used for the study of equation (1.2), taking into account a general nonlinearity f(u,r). So we study the equation

(3.1) u ′′ + n - 1 r u + λ ( r ) r 2 u + f ( u , r ) = 0 .

For l>2 fixed, we apply the Fowler transformation

(3.2) { α l = 2 l - 2 and γ l = α l - ( n - 2 ) , r = e t , x l = u ( r ) r α l , y l = u ( r ) r α l + 1 ,

obtaining then the nonautonomous system

(3.3) ( x ˙ l y ˙ l ) = ( α l 1 - λ ( e t ) γ l ) ( x l y l ) - ( 0 g l ( x l , t ) ) ,

where

g l ( x , t ) = f ( x e - α l t , e t ) e ( α l + 2 ) t .

Notice that in this way the singularity has been removed. Our focus will be on asymptotically autonomous systems; see (Gu) and (Gs) below. Moreover, we introduce a further standard assumption ensuring forward and backward continuability of the trajectories of (3.3) for any t, which will be in force through the whole paper:

  1. Either f has the form f(u,r)=k(r)n(u), where the functions k and n are of class C1 and k(r)>0 for r>0, or there exists h:(0,+) such that

    0 x r f ( u , r ) 𝑑 u h ( r ) 0 x f ( u , r ) 𝑑 u for all  ( x , r ) × ( 0 , + ) .

The proof of the global continuability under this assumption is based on an appropriate energy estimate combined with the Gronwall Lemma (cf., among others, [4] and [21, Section 2.1]).

Notice that (3.3) can be recasted as the second-order ODE

(3.4) x ¨ l = ( α l + γ l ) x ˙ l - ( λ ( e t ) + α l γ l ) x l - g l ( x l , t ) ,

where the Hardy term is taken into account in the coefficient of xl. Equation (3.4) has been largely investigated and several results are available in literature.

Remark 3.1.

We can use the results obtained for (3.4) in the case λ(0)=0 to obtain information for the case λ(0)0, λ(et)<(n-2)2/4. In fact, if (xl(t),x˙l(t)) is a solution of (3.4) where λ(0)=0, α=αl and γ=γl, then it also solves (3.4) where λ(0)0, α=αl-κ(λ(0)) and γ=γl+κ(λ(0)). However, notice that in the former case for (3.3) we have yl(t)=x˙l(t)-αlxl(t), while in the latter we have yl(t)=x˙l(t)-(αl-κ(λ(0)))xl(t); so the presence of the Hardy term simply determines a deflection of the yl axis in (3.3). For this reason, a solution of (3.4) can be interpreted as a positive and decreasing solution for (3.1) when λ(0)=0, and, e.g., a positive, but non-monotone solution for (3.1) when λ(0)<0.

The results which we are going to present through the section are based on two main ingredients: we use invariant manifold theory to develop the asymptotic estimates, and the Pohozaev function to qualitatively sketch the phase portraits. In the following subsections, we briefly review these tools and we provide general statements and proofs of the structure results for equation (1.2). A further well-known tool we use is Kelvin inversion, which establishes a duality between subcritical and supercritical problems, see, e.g., [8, Section 2.1] and [14].

3.1 The Autonomous Case

Before discussing the nonautonomous system (3.3), we briefly review the situation for the autonomous case (i.e., λ constant), referring the reader to [14] for all the details. Moreover, for simplicity we choose to consider the nonlinearity f(u,r)=Krδu|u|q-2, where δ>-2 and q>2 are given. This means that the first-order system to be dealt with is given by

(3.5) ( x ˙ l y ˙ l ) = ( α l 1 - λ γ l ) ( x l y l ) - ( 0 K x l | x l | q - 2 ) ,

with λ being a constant.

Denoting by ϕl(t,τ;Q)=(xl(t,τ;Q),yl(t,τ;Q)) the trajectory of (3.5) which takes the value Q=(Qx,Qy) at t=τ, we notice that positive solutions correspond to xl(t,τ;Q)>0, while decreasing ones correspond to yl(t,τ;Q)<0. More correspondences can be drawn by looking at the nature of the origin as a critical point, which is determined by the eigenvalues of the linear part of (3.5), given by

Λ 1 = α l + γ l - ( n - 2 ) 2 - 4 λ 2 < Λ 2 = α l + γ l + ( n - 2 ) 2 - 4 λ 2 .

We stress that Λ1 and Λ2 are both real in view of the assumption λ<(n-2)2/4. This restriction arises naturally when dealing with positive solutions since if λ>(n-2)2/4, the origin is a focus (stable if l>2*, unstable if 2<l<2*) and positive solutions cannot exist. In particular, we can distinguish a strongly unstable manifold Mu and a strongly stable one Ms, corresponding respectively to Λ2 if Λ2>0, i.e., l<I(λ), and to Λ1 if Λ1<0, i.e., l>2*(λ). In fact, for 2*(λ)lI(λ), both the unstable and the stable manifold Mu and Ms, respectively, are nontrivial, while if l>I(λ) (resp. l<2*(λ)), then the origin is stable and Ms is a strongly stable manifold (resp. the origin is unstable and Mu is a strongly unstable manifold). Now, [14, Lemma 2.1] guarantees that

  1. QMulimt-ϕl(t,τ;Q)=Ou(r) is regular;

  2. QMslimt+ϕl(t,τ;Q)=Ou(r) is fast decaying.

Figure 1 
						Phase portrait of system (3.5), varying l>2{l>2}. Some level curves of the function H2*{H_{2^{*}}} defined in (3.11) are drawn. The origin is a global repeller for 2<l≤2*⁢(λ){2<l\leq 2_{*}(\lambda)} and we can identify the strongly unstable manifold Mu{M^{u}}. If 2*⁢(λ)<l<I⁢(λ){2_{*}(\lambda)<l<I(\lambda)}, the origin is a saddle and we can recognize an unstable manifold Mu{M^{u}} and a stable manifold Ms{M^{s}}. If l=2*{l=2^{*}}, the system is Hamiltonian and presents periodic orbits and two homoclinic trajectories, while if l≠2*{l\neq 2^{*}}, there exists two heteroclinic trajectories. Finally, if l≥I⁢(λ){l\geq I(\lambda)}, the origin is a global attractor and we can identify the strongly stable manifold Ms{M^{s}}. We have denoted by 𝑷+{{\boldsymbol{P^{+}}}} and 𝑷-{{\boldsymbol{P^{-}}}} the critical points P and -P{-P}, respectively.
Figure 1

Phase portrait of system (3.5), varying l>2. Some level curves of the function H2* defined in (3.11) are drawn. The origin is a global repeller for 2<l2*(λ) and we can identify the strongly unstable manifold Mu. If 2*(λ)<l<I(λ), the origin is a saddle and we can recognize an unstable manifold Mu and a stable manifold Ms. If l=2*, the system is Hamiltonian and presents periodic orbits and two homoclinic trajectories, while if l2*, there exists two heteroclinic trajectories. Finally, if lI(λ), the origin is a global attractor and we can identify the strongly stable manifold Ms. We have denoted by 𝑷+ and 𝑷- the critical points P and -P, respectively.

In Lemma 3.2, such correspondences will be extended to the more general nonautonomous context (see also [16], where this is proved by using some standard facts from dynamical systems theory; cf. [7]). Once these correspondences have been established, to draw a complete picture of all the possible phase portraits we need to use an energy function closely related to the Pohozaev identity; see [14]. Another possibility is to deduce the results for the autonomous context as a particular case of the ones obtained in the nonautonomous setting, stated in Sections 3.2 and 3.3. We refer the reader to Figure 1, where we highlight the different phase-portraits for the autonomous equation according to the value of l; see [14, Figure 1]. Theorems 1.1, 1.2 and 1.3 then follow almost directly.

We remark that we may have considered more general nonlinearities of the kind

(3.6) f ( u ) = i = 1 j c i r δ i u | u | q i - 2

(compare with Remark 2.5), with ci and δi=2(qi-l)/(l-2), i=1,,j, for a suitable l>2. Actually, the same arguments work also in this case. Even more, we can extend our discussion to general autonomous nonlinearities gl(x) satisfying the following assumption (for a proof in the λ=0 setting, see, e.g., [8]):

  1. There exists l>2 such that gl(x)x is decreasing for x<0 and increasing for x>0, and

    lim x 0 g l ( x ) x = 0 , lim x + g l ( x ) x = + .

Assumption (G0), together with λ<(n-2)2/4, ensures the existence and the uniqueness of the critical point P=(Px,Py) in the half-plane {x>0} whenever 2*(λ)<l<I(λ), and this is enough to extend all the results of the previous sections. Of course, it is satisfied by the function f appearing in (1.3), but also by more general nonlinearities of the form (3.6).

3.2 Nonautonomous Invariant Manifold Theory: Asymptotic Estimates

We now pass to consider (3.3), with the aim of obtaining asymptotic estimates of the solutions through the use of invariant manifold theory for nonautonomous systems. The approach we follow is simply to add a function of t as a variable, so as to get a 3-dimensional autonomous system: this is the simplest way of proceeding and it gives sharper results, requiring on the other hand stronger assumptions. Namely, we need the system to be asymptotically autonomous (see [16, 17]).

Without further mentioning, in the whole section we assume that λ(0)<(n-2)2/4, λ(+)<(n-2)2/4. Moreover, we introduce the following hypotheses:

  1. There is lu>2 such that

    lim t - g l u ( x , t ) = g l u ( x , - )

    uniformly for x>0 belonging to a compact interval, where glu(x,-)0 is a C1 function satisfying (G0). Moreover, there is ϖ>0 such that

    lim t - e - ϖ t t [ | g l u ( x , t ) | + | λ ( e t ) | ] = 0 .

  2. There is ls>2 such that

    lim t + g l s ( x , t ) = g l s ( x , + )

    uniformly for x>0 belonging to a compact interval, where gls(x,+)0 is a C1 function satisfying (G0). Moreover, there is ϖ>0 such that

    lim t + e ϖ t t [ | g l s ( x , t ) | + λ ( e t ) | ] = 0 .

Assumptions (Gu) and (Gs) ensure that the considered system is at least C1; they are needed to construct the unstable and the stable manifolds, so as to determine the asymptotic behavior of positive solutions of (1.2), thus allowing to extend the results to the nonautonomous case. We emphasize that if (Gu) (resp. (Gs)) holds, then lu (resp. ls) is uniquely defined: for instance, if f=u|u|p-2+u|u|q-2 and 2<p<q, then lu=q and ls=p.

The construction of unstable and stable manifolds via (Gu) and (Gs) is standard, and follows closely the ideas used in [16, 17] and exploited in [1, 13]. For the reader’s convenience, we now briefly review such a construction, which is based on standard facts of invariant manifold theory for autonomous systems; see, e.g., [7, Section 13].

Assume (Gu). Following [17], we set z=eϖt and we introduce the following 3-dimensional system, which is useful to determine the behavior of trajectories as t-:

(3.7) ( x ˙ l u y ˙ l u z ˙ ) = ( α l u 1 0 - λ ( z 1 / ϖ ) γ l u 0 0 0 ϖ ) ( x l u y l u z ) - ( 0 g l u ( x l u , ln ( z ) ϖ ) 0 ) .

Namely, system (3.7) is obtained by setting l=lu in (1.2) and adding the variable z. With respect to the case analyzed in the previous section, the 3-dimensional dynamics in (3.7) has a further unstable direction along z in the origin; notice that all the bounded trajectories converge to the plane z=0, and the dynamics in such a plane is reduced to the one of the autonomous system

(3.8) ( x ˙ l u y ˙ l u ) = 𝒜 ( l u ) ( x l u y l u ) - ( 0 g l u ( x l u , - ) ) ,

where

𝒜 ( l u ) = ( α l u 1 - λ ( 0 ) γ l u ) .

We see that the eigenvalues of 𝒜(lu) are given by

1 = γ l u + κ ( λ ( 0 ) ) < α l u - κ ( λ ( 0 ) ) = 2 ,

so that the following assertions hold:

  1. if 2<lu<2*(λ(0)), then 0<1<2;

  2. if 2*(λ(0))lu<I(λ(0)), then 10<2 (1=0 if and only if lu=2*(λ(0)));

  3. if luI(λ(0)), then 1<20 (2=0 if and only if lu=I(λ(0))).

Hence, for system (3.7) the origin has an unstable manifold Wu which is 3-dimensional if 2<lu<2*(λ(0)) (O being an unstable node for (3.8)), 2-dimensional if 2*(λ(0))lu<I(λ(0)) (in this case, O is a saddle for (3.8)) and 1-dimensional if luI(λ(0)) (here O is a stable node for (3.8)). The manifold Wu crosses transversally the plane {z=0} and intersects it exactly in the unstable manifold Mu of the reduced 2-dimensional autonomous system (3.8).

Correspondingly, we can define the unstable manifold as the set of initial conditions at time t=τ going to the origin at t-; since we want to avoid the trivial (for our problem) dynamics appearing along the z-axis, we define

w u ( τ ) = { ( x , y ) ( x , y , e ϖ τ ) W u } .

The manifold wu(τ) is an open set if 2<lu<2*(λ(0)), and it reduces to the origin if luI(λ(0)). By construction, ϕ(t,τ;Q)(0,0) as t- if and only if Qwu(τ); see [17], [7, Section 13] and [3, Appendix].

More precisely, when at least one of the two eigenvalues 1 and 2 of 𝒜(lu) is positive, namely if 2<lu<I(λ(0)), we can also construct the set

W l u u ( τ ) = { Q lim t - ϕ l u ( t , τ ; Q ) e ( κ ( λ ( 0 ) ) - α l u ) t = d ( Q ) ( 1 , - κ ( λ ( 0 ) ) ) } ,

where d:Wluu(τ) is an invertible function, implicitly giving a parametrization for Wluu(τ). This set turns out to be a 1-dimensional immersed manifold, corresponding to a strongly unstable dynamics. Of course, wu(τ)=Wluu(τ) when 2*(λ(0))<lu<I(λ(0)), but if 2<lu<2*(λ(0)), we have wu(τ)Wluu(τ) since they are respectively 2- and 1-dimensional. In fact, from standard facts in invariant manifold theory we see that if 𝑸wu(τ), then

lim t - | ϕ l u ( t , τ ; 𝑸 ) | e ( κ ( λ ( 0 ) ) - γ l u ) t = c ( 𝑸 ) 0 .

In a similar manner, to examine the behavior of the solutions in the future we can consider the system

(3.9) ( x ˙ l s y ˙ l s ζ ˙ ) = ( α l s 1 0 - λ ( ζ - 1 / ϖ ) γ l s 0 0 0 - ϖ ) ( x l s y l s ζ ) - ( 0 g l s ( x l s , - ln ( ζ ) ϖ ) 0 ) .

Again, all the trajectories bounded for t>0 converge to the plane ζ=0, where the dynamics is the one of the autonomous system

(3.10) ( x ˙ l s y ˙ l s ) = 𝒜 ( l s ) ( x l s y l s ) - ( 0 g l s ( x , + ) ) ,

where

𝒜 ( l s ) = ( α l s 1 - λ ( + ) γ l s ) .

Similarly to before, we see that the eigenvalues of 𝒜(ls) are given by

1 = γ l s + κ ( λ ( + ) ) < α l s - κ ( λ ( + ) ) = 2 ,

so that the following assertions hold:

  1. if 2<ls<2*(λ(+)), then 0<1<2;

  2. if 2*(λ(+))ls<I(λ(+)), then 10<2 (1=0 if and only if ls=2*(λ(+)));

  3. if lsI(λ(+)), then 1<20 (2=0 if and only if ls=I(λ(+))).

Hence, for system (3.9) the origin has a stable manifold Ws which is 1-dimensional if 2<ls<2*(λ(+)) (O being an unstable node for (3.10)), 2-dimensional if 2*(λ(+))ls<I(λ(+)) (in this case, O is a saddle for (3.10)) and 3-dimensional if lsI(λ(+)) (here O is a stable node for (3.10)). The manifold Ws crosses transversally the plane {ζ=0} and intersects it exactly in the stable manifold Ms of the reduced 2-dimensional autonomous system (3.10).

Correspondingly, we can define the stable manifold as the set of initial conditions at time t=τ going to the origin as t+:

w s ( τ ) = { ( x , y ) ( x , y , e - ϖ τ ) W s } .

By construction, ϕ(t,τ;Q)(0,0) as t+ if and only if Qws(τ).

More precisely, when at least one of the two eigenvalues 1 and 2 of 𝒜(ls) is negative, namely if 2*(λ(+))<ls, we can also construct the set

W l s s ( τ ) = { Q lim t + ϕ l s ( t , τ ; Q ) e ( γ l s - κ ( λ ( + ) ) ) t = L ( Q ) ( 1 , - ( n - 2 - κ ( λ ( + ) ) ) ) } ,

where L:Wlss(τ) again is invertible and gives a parametrization for Wlss(τ). This set turns out to be the 1-dimensional immersed manifold (see [17], [7, Section 13] and [3, Appendix]) corresponding to strongly stable solutions. In fact, if 2*(λ(+))<lsI(λ(+)), then Wlss(τ) coincides with the stable manifold ws(τ), while if ls>I(λ(+)), then ws(τ)Wlss(τ) since they are 2- and 1-dimensional, respectively. Further, if 𝑸ws(τ)Wlss(τ), then

lim t + | ϕ l s ( t , τ ; 𝑸 ) | e ( α l s - κ ( λ ( + ) ) ) t = c ( 𝑸 ) > 0 .

So we can reformulate the correspondences which we have seen in the previous sections. First, we look at strongly stable and unstable solutions.

Lemma 3.2.

Assume (Gu) with 2<lu<I(λ(0)) and (Gs) with ls>2*(λ(+)); let u(r) be a solution of (1.2), and let ϕlu(t,τ;Q), ϕls(t,τ;R) be the corresponding trajectories of (3.3). Then u(r) is regular if and only if QWluu(τ), and it has fast decay if and only if RWlss(τ).

Proof.

If QWluu(τ), since by definition we have

lim t - ϕ l u ( t , τ ; Q ) e ( κ ( λ ( 0 ) ) - α l u ) t = d ( Q , τ ) ( 1 , - κ ) ,

the corresponding solution u(r) of (1.2) is regular (according to the previous notation, it is u=u(r,d)).

Conversely, assume QWluu(τ). Here we distinguish three cases:

  1. If 2*(λ(0))<lu<I(λ(0)), then the fact that ϕlu(t,τ;Q)(0,0) as t- implies that, for every n,

    u ( r n ) > c r n α l u

    for a suitable sequence rn0 and a real number c>0. It follows that u(r) is not regular.

  2. If lu=2*(λ(0)) (here 1 vanishes), then limt-|ϕlu(t,τ;Q)|e-εt=+ for any ε>0 (since it converges to the origin along the central direction; see [7, 13]). So

    u ( r ) > c r - ( α l u - ε ) = c r - ( n - 2 - κ ( λ ( 0 ) ) - ε ) as  r 0

    and it cannot be a regular solution.

  3. If 2<lu<2*(λ(0)), then

    lim t - | ϕ l u ( t , τ ; Q ) | e - 1 t = c ( 𝑸 ) ( 1 , - ( n - 2 - κ ( λ ( 0 ) ) ) ) ,

    where c(𝑸)>0. Hence,

    u ( r ) c ( 𝑸 ) r γ l u - α l u + κ ( λ ( 0 ) ) = c ( 𝑸 ) r - ( n - 2 ) + κ ( λ ( 0 ) ) as  r 0 .

The proof for fast-decay solutions is analogous. ∎

We can also prove the existence of singular solutions, either when a second equilibrium 𝒫- of system (3.7) appears in the half-plane {x>0}, or when the origin is globally unstable, i.e., wu(τ) is 2-dimensional. Analogously, we can prove the existence of slow-decay solutions, either when a second equilibrium 𝒫+ of system (3.9) appears in the half-plane {x>0}, or when the origin is globally stable, i.e., ws(τ) is 2-dimensional. We emphasize that in this case we have 𝒫-=(Px-,Py-,0) and 𝒫+=(Px+,Py+,0), where P-=(Px-,Py-) is the critical point of the 2-dimensional autonomous system obtained by setting l=lu and t=- in (3.3), while P+=(Px+,Py+) is the critical point obtained by setting l=ls and t=+ in (3.3).

This fact is explained in the next two lemmas.

Lemma 3.3.

Assume (Gu) with 2*(λ(0))<lu<I(λ(0)). Then there exists at least one singular solution u¯(r) of (1.2) and it satisfies

lim r 0 u ¯ ( r ) r α l u = P x - .

This solution corresponds to a trajectory ϕlu(t) of (3.3) such that limt-ϕlu(t)=P-. Moreover,

  1. if 2 * < l u < I ( λ ( 0 ) ) , then u ¯ ( r ) is the unique singular solution of ( 1.2 );

  2. if 2 * ( λ ( 0 ) ) < l u < 2 * , then there are uncountably many singular solutions u ( r ) of ( 1.2 ), and they all satisfy

    lim r 0 u ( r ) r α l u = P x - .

Similarly, assume (Gs) with 2*(λ(+))<ls<I(λ(+)). Then there exists at least one singular solution v¯(r) of (1.2) and it satisfies

lim r + v ¯ ( r ) r α l s = P x + .

Moreover,

  1. if 2 * ( λ ( + ) ) < l s < 2 * , then v ¯ ( r ) is the unique slow-decay solution of ( 1.2 );

  2. if 2 * < l s < I ( λ ( + ) ) , then there exist uncountably many slow-decay solutions v ( r ) of ( 1.2 ), and they all satisfy

    lim r + v ( r ) r α l s = P x + .

Proof.

Let (Gu) hold: then 𝒫- admits an unstable manifold which is 1-dimensional if 2*lu<I(λ(0)) and 3-dimensional if 2*(λ(0))<lu<2*. Since all the trajectories must converge to the z=0 plane as t-, if lu2* they either cross indefinitely the coordinate axes x and y (clockwise), or they converge to 𝒫- or to the origin: correspondingly, u(r) either changes sign indefinitely, or it is singular or regular. The first two facts are trivial, the last one follows from Lemma 3.2.

The proof if (Gs) is assumed is analogous. ∎

Lemma 3.4.

Assume (Gu) with 2<lu<2*(λ(0)). Then there are uncountably many singular solutions u(r) of (1.2) satisfying

lim r 0 u ( r ) r n - 2 - κ ( λ ( 0 ) ) = L ( u )

for a suitable constant L(u)>0.

Similarly, assume (Gs) with ls>I(λ(+)). Then there are uncountably many slow-decay solutions u(r) of (1.2) satisfying

lim r + u ( r ) r κ ( λ ( + ) ) = L ¯ ( u )

for a suitable constant L¯(u)>0.

Proof.

We just prove the first statement, the other one is analogous. Assuming 2<lu<2*(λ(0)), for every τ the origin possesses an unstable manifold wu(τ) which is 2-dimensional. If Qwu(τ)Wluu(τ), then

lim t - | ϕ l u ( t , τ ; Q ) | e ( n - 2 - α l u - κ ( λ ( 0 ) ) ) t { 0 } ,

hence u(r)rn-2-κ(λ(0))L0 for r0, so u(r) is singular. ∎

Remark 3.5.

We emphasize that if (Gu) and (Gs) hold and lu,ls2*, then definitively positive solutions may have just the asymptotic behaviors described in Lemmas 3.2, 3.3 and 3.4, i.e., they may be either regular or singular as r0, and they may have either fast or slow decay as r+. When ls=2*, the plane ζ=0 is a central manifold filled with periodic and homoclinic trajectories. Thus bounded trajectories of (3.9) may have these periodic trajectories or even the homoclinic as ω-limit set; we refer the interested reader to [5, Theorems 1.4 and 1.6] for a discussion of this point in absence of Hardy terms.

Remark 3.6.

Using nonautonomous invariant manifold theory, all the construction may be repeated in a slightly more general setting, see, e.g., [7, Section 13]. However, we have preferred to maintain a basic approach not to overload much the technicalities.

3.3 Phase Portraits and the Pohozaev Function

In this section, we aim at showing that the manifolds Wluu(τ) and Wlss(τ) have a shape similar to the autonomous case. Our main tool will be represented by the use of a sort of energy function Hl, which is closely related to the Pohozaev identity [22]; this function gives a translation of the Pohozaev function which is a standard tool in this context. Setting Gl(x,t)=0xgl(s,t) (notice that if gl(s,t)=k(es)xq-1, then Gl(x,t)=k(es)xq/q), we define Hl by

(3.11) H l ( x , y , t ) = λ ( e t ) x 2 2 + n - 2 2 x y + y 2 2 + G l ( x , t ) .

We begin by recalling some technical results.

Remark 3.7.

Assume (Gu) with 2<lu<I(λ(0)); let u(r) be a regular solution for (1.2) and let ϕlu(t), ϕ2*(t) be the corresponding trajectories of (3.3). Then ϕ2*(t)0 and H2*(ϕ2*(t),t)0 as t-.

Analogously, assume (Gs) with 2<ls<I(λ(+)); let u(r) be a fast-decay solution for (1.2) and let ϕls(t), ϕ2*(t) be the corresponding trajectories of (3.3). Then ϕ2*(t)0 and H2*(ϕ2*(t),t)0 as t+.

Proof.

We just consider the first statement since the second one is analogous. From the definition of regular solution, we have that

ϕ l u ( t , τ ; Q ) d e ( α l u - κ ( λ ( 0 ) ) ) t ( 1 , - κ ( λ ( 0 ) ) ) ( 0 , 0 )

as t-, where d=d(Q,τ). Since α2*-αlu=(αlu+γlu)/2, we find

(3.12) ϕ 2 * ( t ) = e ( α l u + γ l u ) t / 2 ϕ l u ( t ) d e [ ( α l u - γ l u ) / 2 - κ ( λ ( 0 ) ) ] t 0 .

We now compute

H 2 * ( x 2 * ( t ) , t ) = e - ( α l u + γ l u ) t H l u ( x l u ( t ) , t )
= d 2 [ λ ( e t ) 2 - κ ( λ ( 0 ) ) ( n - 2 ) 2 + κ ( λ ( 0 ) ) 2 2 ] e - ( α l u + γ l u ) t e ( 2 α l u - 2 κ ( λ ( 0 ) ) ) t + e - ( α l u + γ l u ) t G l u ( x l u ( t ) , t )
= d 2 [ λ ( e t ) 2 - κ ( λ ( 0 ) ) ( n - 2 ) 2 + κ ( λ ( 0 ) ) 2 2 ] e t ( n - 2 ) 2 - 4 λ ( 0 ) + e - ( α l u + γ l u ) t G l u ( x l u ( t ) , t ) .

Since the first exponential goes to 0 as t-, and Glu(xlu(t),t)/(xlu(t))20 as xlu0 uniformly for t0, using (3.12), we see that H2*(x2*(t),t) also goes to 0 as t-, yielding the statement. ∎

Remark 3.8.

It may be useful to compare the solutions obtained with two different values of l: if u(r) is a solution of (1.2) and ϕ(t,τ;Q), ϕL(t,τ;R) are the corresponding trajectories of (1.2) with l= and l=L, respectively, then

ϕ ( t , τ ; Q ) = ϕ L ( t , τ ; R ) e ( α - α L ) t and Q = R e ( α - α L ) t .

It follows that ϕ(t,τ;Q) and ϕL(t,τ;R) are homothetic.

Remark 3.9.

Let (C) and (Gu) hold; if |ϕlu(t)| becomes unbounded as t-, then it has to cross the coordinate axes x and y indefinitely (clockwise). Analogously, let (C) and (Gs) hold; if |ϕls(t)| becomes unbounded as t+, then it has to cross the coordinate axes indefinitely (clockwise).

Proof.

The proof can be adapted trivially from [12, Lemma 3.9], but we sketch it here for completeness. From (C) it follows that any trajectory is continuable for any t. We assume (Gs), the case of (Gu) is completely analogous. Passing to polar coordinates by writing xls=ρcos(θ), yls=ρsin(θ), we observe that

(3.13) θ ˙ = - [ ( n - 2 ) cos ( θ ) sin ( θ ) + sin 2 ( θ ) + λ ( e t ) cos 2 ( θ ) ] - g l s ( ρ cos ( θ ) , t ) ρ cos ( θ ) .

Since the term gls is superlinear, we see that the right-hand side of (3.13) goes to - as xls+. Moreover, as θ˙=-1 whenever cos(θ)=0, we infer that for ρ large enough it holds that θ˙<0, and thus the statement follows. ∎

We now observe that

(3.14) H 2 * ( ϕ 2 * ( t ) , t ) = e - ( α l + γ l ) t H l ( ϕ l ( t ) , t ) ;

further, we have that

(3.15) d H 2 * d t ( ϕ 2 * ( t ) , t ) = e t λ ( e t ) [ x 2 * ( t ) ] 2 2 + G 2 * t ( x 2 * ( t ) , t ) .

In the autonomous case (1.3), in particular, we have that

d H 2 * d t ( ϕ 2 * ( t ) , t ) = G 2 * t ( x 2 * ( t ) , t ) = ( δ + α 2 * ( 2 * - q ) ) e [ δ + α 2 * ( 2 * - q ) ] t x 2 * q ( t ) q .

Since δ+α2*(2*-q)=(q-2)(α2*-αl), the function H2*(ϕ2*(t),t) is increasing in t (resp. decreasing) along the trajectories ϕ2*(t) of (3.5) whenever l<2* (resp. l>2*), while it is a first integral in the critical case l=2*. In other words, the monotonicity of H2* determines whether the system is subcritical, critical or supercritical. In this case, the level sets of H2* are the ones sketched in Figure 1; in general, for any fixed value of t the 0-level set of the function Hl is made up by an 8-shaped curve with center in the origin; see Figure 1. The dynamics is here very neat to interpret, as we briefly comment in the caption.

The plan is now to rephrase these considerations in the more general nonautonomous context. To this end, we introduce the following assumptions:

  1. λ(et)+tG2*(x,t)0 for any t and any x>0, strictly for a certain t and any x>0.

  2. λ(et)+tG2*(x,t)0 for any t and any x>0, strictly for a certain t and any x>0.

Assumptions (H’+) and (H’-) determine if the system is sub- or super-critical as we will see below. Clearly, if (F0) holds, then (H’+) and (H’-) reduce respectively to (H+) and (H-).

Now, reasoning as in the previous section, we easily get the following results, from which the first part of Theorem 2.3 follows.

Theorem 3.10.

Assume (C), (H’+), (Gu) with lu(2,2*] and (Gs) with ls(2,2*]; then all the regular solutions are oscillatory.

Further, if ls(2*(λ(+)),2*], all the singular solutions are positive: there is at least one S.G.S. with slow decay, and there are uncountably many S.G.S. with fast decay. If ls(2*(λ(+)),2*) the S.G.S. with slow decay is unique. The asymptotic behavior of the solutions is the one given by Lemma 3.3 for lu>2*(λ(0)), ls>2*(λ(+)), and by Lemma 3.4 for lu(2,2*(λ(0))].

Finally, if ls(2,2*(λ(+))], then singular solutions are oscillatory too.

Proof.

Let u(r) be a solution of (1.2) and let ϕlu(t,τ,Q), ϕ2*(t,τ,R), ϕls(t,τ,S) be the corresponding trajectories of (3.3) for l=lu,2*,ls, respectively (with the obvious relationships already highlighted in Remark 3.8). From (H’+) and (3.15) we see that H2*(ϕ2*(t,τ,R),t) is increasing in t, so that we have

lim t + H 2 * ( ϕ 2 * ( t , τ , R ) , t ) = C { + } .

Thus, if ϕlu(t,τ;Q) is a regular solution, then QWluu(τ); in view of Remark 3.7, we have that

H 2 * ( ϕ 2 * ( t , τ , R ) , t ) > 0

and, from (3.14),

H l s ( ϕ l s ( t , τ , S ) , t ) > 0 ,

for any t. Since αls+γls0, from (3.14) we get

lim t + H l s ( ϕ l s ( t , τ , S ) , t ) = lim t + C e ( α l s + γ l s ) t = C ¯ ( 0 , + ] .

In particular, if ls<2*, then C¯=+ so ϕls(t,τ,S) becomes unbounded as t+, and u(r) is oscillatory; see Remark 3.9. If ls=2* and C¯=+, we can conclude as above, while if C¯ is a positive constant, then (ϕls(t,τ,S),ζ(t)) converges to a periodic solution of (3.9) with H>0, so it must cross the coordinate axes indefinitely, and u(r) is oscillatory.

If SWlss,+(τ), then H2*(ϕ2*(t,τ,R),t)<0, and from (3.14) we get Hlu(ϕlu(t,τ,Q),t)<0 for any t. It follows that ϕlu(t,τ,Q) is forced to belong to the set {x>0} for any t (and to the set {y<0<x} if λ(r)0 for any r>0). So u(r) is positive for any r>0 (and decreasing if λ(r)0 for any r>0). So ϕlu(t,τ,Q) is bounded for t0, but QWluu(τ) since all the regular solutions are oscillatory: it follows that u(r) is a singular solution, hence it is an S.G.S. with fast decay. The asymptotic behavior of u(r) is the one given in Lemma 3.3.

Further, observe that if 2*(λ(+))<ls2*, then the critical point (P+,0) of (3.9) admits a 1-dimensional stable manifold, corresponding to a unique trajectory of (3.9), and a unique slow-decay solution v(r) of (1.2); from Lemma 3.3 we also see that if ls<2*, this is the unique slow-decay solution. On the contrary, if ls=2*, there may also be uncountably many slow-decay solutions corresponding to trajectories converging to periodic solutions of (3.9) with H<0.

Repeating the argument developed for fast-decay solutions, we see that all the slow-decay solutions are positive (and decreasing if λ(r)0 for any r>0), so they are S.G.S. with slow decay. Finally, observe that if 2<ls2*(λ(+)), then the origin is the unique critical point of (3.10), so all the solutions are oscillatory for large r. ∎

With the same argument we also prove the dual result concerning the supercritical case.

Theorem 3.11.

Assume (C), (H’-), (Gu) and (Gs) with ls2*.

If lu[2*,I(λ(0))), then all the regular solutions are G.S. with slow decay, and there is at least one S.G.S. with slow decay, which is unique if lu(2*,I(λ(0))).

The asymptotic behavior of the solutions is the one given by Lemmas 3.3 and 3.4. If luI(λ(0)), then all the solutions are oscillatory for small r.

Finally, we give a proof of the statement in the asymptotically critical case.

Proposition 3.12.

Assume (H’+) and (Gu) with lu=2*(λ(0)), and suppose that there is q>2 such that

(3.16) g l u ( x , t ) = c ( t ) x | x | q - 2 + o ( x | x | q - 2 ) as  x 0

uniformly for t0, where c(t)c¯>0 exponentially. Then singular solutions u(r) of (1.2) are such that

(3.17) lim r 0 u ( r ) r n - 2 - κ ( λ ( 0 ) ) | ln ( r ) | 1 / ( q - 2 ) C = [ n - 2 - κ ( λ ( 0 ) ) c ¯ ( q - 2 ) ] 1 / ( q - 2 ) > 0 .

Analogously, assume (H’-) and (Gs) with ls=I(λ(+)), and suppose that there is q>2 (possibly different) such that

g l s ( x , t ) = c ( t ) x | x | q - 2 + o ( x | x | q - 2 ) as  x 0

uniformly for t0, where c(t)c¯>0 exponentially. Then slow-decay solutions are such that

lim r + u ( r ) r κ ( λ ( + ) ) | ln ( r ) | 1 / ( q - 2 ) C = [ n - 2 - κ ( λ ( + ) ) c ¯ ( q - 2 ) ] 1 / ( q - 2 ) > 0 .

Proof.

In this proof, we slightly refine and generalize the argument developed in [14, Lemma 2.8] for the corresponding problem in the autonomous case.

Assume (H’+) and (Gu) with lu=2*(λ(0)) and (3.16). We recall that all the trajectories of (3.7) converge to the plane z=0 as t-. Moreover, since H2*(ϕ2*(t),t) is increasing in t, all the trajectories are bounded for t0, so they converge to the origin as t-, the unique critical point of (3.7). The eigenvalues of 𝒜lu(τ) are 1=0<2, so that if QWluu(τ), then ϕlu(t,τ;Q)=(X(t),Y(t)) converges to the origin polynomially as t-, and approaches T0={(x,y)y=-αlux}. So, for any ε>0, setting σ=ϖq-2>0, there is N>0 large enough so that

(3.18) ε 1 / ( q - 2 ) | X ( t ) | > e σ t , ε 1 / ( q - 2 ) | Y ( t ) | > e σ t    for  t < - N .

Assume for definiteness that Y(t,τ;Q)<0<X(t,τ;Q) for t<-N, and let us set

T + = { ( x , y ) y > - α l u x } and T - = { ( x , y ) y < - α l u x } .

We claim that there exists t1 such that ϕlu(t,τ;Q)=(X(t),Y(t))T+ for t<t1.

In fact, the flow of (3.7) on T0 points towards T-, so that two alternatives are possible: either there exists t¯ such that ϕlu(t,τ;Q)T- for any t<t¯ and X˙(t¯)=0, or there is t1 such that ϕlu(t,τ;Q)T+ for any t<t1 and ϕlu(t1,τ;Q)T0. In the latter case, the claim is proved, while in the former X˙(t)<0<X(t) holds for t<t¯, but X(t)=-tX˙(s)𝑑s<0, and a contradiction is immediately achieved.

It follows that there is t0<t1 such that X˙(t)>0 and Y(t)=-αluX(t)+h(X(t),t), where h(x,t)>0 and h(x,t)=o(x) as x0, uniformly for t<t0. So, setting Z(t)=-Y(t), for any ε>0 we can choose t0=t0(ε) such that

(3.19) ( α l u - ε ) X ( t ) < Z ( t ) < α l u X ( t )

for t<t0. The plan is now to use such an inequality to estimate X(t) via the second equation in (3.7).

(1) Assume first λ(0)=0.

From (Gu) we immediately deduce that |λ(et)|eϖt for t<t0; up to enlarging |t0|, we can assume that also (3.18) holds. Moreover, since αlu=n-2, we have that γlu=0. Then, using (3.19) in (3.7) together with (3.18), we find that

(3.20) Z ˙ ( t ) - e ϖ t X ( t ) + ( c ¯ - ε ) X q - 1 ( t ) > c ¯ - 2 ε α l u q - 1 Z q - 1 ( t ) ,

where we used the fact that X(t) goes polynomially to 0 in the estimate of eϖtX(t). Analogously, we find

(3.21) Z ˙ ( t ) e ϖ t X ( t ) + ( c ¯ + ε ) X q - 1 ( t ) < c ¯ + 2 ε ( α l u - ε ) q - 1 Z q - 1 ( t ) .

Integrating (3.20) and (3.21), we find that, for t<t0,

[ 1 Z ( t 0 ) q - 2 + ( c ¯ + 2 ε ) ( 2 - q ) α l u q - 1 ( t - t 0 ) ] - 1 / ( q - 2 ) Z ( t ) [ 1 Z ( t 0 ) q - 2 + ( c ¯ - 2 ε ) ( 2 - q ) α l u q - 1 ( t - t 0 ) ] - 1 / ( q - 2 ) .

So, letting ε go to 0, we find

Z ( t ) [ α l u q - 1 c ¯ ( q - 2 ) | t | ] 1 / ( q - 2 ) .

Since Z(t) has the same order of X(t) and X(t)=u(et)e(n-2)t, we find u(r)Cr-(n-2), where

C = [ α l u c ¯ ( q - 2 ) ] 1 / ( q - 2 ) .

(2) Now assume 0λ(0)<(n-2)2/4.

Using (3.2), we pass again from equation (1.2) to (3.4). Then, using the observation of Remark 3.1, we can reduce to consider the previous case. Eventually we find that X(t)C|t|1/(q-2) as t-, and, since

X ( t ) = u ( e t ) e α l u t

and recalling that αlu=n-2-κ(λ(0)), we get (3.17).

When ls=I(λ(+)), we find that the origin is stable even in its central direction, and the statement can be obtained by reasoning as above, but reversing the direction of t. ∎


Dedicated to the memory of Professor Russell Johnson



Communicated by Kenneth Palmer


Funding statement: Both authors were supported by GNAMPA-INdAM (Italy).

Acknowledgements

Matteo Franca is indebted to Prof. Russell A. Johnson (formerly University of Florence) for having introduced him to the subject of the article, and for the many helpful discussions on the topic.

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Received: 2017-03-04
Accepted: 2017-08-21
Published Online: 2017-09-23
Published in Print: 2018-02-01

© 2017 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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