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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Sergio RolandoORCID iD: https://orcid.org/0000-0002-1679-8198
Published Online: 2017-11-21 | DOI: https://doi.org/10.1515/anona-2017-0177

Abstract

Many existence and nonexistence results are known for nonnegative radial solutions to the equation

-u+A|x|αu=f(u)in N,N3,A,α>0,uD1,2(N)L2(N,|x|-αdx),

with the nonlinearities satisfying |f(u)|(const.)up-1 for some p>2. The existence of nonradial solutions, by contrast, is known only for N4, α=2, f(u)=u(N+2)/(N-2) and A large enough. Here we show that the above equation has multiple nonradial solutions as A+ for N4, 2/(N-1)<α<2N-2 and α2, with the nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

Keywords: Semilinear elliptic PDE; singular vanishing potential; symmetry breaking

MSC 2010: 35J60; 35Q55; 35J20

1 Introduction and main result

This paper is concerned with the following semilinear elliptic problem:

{-u+A|x|αu=f(u)in N,N3,u0in N,uHα1,u0.(1.1)

where A,α>0 are real constants, f: is a continuous nonlinearity satisfying |f(s)|(const.)sp-1 for some p>2 and all s0, and Hα1:=D1,2(N)L2(N,|x|-αdx) is the natural energy space related to the equation. We will deal with problem (1.1) in the weak sense, that is, when we speak about solutions to (1.1) we will always mean weak solutions, i.e., functions uHα1{0} such that u0 almost everywhere in N and

Nuvdx+NA|x|αuv𝑑x=Nf(u)v𝑑xfor all vHα1.(1.2)

As is well known, problems like (1.1) are models for stationary states of reaction diffusion equations in population dynamics (see, e.g., [17]). They also arise in many other branches of mathematical physics, such as nonlinear optics, plasma physics, condensed matter physics and cosmology (see, e.g., [11, 27]), where its nonnegative solutions lead to special solutions (solitary waves and solitons) for several nonlinear field theories like nonlinear Schrödinger (or Gross–Pitaevskii) and Klein–Gordon equations. In this context, (1.1) is a prototype for problems exhibiting radial potentials which are singular at the origin and/or vanishing at infinity (sometimes called the zero mass case; see, e.g., [22, 8]).

Although it can be considered as a quite recent investigation, the study of problem (1.1) has already some history, which probably started in [26] and continued in [16, 9, 24, 25, 5, 15] (see [4] for a similar cylindrical problem). Currently, the problem of existence and nonexistence of radial solutions is essentially solved in the pure-power case f(u)=up-1, where the results obtained rest upon compatibility conditions between α and p. These can be summarized as follows (for a chronological overview of these results see [5]): the problem has a radial solution for (α,p)=(2,2*) (see [26]) and for all the pairs (α,p) satisfying

{0<α<2,2α*<p<2*,or{2<α<2N-2,2*<p<2α*,or{α2N-2,p>2*,with 2α*:=22N-2+α2N-2-α(1.3)

(see [25]), while it has no solution if

{0<α<2,p(2α,2*),or{α=2,p2*,or{2<α<N,p(2*,2α),or{αN,p2*,with 2α:=2NN-α

(see [9]) and no radial solution for both

{0<α<2,2α<p2α*,and{2<α<2N-2,2α*p<2α

(see [5] and [15], respectively). As usual, 2*:=2N/(N-2) denotes the critical exponent for the Sobolev embedding in dimension N3. All these results are portrayed in the picture of the αp-plane given in Figure 1, where nonexistence regions are shaded in gray (nonexistence of radial solutions) and light gray (nonexistence of solutions at all, which includes both the lines p=2* and p=2α except for the pair (α,p)=(2,2*)), whereas white color (of course above the line p=2) means existence of radial solutions. As to nonradial solutions, the only result available is the one contained in [26, Theorem 0.5], where Terracini proves that problem (1.1), with N4, α=2 and f(u)=u2*-1, has at least a nonradial solution for every A large enough. This brought Catrina to say, in the introduction of his paper [15]: “Two questions still remain: whether one can find non-radial solutions in the case when radial solutions do not exist, or in the case when radial solutions exist”.

Regions of nonexistence of solutions (light gray), and existence(white with p>2{p>2}) and nonexistence (dark gray) of radial solutions.
Figure 1

Regions of nonexistence of solutions (light gray), and existence(white with p>2) and nonexistence (dark gray) of radial solutions.

Su, Wang and Willem [25] covered also the case where problem (1.1) has general nonlinearities satisfying the power growth condition |f(u)|(const.)up-1 for some p>2, and ensured that, under some rather standard additional assumptions on f (precisely (f1) and (f2) below), problem (1.1) has a radial solution for all the pairs (α,p) satisfying (1.3). To be precise, they only concerned themselves with radial weak solutions in the sense of the dual space of the radial subspace of Hα1 (where the energy functional of the problem is well defined by the embeddings they proved), but the symmetric criticality type results of [6] actually apply, yielding solutions in the sense of our definition (1.2). No results are known in the literature about nonradial solutions.

This general lack of symmetry breaking results is the motivation of this paper, where we prove that problem (1.1) has multiple nonradial solutions as A+, provided that N4, α(2/(N-1),2N-2){2} and f belongs to a suitable class of nonlinearities satisfying a power growth condition. We observe straight away that such a class of nonlinearities does not unfortunately contain pure powers (which does not satisfy our assumption (f0), where p1p2).

The main assumptions characterizing our class of nonlinearities are the following, where we define F(s):=0sf(t)𝑑t:

  • (f0)

    sups>0|f(s)|min{sp1-1,sp2-1}<+ for some 2<p1<2*<p2.

  • (f1)

    There exists θ>2 such that θF(s)f(s)s for all s>0.

  • (f2)

    F(s)>0 for all s>0.

  • (f3)

    The function f(s)s is strictly increasing on (0,+).

  • (f4)

    There exists μ>2 such that the function F(s)sμ is decreasing on (0,+).

For N3, α(0,2N-2), α2 and 2<p1<2*<p2, we define

ν:=νN,α,p1,p2:={2min{N-1α,N-22-α2*-p1p1-2}-2N(1α-12)-1if 0<α<2,2min{N-1α,N-2α-2p2-2*p2-2}-1if 2<α<2N-2,(1.4)

where denotes the ceiling function (i.e., x:=min{n:nx}).

ν as a function of p1∈(2,p1*){p_{1}\in(2,p_{1}^{*})} for N≥3{N\geq 3} and α∈(0,2){\alpha\in(0,2)} fixed.
Figure 2

ν as a function of p1(2,p1*) for N3 and α(0,2) fixed.

ν as a function of p2>p2*{p_{2}>p_{2}^{*}} for N≥3{N\geq 3} and α∈(2,2⁢N-2){\alpha\in(2,2N-2)} fixed.
Figure 3

ν as a function of p2>p2* for N3 and α(2,2N-2) fixed.

Our main result is the following theorem.

Theorem 1.1.

Let N4 and α(2/(N-1),2N-2), α2. Let f:RR be a continuous function satisfying (f1)(f4). Assume that (f0) holds with

p1<p1*:=2α2(N-1)-2α(N-1)+4Nα2(N-1)-2α(N+1)+4N𝑜𝑟p2>p2*:=22N+2-α2N-2-α(1.5)

if α(2/(N-1),2) or α(2,2N-2), respectively. Then there exists A*>0 such that for every A>A*, problem (1.1) has both a radial solution and ν different nonradial solutions.

Some comments on Theorem 1.1 are in order. First of all, under the assumptions of the theorem, ν is positive (see Lemma 5.2 below), and so at least one nonradial solution actually exists. On the other hand, it is easy to check that for every fixed N and α, the behavior of ν as a function of p1 and p2 is the one portrayed in Figures 2 and 3, respectively, whence one sees that the number ν of nonradial solutions may assume every natural value (as N).

Regarding assumptions (1.5) and (f0), it is worth observing that α(0,2) implies 2<p1*<2*, while 2<α<2N-2 implies p2*>2*, and so (1.5) and (f0) are consistent with each other. Assumption (f0) is the so-called double-power growth condition and seems to be typical in nonlinear problems with potentials vanishing at infinity (see, e.g., [12, 13, 2, 19, 6, 7, 8, 18, 22, 14, 3]). Such an assumption obviously implies the single-power growth condition |f(s)|(const.)sp-1 for all p[p1,p2] and s0, but it is actually more stringent than that, since it requires p1p2. We finally observe that (f0) still remains true if one raises p1 and lowers p2, but this decreases ν (see Figures 2 and 3), and therefore it is convenient to apply Theorem 1.1 with p1 as small as possible and p2 as large as possible (which is also consistent with assumption (1.5)).

The plan of the paper is the following. In Section 2 we define the variational setting and introduce the argument we will use in the proof of Theorem 1.1, which will be given in Section 5. Observe that we cannot use the technique used in [26], where the homogeneity of the nonlinearity is exploited and a nonradial solution is obtained as a global minimizer of the Sobolev type quotient associated to the problem. Our argument, instead, essentially relies on the following two main elements:

  • (i)

    The compact embeddings between some suitable functional spaces of symmetric functions, which yield the existence of ν different solutions of mountain-pass type.

  • (ii)

    The separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

Sections 3 and 4 are devoted to the estimation of these levels in order to separate them. As a conclusion, we get ν nonradial solutions on which the energy functional of the equation has a lower value than the energy levels of radial solutions.

We end this introductory section by giving some examples of nonlinearities to which Theorem 1.1 applies, and by presenting some notations we use throughout the paper.

Example 1.2.

Let N4 and α(2/(N-1),2N-2), α2, and let 2<p1<2*<p2 be such that (1.5) holds. The most obvious nonlinearity to which Theorem 1.1 applies is f(s)=min{|s|p1-1,|s|p2-1}, which satisfies (f1) and (f4) for θ=p1 and any μ>p2. Other simple examples are

f(s)=|s|p2-11+|s|p2-p1,f(s)=dds(|s|p21+|s|p2-p1),

both of which satisfy (f1) with θ=p1. In the latter case, (f4) clearly holds for any μ>p2. We leave it to the reader to check that (f4) also holds in the former case for μ large enough.

Notations.

  • σd denotes the (d-1)-dimensional measure of the unit sphere of d.

  • Cc(Ω) is the space of infinitely differentiable real functions with compact support in the open set Ωd.

  • D1,2(N)={uL2*(N):uL2(N)} is the usual Sobolev space, which identifies with the completion of Cc(N) with respect to the norm of the gradient.

2 Preliminaries

Let N3 and A,α>0. Let f: be a continuous function satisfying (f0)(f2). In this section we define the functional setting and introduce the argument we will use in proving Theorem 1.1.

As already mentioned in the introduction, we define the Hilbert space

Hα1:={uD1,2(N):Nu2|x|α𝑑x<},

which we endow with the following scalar product and related norm:

(u,v)A:=N(uv+A|x|αuv)𝑑x,uA2:=N(|u|2+A|x|αu2)𝑑x.

Of course, the embedding Hα1D1,2(N) is continuous.

Given any integer K such that 1KN-1, we write every xN as x=(y,z)K×N-K, and in the space Hα1, we consider the following closed subspaces of symmetric functions:

Hr:={uH:u(x)=u(|x|)}andHK:={uH:u(x)=u(y,z)=u(|y|,|z|)}.

Of course u(y,z)=u(|y|,|z|) naturally means that u(y,z)=u(S1y,S2z) for all isometries S1 and S2 of K and N-K, respectively. Similarly for u(x)=u(|x|). Note that HrHK for every K, since |x|2=|y|2+|z|2. The next lemma clarifies better the relation between the spaces HK and Hr.

Lemma 2.1.

Let 1K1<K2N-1. Then HK1HK2=Hr.

Proof.

The proof is essentially an adaptation of the one of [21, Lemma 3.3]. For any xN, we will denote by (y1,z1) its decomposition in K1×N-K1, and by (y2,z2) its decomposition in K2×N-K2. Let uHK1HK2, and for every s,t0, define

u~(s,t):=u(s,0,,0,t).

Then we clearly have u(x)=u~(|y1|,|z1|)=u~(|y2|,|z2|) for every x=(y1,z1)=(y2,z2)N. Let x=(y1,z1)K1×N-K1 and x=(y2,z2)K2×N-K2 be such that |x|=|x|, i.e., |y1|2+|z1|2=|y2|2+|z2|2. Suppose that |y1||y2| and define x′′N by setting

x′′:=(|y1|,0,,0,|y2|2-|y1|2,0,,0,|z2|,0,,0),

where the first block of zeros has K1-1 zeros, the second K2-K1-1, and the third N-K2-1. Then

u(x′′)=u~(|y1′′|,|z1′′|)=u~(|y1|,|y2|2-|y1|2+|z2|2)=u~(|y1|,|z1|)

and

u(x′′)=u~(|y2′′|,|z2′′|)=u~(|y1|2+|y2|2-|y1|2,|z2|)=u~(|y2|,|z2|),

which implies u~(|y1|,|z1|)=u~(|y2|,|z2|), i.e., u(x)=u(x). If |y1|>|y2|, we repeat the argument with x′′ defined by

x′′:=(|y2|,0,,0,|y1|2-|y2|2,0,,0,|z1|,0,,0)

(with the same lengths of blocks of zeros) and get the same result. Hence, |x|=|x| implies u(x)=u(x), i.e., uHr. ∎

We modify the function f by setting f(s)=0 for all s<0 and, with a slight abuse of notation, we still denote by f the modified function. Then, by (f0), there exist M1,M2>0 such that

|f(s)|M1min{|s|p1-1,|s|p2-1}and|F(s)|M2min{|s|p1,|s|p2}for all s,

which, in particular, yields

|f(s)|M1|s|p-1and|F(s)|M2|s|pfor all p[p1,p2] and s.(2.1)

By the continuous embeddings Hα1D1,2(N)L2*(N), one can check (see, for example, [20]) that condition (2.1) with p=2* implies that the energy functional associated to the equation of (1.1), i.e.,

I(u):=12uA2-NF(u)𝑑x,(2.2)

is of class C1 on Hα1 and has Fréchet derivative I(u) at any uHα1 given by

I(u)v=(u,v)A-Nf(u)v𝑑xfor all vHα1.(2.3)

This yields that the critical points of I:Hα1 satisfy (1.2). A standard argument shows that such critical points are nonnegative (see the proof of Theorem 1.1 in Section 5), and therefore we conclude that the nonzero critical points of I are weak solutions to problem (1.1).

Accordingly, our argument in proving Theorem 1.1 will be essentially the following. The existence of a critical point for the restriction I|Hr readily follows from the results of [25]. By exploiting the compact embeddings of [2] and the results of [8] about Nemytskiĭ operators on the sum of Lebesgue spaces, we will show in Section 5 that I|HK has a nonzero critical point uK for every 2KN-2. Thanks to the classical Palais principle of symmetric criticality [23], all these critical points are also critical points of I, and thus weak solutions to (1.1). Hence, Theorem 1.1 is proved if we show that uKHr for every K, which also implies uK1uK2 for K1K2, by Lemma 2.1. This will be achieved by showing that the critical levels I(uK) are lower than all the nonzero critical levels of I|Hr. The starting points in proving this are the following lemmas.

Lemma 2.2.

For every uHr{0}, u0, there exists tu>0 such that I(tuu)=maxt0I(tu).

Proof.

Since u0 and u0, we can fix δ>0 such that the set {xN:uδ} has positive measure. From assumptions (f1) and (f2), we deduce that there exists a constant C>0 such that F(s)Csθ for all sδ. Then, for every t>1, one has

NF(tu)𝑑x={xN:tuδ}F(tu)𝑑x+{xN:tu<δ}F(tu)𝑑x{xN:tuδ}F(tu)𝑑xCtθ{xN:tuδ}uθ𝑑xCtθ{xN:uδ}uθ𝑑x,

and therefore

I(tu)12t2uA2-Ctθ{xN:uδ}uθ𝑑x-as t+,(2.4)

since θ>2. On the other hand, condition (2.1) with p=2* and the continuous embeddings HrHα1D1,2(N)L2*(N) imply that there exists a constant C>0 such that I(tu)uA2t22-CuA2*t2* for all t0, which in turn yields I(tu)>0 for all t>0 small enough. Together with (2.4), this gives the result. ∎

According to Lemma 2.2, define

mA:=infuHr{0},u0maxt0I(tu).

Lemma 2.3.

Assume (f3) and let uHr{0}. If u is a critical point for I, then I(u)mA.

Proof.

As already observed, if u is a critical point for I, then u is nonnegative. We shall now prove that I(u)=maxt0I(tu), which obviously yields the result. For t0, define

g(t):=I(tu)=12t2uA2-NF(tu)𝑑x.

As u is a critical point for I, we readily have that t=1 is a critical point for g. Indeed, g(t)=I(tu)u, and thus g(1)=I(u)u=0. We now show that, on the other hand, g has at most one critical point in (0,+). We have g(t)=0 if and only if I(tu)u=0, i.e.,

I(tu)u=tuA2-Nf(tu)u𝑑x=0.

So, if 0<t1<t2 are critical points for g, then one has

uA2=1t1Nf(t1u)u𝑑x=1t2Nf(t2u)u𝑑x,

which implies

Eu(f(t2u)t2u-f(t1u)t1u)u2𝑑x=0,(2.5)

where Eu:={xN:u>0}. Since the integrand in (2.5) is nonnegative by assumption (f3), we have that f(t2u)t2u-f(t1u)t1u=0 almost everywhere on Eu. Since Eu has positive measure (because 0u0), this implies t1=t2, again by assumption (f3). As a conclusion, according to Lemma 2.2, we deduce that tu=1 and the claim ensues. ∎

Lemma 2.4.

There exists R>0 such that for every 1KN-1, one has

infuHK,uARI(u)=0𝑎𝑛𝑑infuHK,uA=RI(u)>0.(2.6)

Proof.

The claim readily follows from (2.1) with p=2* and the continuous embeddings HKHα1D1,2(N)L2*(N), which imply that there exists a constant C>0 such that I(u)uA2/2-CuA2* for all uHK. ∎

In Section 4 we will see that I|HK takes negative values by choosing a suitable u¯KHK such that I(u¯K)<0. This implies u¯KA>R, by (2.6), and therefore the functional I|HK has a mountain-pass geometry. In Section 5 we will see that it also satisfies the Palais–Smale condition for 2KN-2, and so it admits a (nonnegative) critical point uK at the mountain-pass level

cA,K:=infγΓmaxt[0,1]I(γ(t))>0,where Γ:={γC([0,1];HK):γ(0)=0,γ(1)=u¯K}.(2.7)

With a view to obtaining the separation inequality cA,K<mA, Sections 3 and 4 are devoted to estimating mA and cA,K.

3 Estimate of mA

Let N3 and α,A>0. Let f: be a continuous function satisfying (f0)(f2). This section is devoted to deriving the estimate of mA given in Proposition 3.2 below, which relies on the following radial lemma (see also [6, Appendix] and [25, Lemmas 4 and 5] for similar results).

Lemma 3.1.

Every uHr satisfies

|u(x)|2/σNA1/4uA|x|(2N-2-α)/4almost everywhere in N

(recall that σN denotes the (N-1)-dimensional measure of the unit sphere of RN).

Proof.

Let uHr and let u~:(0,+) be continuous and such that u(x)=u~(|x|) for almost every xN. Set

v(r):=rN-1-α/2u~(r)2for all r>0.

By [6, Lemma 27], we have that u~W1,1((a,b)) for every 0<a<b<+, whence vW1,1((a,b)) and

v(b)-v(a)=abv(r)𝑑r.

Moreover, for almost every r(a,b), one has

v(r)=(N-1-α2)rN-2-α/2u~(r)2+2rN-1-α/2u~(r)u~(r).(3.1)

If α<2N-2, this implies v(r)2rN-1-α/2u~(r)u~(r), and therefore

v(a)v(b)-abv(r)𝑑rv(b)-ab2rN-1-α/2u~(r)u~(r)𝑑rv(b)+2abrN-1-α/2|u~(r)||u~(r)|𝑑r=v(b)+2abr(N-1)/2|u~(r)|r(N-1)/2-α/2|u~(r)|𝑑rv(b)+2(0+rN-1|u~(r)|2𝑑r)1/2(0+rN-1-α|u~(r)|2𝑑r)1/2v(b)+2σNA(N|u|2dx)1/2(NAu2|x|αdx)1/2v(b)+2σNAuA2.(3.2)

If α2N-2, then (3.1) gives v(r)2rN-1-α/2u~(r)u~(r), and thus we have

v(b)v(a)+ab2rN-1-α/2u~(r)u~(r)𝑑rv(a)+2σNAuA2,(3.3)

as before. Now observe that there exist 0<an0 and bn+ such that v(an)0 and v(bn)0. Indeed, if l:=lim infr0+v(r)>0, then for every r smaller than some suitable r0>0, one has |u~(r)|l/2r-(N-1-α/2)/2, and therefore one of the following contradictions ensues:

Nu2|x|α𝑑xl2Br01|x|N-1+α/2𝑑x=+if α2

or

N|u|2*dx(l2)2*/2Br01|x|N-1-α/2N-2Ndx=+if α2.

Similarly, if lim infr+v(r)>0, then one obtains

Nu2|x|αdx=+if α2  and  N|u|2*dx=+if α2.

Therefore, the claim follows by letting n in (3.2) with a=r and b=bn, and in (3.3) with a=an and b=r. ∎

We can now prove our estimate for mA.

Proposition 3.2.

Assume 0<α<2N-2, α2, and let p=max{2α*,p1} if 0<α<2, or p=min{2α*,p2} if 2<α<2N-2. Then there exists a constant C0>0, independent from A, such that

mAC0AN-2α-2p-2*p-2.

Proof.

Let uHr{0}. By Lemma 3.1, we have

N|u|2α*dx=N|u|2α*-2u2dx(2σN)(2α*-2)/2A(2α*-2)/4uA2α*-2Nu2|x|[(2N-2-α)/4](2α*-2)𝑑x=(2σN)2α/(2N-2-α)Aα/(2N-2-α)uA2α*-2ANAu2|x|α𝑑x(2σN)2α/(2N-2-α)A(2N-2)/(2N-2-α)uA2α*,

since 2α*-2=4α/(2N-2-α). On the other hand, one has

N|u|2*dxSN2*(N|u|2dx)2*/2SN2*uA2*,

where SN denotes the Sobolev constant in dimension N. Then, in either case p=max{2α*,p1}<2* or p=min{2α*,p2}>2*, we can argue by interpolation: there exists λ[0,1) such that p=λ2*+(1-λ)2α*, and by Hölder inequality, we get

N|u|pdx=N|u|λ2*+(1-λ)2α*dx(N|u|2*dx)λ(N|u|2α*dx)1-λCuApA(2N-2)(1-λ)/(2N-2-α),

where C:=SNλ2*(2σN)2α(1-λ)/(2N-2-α) only depends on N,α,p. Recalling condition (2.1), this implies

|NF(u)dx|M2N|u|pdxM2CuApA(2N-2)(1-λ)/(2N-2-α),

and therefore

I(u)12uA2-auAp,

where we set a=M2CA-(2N-2)(1-λ)/(2N-2-α) for brevity. Hence,

I(tu)12t2uA2-atpuAp=:gu(t)for every t0.

The function gu:[0,+) attains its maximum in tu:=(ap)-1/(p-2)uA and since

1-λ=p-2*2α*-2*=(p-2*)(2N-2-α)(N-2)4(α-2)(N-1),

one computes

gu(tu)=(1ap)2/(p-2)(12-1p)=p-22pp/(p-2)(1a)2/(p-2)=p-22pp/(p-2)(A(2N-2)(1-λ)/(2N-2-α)M2C)2/(p-2)=p-22pp/(p-2)AN-2α-2p-2*p-2(M2C)2/(p-2).

Hence,

maxt0I(tu)maxt0gu(t)=gu(tu)=C0AN-2α-2p-2*p-2,

with the definition of C0 being obvious. Since uHr{0} is arbitrary, we conclude

mA=infuHr{0},u0maxt0I(tu)infuHr{0}maxt0I(tu)C0AN-2α-2p-2*p-2,

and the proof is complete.∎

Remark 3.3.

If p is as in Proposition 3.2, it is easy to check that

N-2α-2p-2*p-2={min{N-1α,N-22-α2*-p1p1-2}if 0<α<2,min{N-1α,N-2α-2p2-2*p2-2}if 2<α<2N-2.

4 Estimate of cA,K

Let N3, 2KN-2 and α>0, α2. Let f: be a continuous function satisfying (f0)(f2). In this section we define a suitable u¯KHK such that I(u¯K)<0 and estimate the corresponding mountain-pass level (2.7).

In defining u¯K, we will use the following construction of positive HK functions, which is inspired by [10]. Denote by ϕ:D2{0} the change to polar coordinates in 2{0}, namely, ϕ(ρ,θ)=(ρcosθ,ρsinθ) for all (ρ,θ)D:=(0,+)×[0,2π). Define

E:=(14,34)×(π6,π3)

and take any ψ:2 such that ψCc(E) and ψ>0. For 0<ε<1 and (ρ,θ)2, define

ψε(ρ,θ):=ψ(ρ1/ε,θε),

in such a way that ψεCc(Eε), where

Eε:={(ρ,θ)2:(ρ1/ε,θε)E}={(ρ,θ)2:(14)ε<ρ<(34)ε,πε6<θ<πε3}.

Finally, define

vε(y,z):=ψε(ϕ-1(|y|,|z|))for x=(y,z)(K×N-K){0},vε(0):=0.

Then vεCc(Ωε)HK, where Ωε:={(y,z)K×N-K:(|y|,|z|)ϕ(Eε)}.

For future reference, we now compute the relevant integrals of vε. By means of spherical coordinates in K and N-K, one has

Nvε2|x|α𝑑x=Ωεψε(ϕ-1(|y|,|z|))2|x|α𝑑x=σKσN-Kϕ(Eε)ψε(ϕ-1(s,t))2(s2+t2)α/2sK-1tN-K-1𝑑s𝑑t=σKσN-KEεψε(ρ,θ)2ρα-N+1H(θ)𝑑ρ𝑑θ=σKσN-KEεψ(ρ1/ε,θ/ε)2ρα-N+1H(θ)𝑑ρ𝑑θ,

where H(θ):=(cosθ)K-1(sinθ)N-K-1, and by the change of variables

r=ρ1/ε,φ=θε,

one obtains

Nvε2|x|α𝑑x=σKσN-KEψ(r,φ)2r(α-N+1)εH(εφ)ε2rε-1𝑑r𝑑φ=σKσN-Kε2Eψ(r,φ)2r(α-N)ε+1H(εφ)𝑑r𝑑φ.(4.1)

Similarly (recall that F(0)=0),

NF(vε)𝑑x=ΩεF(ψε(ϕ-1(|y|,|z|)))𝑑x=σKσN-Kϕ(Eε)F(ψε(ϕ-1(s,t)))sK-1tN-K-1𝑑s𝑑t=σKσN-KEεF(ψε(ρ,θ))ρN-1H(θ)𝑑ρ𝑑θ=σKσN-Kε2EF(ψ(r,φ))rNε-1H(εφ)𝑑r𝑑φ(4.2)

and

N|vε|2dx=Ωε|ψε(ϕ-1(|y|,|z|))Jϕ-1(|y|,|z|)|2dx=σKσN-Kϕ(Eε)|ψε(ϕ-1(s,t))Jϕ-1(s,t)|2sK-1tN-K-1dsdt=σKσN-KEε|ψε(ρ,θ)Jϕ-1(ρ,θ)|2ρN-1H(θ)dρdθ=σKσN-KEε(ψερ(ρ,θ)2+1ρ2ψεθ(ρ,θ)2)ρN-1H(θ)𝑑ρ𝑑θ=σKσN-KEε1ε2(ρ2/εψr(ρ1/ε,θε)2+ψφ(ρ1/ε,θε)2)ρN-3H(θ)𝑑ρ𝑑θ=σKσN-KE(ψr(r,φ)2+1r2ψφ(r,φ)2)r(N-2)ε+1H(εφ)𝑑r𝑑φ,(4.3)

where we set ψr=ψr and ψφ=ψφ for brevity.

Lemma 4.1.

The mapping wA:=vA-1/2HK, A>1, is such that

limA+wAA2NF(wA)𝑑x=+.

Proof.

According to the previous computations, for ε=A-1/2<1, we have

vεA2NF(vε)𝑑x=N|vε|2dx+ANvε2|x|αdxNF(vε)𝑑x=E((ψr2+1r2ψφ2)r(N-2)ε+1+Aε2ψ2r(N-α)ε-1)H(εφ)𝑑r𝑑φε2EF(ψ)rNε-1H(εφ)𝑑r𝑑φ=AE((ψr2+1r2ψφ2)r(N-2)ε+1+ψ2r(N-α)ε-1)H(εφ)𝑑r𝑑φEF(ψ)rNε-1H(εφ)𝑑r𝑑φ.(4.4)

In the integration set E, one has επ6<εφ<επ3, and thus, for ε>0 small enough (i.e., A>1 large enough), we get that εφ2<sinεφ<εφ and 12<cosεφ<1. Hence, there exist two constants C¯1,C¯2>0 such that

C¯1εN-K-1<H(εφ)<C¯2εN-K-1.

Similarly, since 14<r<34 in E, all the terms r(N-2)ε+1, r(N-α)ε-1 and rNε-1 are bounded, and bounded away from zero by positive constants independent of ε(0,1) (i.e., of A>1), say C¯3 and C¯4, respectively. Using these bounds in (4.4), we have

vεA2NF(vε)𝑑xAC¯1E((ψr2+1r2ψφ2)r(N-2)ε+1+ψ2r(N-α)ε-1)𝑑r𝑑φC¯2EFrNε-1𝑑r𝑑φAC¯1C¯4E(ψr2+1r2ψφ2+ψ2)𝑑r𝑑φC¯2C¯3EF(ψ)𝑑r𝑑φ.

The last ratio is positive and independent of A, whence the claim follows. ∎

According to Lemma 4.1, we fix A0>1 so that

wAA2NF(wA)𝑑x>1for every A>A0.(4.5)

We now distinguish the cases 0<α<2 and α>2.

Proposition 4.2.

Assume (f4) and 0<α<2. Let A>A0 and define u¯KHK by setting

u¯K(x):=wA(xλ),with λ:=wAA2/α(NF(wA)𝑑x)1/α.

Then I(u¯K)<0 and the corresponding mountain-pass level (2.7) satisfies

cA,KC1A(K-1)/2+N(1/α-1/2),

where the constant C1>0 does not depend on A.

Proof.

Since A>A0, one has λ>1. Then an obvious change of variables yields

I(u¯K)=λN-22N|wA|2dx+λN-α2NA|x|αwA2dx-λNNF(wA)dxλN-α2(N|wA|2dx+NA|x|αwA2dx)-λNNF(wA)dx=λN2(λ-αwAA2-2NF(wA)𝑑x)=-λN2NF(wA)𝑑x<0,

where the last inequality follows from assumption (f2), since wA>0 almost everywhere. In order to estimate cA,K, consider the straight path γ(t):=tu¯K, t[0,1]. Clearly, cA,Kmaxt[0,1]I(γ(t)). Thanks to assumption (f4), which implies F(ts)tμF(s) for all s>0 and t[0,1], we have

I(γ(t))=12t2u¯KA2-NF(tu¯K)𝑑x12t2u¯KA2-tμNF(u¯K)𝑑x=12t2a-tμb,

where we set a:=u¯KA2 and b:=NF(u¯K)𝑑x for brevity. The function g(t):=12t2a-tμb reaches its maximum in t=(abμ)1/(μ-2), and so we get

I(γ(t))g((abμ)1/(μ-2))=a(abμ)2/(μ-2)(12-1μ).

Hence, setting m:=(1μ)2/(μ-2)(12-1μ) for brevity and recalling that λ>1, we obtain

cA,Kmu¯KA2μ/(μ-2)(NF(u¯K)𝑑x)2/(μ-2)=m(λN-2N|wA|2dx+λN-αNA|x|-αwA2dx)μ/(μ-2)(λNNF(wA)𝑑x)2/(μ-2)mλμ(N-α)/(μ-2)(N|wA|2dx+NA|x|-αwA2dx)μ/(μ-2)λ2N/(μ-2)(NF(wA)𝑑x)2/(μ-2)=mλμ(N-α)-2Nμ-2wAA2μ/(μ-2)(NF(wA)𝑑x)2/(μ-2).(4.6)

Inserting the definition of λ into (4.6), we get

cA,KmwAA2μμ-2+2αμ(N-α)-2Nμ-2(NF(wA)𝑑x)2(μ-2)+1αμ(N-α)-2Nμ-2=mwAA2N/α(NF(wA)𝑑x)(N-α)/α

and therefore, using computations (4.1)–(4.3) with ε=A-1/2, we have

C¯1εN-K-1<H(εφ)<C¯2εN-K-1

and

cA,KmσKσN-K(E((ψr2+1r2ψφ2)r(N-2)ε+1+ψ2r(N-α)ε-1)H(εφ)𝑑r𝑑φ)N/αε2(N-α)/α(EF(ψ)rNε-1H(εφ)𝑑r𝑑φ)(N-α)/α.

As in the proof of Lemma 4.1, we take four constants C¯1,,C¯4>0 independent of A such that for every (r,φ)E, one has C¯1εN-K-1<H(εφ)<C¯2εN-K-1 and the terms r(N-2)ε+1, r(N-α)ε-1 and rNε-1 are bounded, and bounded away from zero by C¯3 and C¯4, respectively. Hence, we conclude

cA,KmσKσN-K(C¯2C¯3E((ψr2+1r2ψφ2)+ψ2r)εN-K-1𝑑r𝑑φ)N/αε2(N-α)/α(C¯1C¯4EF(ψ)εN-K-1𝑑r𝑑φ)(N-α)/α=Cε(N-K-1)N/α(E((ψr2+1r2ψφ2)+ψ2r)𝑑r𝑑φ)N/αε(N-K+1)(N-α)/α(EF(ψ)𝑑r𝑑φ)(N-α)/α=CA(K-1)/2+N(1/α-1/2)(E((ψr2+1r2ψφ2)+ψ2r)𝑑r𝑑φ)N/α(EF(ψ)𝑑r𝑑φ)(N-α)/α,

with the definition of the constant C being obvious. As the last ratio does not depend on A, the conclusion ensues. ∎

Proposition 4.3.

Assume (f4) and α>2. Let A>A0 and define u¯HK by setting

u¯K(x):=wA(xλ),with λ:=wAA(NF(wA)𝑑x)1/2.

Then I(u¯K)<0 and the corresponding mountain-pass level (2.7) satisfies

cA,KC2A(K-1)/2,

where the constant C2>0 does not depend on A.

Proof.

The proof is very similar to the one of Proposition 4.2, so we omit here some computational details. As α>2, we have

I(u¯K)λN-22(N|wA|2dx+NA|x|αwA2dx)-λNNF(wA)dx=-λN2NF(wA)dx<0

and

maxt[0,1]I(tu¯K)mu¯KA2μ/(μ-2)(NF(u¯K)𝑑x)2/(μ-2)=m(λN-2N|wA|2dx+λN-αNA|x|-αwA2dx)μ/(μ-2)(λNNF(wA)𝑑x)2/(μ-2)mλμ(N-2)/(μ-2)(N|wA|2dx+NA|x|-αwA2dx)μ/(μ-2)λ2N/(μ-2)(NF(wA)𝑑x)2/(μ-2)=mλμ(N-2)-2Nμ-2wAA2μ/(μ-2)(NF(wA)𝑑x)2/(μ-2).

Recalling the definition of cA,K and inserting the one of λ, we get

cA,KmwAA2μμ-2+μ(N-2)-2Nμ-2(NF(wA)𝑑x)2μ-2+12μ(N-2)-2Nμ-2=mwAAN(NF(wA)𝑑x)(N-2)/2,

and therefore, using computations (4.1)–(4.3) with ε=A-1/2, we have

cA,KmσKσN-K(E((ψr2+1r2ψφ2)r(N-2)ε+1+ψ2r(N-α)ε-1)H(εφ)𝑑r𝑑φ)N/2εN-2(EF(ψ)rNε-1H(εφ)𝑑r𝑑φ)(N-2)/2C(E((ψr2+1r2ψφ2)+ψ2r)εN-K-1𝑑r𝑑φ)N/2εN-2(EF(ψ)εN-K-1𝑑r𝑑φ)(N-2)/2=CA(K-1)/2(E((ψr2+1r2ψφ2)+ψ2r)𝑑r𝑑φ)N/2(EF(ψ)𝑑r𝑑φ)(N-2)/2,

where C>0 is a suitable constant independent of A. This concludes the proof. ∎

5 Proof of Theorem 1.1

This section is entirely devoted to the proof of Theorem 1.1, so we assume all the hypotheses of the theorem. The proof will be achieved through some lemmas.

Let K be any integer such that 2KN-2. Assume A>AK (where AK is defined by (4.5)) and consider the mountain-pass level cA,K defined by (2.7), with u¯KHK given by Lemma 4.2 if α(2N-1,2), or Lemma 4.3 if α(2,2N-2). We are going to show that cA,K is a critical level for the energy functional I defined in (2.2). To this end, we will make use of the sum space

Lp1+Lp2:={u1+u2:u1Lp1(N),u2Lp2(N)}.

We recall from [8] that such a space can be characterized as the set of measurable mappings u:N for which there exists a measurable set EN such that uLp1(E)Lp2(NE) (see [8, Proposition 2.3]). This is a Banach space with respect to the norm

uLp1+Lp2:=infu1+u2=umax{u1Lp1(N),u2Lp2(N)}

(see [8, Corollary 2.11]), and the continuous embedding Lp(N)Lp1+Lp2 holds for all p[p1,p2] (see [8, Proposition 2.17]), in particular, for p=2*. Moreover, for every uLp1+Lp2 and every φLp1(N)Lp2(N), one has

N|uφ|dxuLp1+Lp2(φLp1(N)+φLp2(N)),(5.1)

where pi=pi/(pi-1) is the Hölder conjugate exponent of pi (see [8, Lemma 2.9]).

Lemma 5.1.

cA,K is a critical level for the functional I|HK.

Proof.

Thanks to Lemma 2.4 (note that I(u¯K)<0 implies u¯KA>R), the claim follows from the mountain pass theorem [1] if we show that I|HK satisfies the Palais–Smale condition. Using the compact embeddings of [2] and the results of [8] about Nemytskiĭ operators on Lp1+Lp2, this is a standard proof but we still give some details for the sake of completeness. Let {un} be a sequence in HK such that {I(un)} is bounded and I(un)0 in the dual space of HK. Then, recalling (2.2) and (2.3), we have

12unA2-NF(un)𝑑x=O(1)andunA2-Nf(un)un𝑑x=o(1)un,

and so assumption (f1) implies

12unA2+O(1)=NF(un)𝑑x1θNf(un)un𝑑x=1θunA2+o(1)un.

This yields that {unA} is bounded, since θ>2. On the other hand, thanks to the fact that p1<2*<p2, the space HK is compactly embedded into Lp1+Lp2, since so is the subspace of D1,2(N) made up of the mappings with the same symmetries of HK (see [2, Theorem A.1]). Hence, there exists uHK such that, up to a subsequence, we have unu in HK and unu in Lp1+Lp2. This implies that {f(un)} is bounded in both Lp1(N) and Lp2(N), since assumption (f0) ensures that the operator vf(v) is continuous from Lp1+Lp2 into Lp1(N)Lp2(N) (see [8, Corollary 3.7]). Then, by (5.1), we get

|Nf(un)(un-u)dx|N|f(un)||un-u|dxun-uLp1+Lp2(f(un)Lp1(N)+f(un)Lp2(N))(const.)un-uLp1+Lp2=o(1),

and therefore

un-uA2=(un,un-u)A-(u,un-u)A=I(un)(un-u)+Nf(un)(un-u)𝑑x-(u,un-u)A=o(1),

where (u,un-u)A=o(1), since unu in HK, and I(un)(un-u)=o(1) because I(un)0 in the dual space of HK and {un-u} is bounded in HK. This completes the proof. ∎

The next lemma clarifies why our separation of cA,K and mA needs assumption (1.5) and the lower bound α>2N-1. Recall the definition (1.4) of ν=νN,α,p1,p2.

Lemma 5.2.

For every α(2N-1,2N-2), α2, we have ν1.

Proof.

Assume 2N-1<α<2. Since α>2N-1, we have

2N-1α-2N(1α-12)=N-2α>1.

On the other hand, by easy computations, condition

2N-22-α2*-p1p1-2-2N(1α-12)>1

turns out to be equivalent to the first inequality of assumption (1.5). This proves that

2min{N-1α,N-22-α2*-p1p1-2}-2N(1α-12)>1,

which means

2min{N-1α,N-22-α2*-p1p1-2}-2N(1α-12)2,

and thus ν1. Similarly, if 2<α<2N-2 , we readily have 2(N-1)α>1, and condition

2N-2α-2p2-2*p2-2>1

turns out to be equivalent to the second inequality of (1.5). This proves again that ν1. ∎

Proof of Theorem 1.1.

On the one hand, the restriction I|Hr has a critical point ur0 thanks to the results of [25], since (f0) ensures that one can find p[p1,p2] such that |f(u)|(const.)up-1 (cf. (2.1)) and (1.3) holds. On the other hand, according to Lemma 5.2, there are ν1 integers K (precisely K=2,,ν+1) such that

K-12+N(1α-12)<min{N-1α,N-22-α2*-p1p1-2}if 2N-1<α<2

and

K-12<min{N-1α,N-2α-2p2-2*p2-2}if 2<α<2N-2.

Let K be any of such integers. By Remark 3.3 and Propositions 3.2, 4.2 and 4.3, there exists A*>AK such that

cA,K<mAfor every A>A*.(5.2)

Then, by Lemma 5.1, there exists uKHK such that I(uK)=cA,K and I|HK(uK)=0, where uK0, since cA,K>0 and I(0)=0. Both ur and uK are also critical points for the functional I:Hα1, by the Palais principle of symmetric criticality [23]. Moreover, it easy to check that they are nonnegative: test I(uK) with the negative part uK-Hα1 of uK and use the fact that f(s)=0 for s<0 to get I(uK)uK-=-uK-A2=0; the same applies for ur. Therefore, ur and uK are weak solutions to problem (1.1). Finally, uK is not radial, because otherwise Lemma 2.3 would imply cA,K=I(uK)mA, which is false by (5.2). This also implies uK1uK2 for K1K2, thanks to Lemma 2.1. ∎

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About the article

Received: 2017-08-03

Revised: 2017-09-01

Accepted: 2017-09-24

Published Online: 2017-11-21


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 885–901, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0177.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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