# Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity  inR3

Authors: Ling Ding; Lin Li; Jin-Ling Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the following nonhomogeneous Kirchhoff equation:-(a+b∫R3‍|∇u|2dx)Δu+u=k(x)f(u)+h(x),x∈R3,u∈H1(R3),u>0,x∈R3, where f is asymptotically linear with respect to t at infinity. Under appropriate assumptions on k,f, and h, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the following nonhomogeneous Kirchhoff equation:(1)-(a+b∫R3‍|∇u|2dx)Δu+u=k(x)f(u)+h(x),000000000000x∈R3,u∈H1(R3),u>0,x∈R3,
where constants a,b>0, and functions k,f and h satisfy the following conditions: k is a positive bounded condition, f∈C(R,R+), f(t)≡0 if t<0 and h∈L2(R3), h≥0. Note that, with a=1,  b=0, and R3 replaced by RN, problem (1) reduces to(2)-Δu+u=k(x)f(u)+h(x)inRN,
which can be looked at as a generalization of the well known Schrödinger equation.WhenΩ is a smooth bounded domain in RN, the problem
(3)-(a+b∫Ω‍|∇u|2dx)Δu=g(x,u),x∈Ω,u=0,x∈∂Ω,
is related to the stationary analogue of the Kirchhoff equation which was proposed by Kirchhoff in 1883 (see [1]) as a generalization of the well known d’Alembert’s wave equation
(4)ρ∂2u∂t2-(P0h+E2L∫0L‍|∂u∂x|2dx)∂2u∂x2=g(x,u)
for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Here, L is the length of the string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density, and P0 is the initial tension. Moreover, Kirchhoff’s type problems also model several physical systems and biological systems and there are many interesting results for problem (3) which can be found in [2–8] and the references therein.Some interesting studies for Kirchhoff-type problem (3) in a bounded domain Ω of RN by variational methods can be found in [2, 9–22]. Very recently, some authors had studied the Kirchhoff equation on the whole space RN and obtained the existence of multiple solutions (see [23–31]). In the same spirit of [24–26, 28–31], we study a nonhomogeneous Kirchhoff equation (1) on the whole space R3. Especially, inspired by the paper [32, 33], we consider the asymptotically linear nonlinearity at infinity of problem (1). For the nonhomogeneous Kirchhoff problem, Chen and Li in [23] study it under the condition of superlinear nonlinearity at infinity. In [33], Wang and Zhou study the existence of two positive solutions for a nonhomogeneous elliptic equation ((1) with a=1 and b=0). In [32], Sun et al. study the existence of a ground state solution for some nonautonomous Schrödinger-Poisson systems involving the asymptotically linear nonlinearity at infinity without the nonhomogeneous term. But we will study the existence of two positive solutions for Kirchhoff-type problem (1) with a,b>0, the asymptotically linear nonlinearity at infinity and the nonhomogeneous term. So, we can not obtain the existence of a ground state solution for Kirchhoff-type problem (1) and the compactness result as in [32] because of the nonhomogeneous term, and we cannot easily obtain the compactness result as in [33] due to the nonlocal term (or b≠0). To our best knowledge, little has been done for nonhomogeneous Kirchhoff problems with respect to the asymptotically linear nonlinearity at infinity.Before stating our main results, we give some notations. For any1≤q≤+∞, we denote by ∥·∥q the usual norm of the Lebesgue space Lq(R3). Define the function space
(5)H1(R3)≔{u∈L2(R3):∇u∈L2(R3)}
with the product and equivalent norm
(6)(u,v)=∫R3‍(a∇u·∇v+uv)dx,∥u∥≔(∫R3‍(a|∇u|2+|u|2)dx)1/2.
Define the function space
(7)D1,2(R3)≔{u∈L6(R3):∇u∈L2(R3)}
with the standard product and norm
(8)(u,v)=∫R3‍∇u·∇vdx,∥u∥D≔(∫R3‍|∇u|2dx)1/2.
Recall that the Sobolev’s inequality with the best constant is
(9)∥v∥6≤S∥v∥.
Moreover, problem (1) has a variational structure. Indeed the corresponding action functional I:H1(R3)→R of (1) is defined by
(10)I(u)=a2∫R3‍|∇u|2dx+b4(∫R3‍|∇u|2dx)2+12∫R3‍|u|2dx-∫R3‍k(x)F(u)dx-∫R3‍h(x)udx.
By Lemma 2.1 in [24] or Lemma 1 in [25], the functional I is C1(H1(R3),R) with the derivative given by
(11)〈I′(u),u〉=(a+b∫R3‍|∇u|2dx)∫R3‍∇u·∇vdx+∫R3‍uvdx-∫R3‍k(x)f(u)udx-∫R3‍h(x)vdx.
Hence, if u∈H1(R3) is a nonzero critical point of I, then it is also a nonnegative solution of (1). In fact, by f(t)≡0 if t<0 and h≥0, we have 〈I′(u),u-〉=-(a+b∥u∥2) ∫R3‍|∇u-|2dx-∫R3‍(u-)2dx -∫R3‍k(x)f(u)u-dx -∫R3‍h(x)u-dx=0, where u-=max⁡{-u,0}. This yields that u-=0; then u=u+-u-=u+≥0, where u+=max⁡{u,0}. By the maximum principle, the nonzero critical point of I is the positive solution for problem (1).Here is the main result of this paper.Theorem 1.
Suppose thath∈L2(R3), h≥0, and the following conditions hold.(f1)
f∈C(R,R+), f(0)=0, and f(t)≡0 for t<0.(f2)
lim⁡t→0(f(t)/t)=0.(f3)
lim⁡t→+∞(f(t)/t)=l<+∞.(k1)
k(x) is a positive continuous function and there exists R0>0 such that
(12)sup⁡{f(t)t:t>0}<inf⁡{1k(x):|x|≥R0}.(k2)
Let(13)l0>μ*≔inf⁡{∫R3‍(a|∇u|2+u2)dx:u∈H1(R3),000000∫R3‍k(x)F(u)dx≥b2l2} hold, where l0=min⁡{l,(b/2)l2}, F(t)=∫0tf(s)ds.
Then problem (1) has at least two positive solutions u0,u∈H1(R3) satisfying I(u0)<0 and I(u)>0 if ∥h∥2<m for some small m>0.Remark 2.
It is not difficult to find some functionsk,f satisfying conditions of Theorem 1. For example, for any R0>0, let
(14)f(t)={R0t21+t,ift≥0,0,ift<0.
Clearly, f satisfies (f1)–(f3) with l=R0. Moreover, F(t)=R0((1/2)t2-t+ln⁡(1+t)) and sup⁡{f(t)/t:t>0}=R0. Taking a positive continuous function k(x),
(15)k(x)={C01+|x|,if|x|≤R02,11+|R0|,if|x|≥R0,
where C0=3M3(1+R0/2)/4π(ln⁡2-1/2) for some M>0. Note that
(16)inf⁡{1k(x):|x|≥R0}=1+R0>R0=sup⁡{f(t)t:t>0};
then (k1) holds. To verify the condition (k2), we have to choose some special R0>0. For any R>0, taking ψ∈C0∞(R3,[0,1]) such that ψ(x)=1 if |x|≤R,  ψ(x)=0 if |x|≥2R and |∇ψ(x)|≤C/aR for all x∈R3, where C>0 is an arbitrary constant independent of x. Then, for any R0>2R, we have
(17)∫R3‍k(x)F(ψ)dx≥∫|x|≤R‍k(x)F(ψ)dx≥R0C01+R(ln⁡2-12)|BR(0)|≥3M3(1+R0/2)4π(ln⁡2-1/2)4πR3R03(1+R0/2)×(ln⁡2-12)=M3R0R3,(18)∫R3‍(a|∇ψ|2+|ψ|2)dx≤∫|x|≤2R‍C2R2dx+∫|x|≤2R‍dx≤(1+C2R2)32π3R3≤32π3R(C2+R2).
Taking R0=l=1, R=(1/M)R0b/23=(1/M)b/23, where M is large enough such that 2R<R0/4, 40π/3M3<1, and (40π/3M3)(b/2)<1. Let C=(1/4M)b/23. Then, we obtain that ∫R3‍k(x)F(ψ)dx≥bR0/2=bl2/2. Moreover, in view of the definition of μ* and (18), one has
(19)μ*≤∫R3‍(a|∇ψ|2+|ψ|2)dx≤32π3R(C2+R2)<40π3M3b2<l0.
So, condition (k2) holds. In particular, the condition (k1) and above examples can also be found in [34] in which the asymptotically linear term k(x)f(u) satisfying (k1) appeared first.Remark 3.
Ifh≡0, we know that problem (1) has a positive ground state solution by using the method in [32] and a trivial solution (u(x)≡0). If h≢0, a trivial solution (u(x)≡0) is replaced by the local minimum solution by Theorem 1. Note that the local minimal solution exists due to the homogeneous term which is looked at as a small perturbation because ∥h∥2<m for small m.
In order to obtain our results, we have to overcome various difficulties. Since the embedding ofH1(R3) into Lp(R3), p∈[2,6], is not compact, condition (k1) and (k2) are crucial to obtain the boundedness of Cerami sequence. Furthermore, in order to recover the compactness, we establish a compactness result ∫|x|≥R‍(|∇un|2+|un|2)dx≤ɛ which is similar to [32] but different from the one in [24–26, 28–31]. In fact, this difficulty can be avoided, when problems are considered, restricting I to the subspace of H1(R3) consisting of radially symmetric functions [23, 24, 29] and constraint potential functions [25, 30], or when one is looking for semiclassical states [28], by using perturbation methods or a reduction to a finite dimension by the projections method. Third, it is not difficult to find that every (PS) sequence is bounded because a variant of Ambrosetti-Rabinowitz condition is satisfied (see [23, 25, 31]). However, for the asymptotically linear case, we have to find another method to verify the boundedness of (PS) sequence.
This paper is organized as follows. In Section2, we manage to give proofs of Theorem 1. In the following discussion, we denote various positive constants as C or Ci  (i=1,2,3,…) for convenience.


## 2. Proof of Main Result

In this section, we prove that problem (1) has a mountain pass type solution and a local minimum solution with h≢0. For this purpose, we use a variant version of Mountain Pass Theorem [35], which allows us to find a so-called Cerami type (PS) sequence (Cerami sequence, in short). The properties of this kind of Cerami sequence sequences are very helpful in showing its boundedness in the asymptotical cases. The following lemmas will show that I has the so-called mountain pass geometry.Lemma 4.
Suppose thath∈L2(R3), h≥0, (f1)–(f3), and (k1) hold. Then there exist ρ,α,m>0 such that I(u)|∥u∥=ρ≥α>0 for ∥h∥2<m.Proof.
For anyɛ>0, it follows from (f1)–(f3) that there exists Cɛ>0 such that
(20)|f(t)|≤ɛ|t|+Cɛ|t|5∀t∈R.
Therefore, we have
(21)|F(t)|≤12ɛ|t|2+Cɛ6|t|6∀t∈R.
Furthermore, by (f1)–(f3) and (k1), there exists C1>0 such that
(22)k(x)≤C1∀x∈R3.
According to (21), (22), and the Sobolev inequality, we deduce that
(23)|∫R3‍k(x)F(u)dx|≤ɛC12∫R3‍|u|2dx+C1Cɛ6∫R3‍|u|6dx≤ɛC12∥u∥2+C2∥u∥6,
where C2=C1CɛS6/6. By b>0, h∈L2(R3), and the Ho¨lder inequality, one has
(24)I(u)=a2∫R3‍|∇u|2dx+b4(∫R3‍|∇u|2dx)2+12∫R3‍|u|2dx-∫R3‍k(x)F(u)dx-∫R3‍h(x)udx≥a2∫R3‍|∇u|2dx+12∫R3‍|u|2dx-ɛC12∥u∥2-C2∥u∥6-∥h∥2∥u∥2≥12∥u∥2-ɛC12∥u∥2-C2∥u∥6-∥h∥2∥u∥≥∥u∥(1-ɛC12∥u∥-C2∥u∥5-∥h∥2).
Taking ɛ=1/2C1 and setting g(t)=(1/4)t-C2t5 for t≥0, we see there exists ρ=(1/20C2)1/4 such that max⁡t≥0g(t)=g(ρ)≔m>0. Then it follows from (24) that there exists α>0 such that I(u)|∥u∥=ρ≥α>0 for ∥h∥2<m. Of course, ρ,m can be chosen small enough; we can obtain the same result: there exists α>0 such that I(u)|∥u∥=ρ≥α>0 for ∥h∥2<m.Lemma 5.
Suppose thath∈L2(R3), h≥0, (f1)–(f3), and (k1)-(k2) hold. Then there exists v∈H1(R3) with ∥v∥>ρ, ρ is given by Lemma 4, such that I(v)<0.Proof.
By (k2) andh≥0, in view of the definition of μ* and l0>μ* with l0=min⁡{l,(b/2)l2}, there is a nonnegative function v∈H1(R3) such that
(25)∫R3‍k(x)F(v)dx≥b2l2,∫R3‍h(x)vdx≥0,
and μ*≤∥v∥2<l0. Then, we have
(26)I(v)=a2∫R3‍|∇v|2dx+b4(∫R3‍|∇v|2dx)2+12∫R3‍|v|2dx-∫R3‍k(x)F(v)dx-∫R3‍h(x)vdx≤12∥v∥2+b4∥v∥4-b2l2≤12∥v∥2-b4l2<0.
Choosing ρ>0 small enough in Lemma 4 such that ∥v∥>ρ, then this Lemma is proved.
From Lemmas4 and 5 and Mountain Pass Lemma in [35], there is a Cerami sequence {un}⊂H1(R3) such that
(27)∥I′(un)∥H-1(1+∥un∥)⟶0,I(un)⟶casn⟶∞,
where H-1 denotes the dual space of H1(R3). In the following Lemmas 6 and 7, we shall prove that I satisfies the Cerami condition, that is; the Cerami sequence {un} has a convergence subsequence.Lemma 6.
Suppose thath∈L2(R3), h≥0, (f1)–(f3), and (k1) hold. Then {un} defined in (27) is bounded in H1(R3).Proof.
By contradiction, let∥un∥→∞. Define wn=un∥un∥-1. Clearly, {wn} is bounded in H1(R3) and there is a w∈H1(R3) such that, up to a sequence,
(28)wn⟶wweaklyinH1(R3),wn⟶wa.e.inR3,wn⟶wstronglyinLloc2(R3)
as n→∞.
Firstly, we claim thatw is nontrivial; that is, w≢0. Otherwise, if w≡0, the Sobolev embedding implies that wn→0 strongly in L2(BR0); R0 is given by (k1). By (f1)–(f3), there exists C3>0 such that
(29)f(t)t≤C3∀t∈R.
Then, for all n∈N, we have
(30)0≤∫|x|<R0‍k(x)f(un)unwn2dx≤C3∥k∥∞∫|x|<R0‍wn2dx⟶0.
This yields
(31)lim⁡n→∞∫|x|<R0‍k(x)f(un)unwn2dx=0.
Furthermore, by (k1), there exists a constant θ∈(0,1) such that
(32)sup⁡{f(t)t:t>0}≤θinf⁡{1k(x):|x|≥R0}.
Then, for all n∈N, we have
(33)∫|x|≥R0‍k(x)f(un)unwn2dx≤θ∫|x|≥R0‍wn2dx≤θ∥w∥2=θ<1.
Combining (31) and (33), we obtain
(34)limsup⁡n→∞∫R3‍k(x)f(un)unwn2dx<1.
By (27), we get
(35)0≤|〈I′(un),un〉|≤∥I′(un)∥H-1∥un∥≤∥I′(un)∥H-1(1+∥un∥)⟶0
as n→∞. Together with ∥un∥→∞ as n→∞, it follows that
(36)〈I′(un),un〉∥un∥2=o(1).
Together with b>0, we have
(37)o(1)=1∥un∥2(a∫R3‍|∇un|2dx+∫R3‍|un|2dx000000000+b(∫R3‍|∇un|2dx)2000000000-∫R3‍k(x)f(un)undx-∫R3‍h(x)undx)≥∥wn∥2-∫R3‍k(x)f(un)unwn2dx≥1-∫R3‍k(x)f(un)unwn2dx,
where, and in what follows, o(1) denotes a quantity which goes to zero as n→∞. Therefore, we deduce that
(38)∫R3‍k(x)f(un)unwn2dx+o(1)≥1,
which contradicts (34). So, w≢0.
Furthermore, because∥un∥→∞ as n→∞, it follows from (35) that
(39)〈I′(un),un〉∥un∥4=o(1);
that is,
(40)o(1)=b(∫R3‍|∇un|2dx)2∥un∥4-1∥un∥2∫R3‍k(x)f(un)unwn2dx.
Together with (22), (29), and b>0, one has
(41)(∫R3‍|∇un|2dx)2∥un∥4=(∫R3‍(|∇un|2+|un|2)dx-∫R3‍|un|2dx)2∥un∥4=o(1).
This yields
(42)(∫R3‍(|∇un|2+|un|2)dx)2∥un∥4-2∫R3‍(|∇un|2+|un|2)dx∫R3‍|un|2dx∥un∥4+(∫R3‍|un|2dx)2∥un∥4=o(1).
This means(43)1-2∫R3‍|wn|2dx+(∫R3‍|wn|2dx)2=(1-∫R3‍|wn|2dx)2=o(1).
Therefore, we have
(44)∫R3‍|wn|2dx⟶1asn⟶∞.
By ∥wn∥=1, we get ∫R3‍|∇wn|2dx→0 as n→∞; thus wn→0 strongly in D1,2(R3); therefore, wn⇀0 weakly in D1,2(R3). Since wn⇀w weakly in H1(R3); we have wn⇀w weakly in D1,2(R3). By the uniqueness of the weak limitation, we have w=0 which contradicts w≠0. Therefore, the Cerami sequence {un} is bounded in H1(R3).Lemma 7.
Suppose thath∈L2(R3), h≥0, (f1)–(f3), and (k1) hold. Then for any ɛ>0, there exist R(ɛ)>R0 and n(ɛ)>0 such that {un} defined in (27) satisfies ∫|x|≥R‍(|∇un|2+|un|2)dx≤ɛ for n>n(ɛ) and R≥R(ɛ).Proof.
LetξR:R3→[0,1] be a smooth function such that
(45)ξR(x)={0,0≤|x|≤R2,1,|x|≥R.
Moreover, there exists a constant C4 independent of R such that
(46)|∇ξR(x)|≤C4R∀x∈R3.
Then, for all n∈N and R≥R0, by (45), (46), and the Ho¨lder inequality, we have
(47)∫R3‍|∇(unξR)|2dx≤∫R3‍|un|2|∇ξR|2dx+∫R3‍|∇un|2|ξR|2dx+2∫R3‍|un||ξR||∇un||∇ξR|dx≤∫R/2<|x|<R‍|∇un|2dx+∫|x|>R‍|∇un|2dx+C42R2∫R3‍|un|2dx+2(∫R3‍|∇un|2|ξR2|dx)1/2×(∫R3‍|un|2|∇ξR|2dx)1/2≤∫R/2<|x|<R‍|∇un|2dx+∫|x|>R‍|∇un|2dx+C42R2∫R3‍|un|2dx+2(∫R/2<|x|<R‍|∇un|2dx0000000+∫|x|>R‍|∇un|2dx)1/2×(C42R2∫R3‍|un|2dx)1/2≤(2+C42R2+22C4R)∥un∥2≤(2+C42R02+22C4R0)∥un∥2.
This implies that
(48)∥unξR∥≤C5∥un∥
for all n∈N and R≥R0, where C5=max⁡{(3+(C42/R02)+22C4/R0)1/2,1}. From Lemma 6, we know that {un} is bounded in H1(R3). Together with (27), we obtain that I′(un)→0 in H-1(R3). Moreover, by (48), for ɛ>0, there exists n(ɛ)>0 such that
(49)〈I′(un),ξRun〉≤C5∥I′(un)∥H-1(R3)∥un∥≤ɛ4
for n>n(ɛ) and R>R0. Note that
(50)〈I′(un),ξRun〉=(a+b∫R3‍|∇un|2dx)×∫R3‍|∇un|2ξRdx+∫R3‍|un|2ξRdx+(a+b∫R3‍|∇un|2dx)×∫R3‍un∇un·∇ξRdx-∫R3‍k(x)f(un)unξRdx-∫R3‍h(x)unξRdx≤ɛ4.
This yields
(51)(a+b∫R3‍|∇un|2dx)∫R3‍|∇un|2ξRdx+∫R3‍|un|2ξRdx+(a+b∫R3‍|∇un|2dx)∫R3‍un∇un·∇ξRdx≤∫R3‍k(x)f(un)unξRdx+∫R3‍h(x)unξRdx+ɛ4.
By (32), we have
(52)k(x)f(un)un≤θun2forθ∈(0,min⁡⁡{1,a}),|x|≥R0.
This yields
(53)∫R3‍k(x)f(un)unξRdx≤θ∫R3‍un2ξRdx
for all n∈N and |x|≥R0. For any ɛ>0, there exists R(ɛ)≥R0 such that
(54)1R2≤4ɛ2C42∀R>R(ɛ).
Because h∈L2(R3), h≥(≢)0, there exists ρ¯=ρ¯(ɛ) such that
(55)∥h∥2,R3∖Bρ(0)<ɛ,∀ρ≥ρ¯.
By the Ho¨lder inequality, (45), (55), and the boundedness of {un} in H1(R3), we have
(56)∫R3‍h(x)unξRdx≤∥h(x)ξR∥2∥un∥2≤∥h(x)∥2,|x|>R/2∥un∥2≤ɛ40000000000000∀R>R(ɛ).
By the Young inequality, (46), and (54), for all n∈N and R>R(ɛ), we obtain
(57)∫R3‍|un∇un·∇ξR|dx≤∫R3‍2ɛ|∇un|12ɛ|un||∇ξR|dx≤ɛ∫R3‍|∇un|2dx+14ɛ∫|x|≤R‍|un|2C42R2dx≤ɛ∫R3‍|∇un|2dx+ɛ∫|x|≤R‍|un|2dx≤ɛmax⁡⁡{a,1}∥un∥2.
Combining b>0, (51), (53), (56), (57), and the boundedness of {un} in H1(R3), there exists C6>0 such that
(58)min⁡⁡{1-θ,1}∫R3‍(a|∇un|2+|un|2)ξRdx≤ɛ2+ɛmax⁡⁡{a,1}∥un∥2(a+b∫R3‍|∇un|2dx)≤C6ɛ∀R>R(ɛ).
Note that C6 is independent of ɛ. So, for any ɛ>0, we can choose R(ɛ)>R0 and n(ɛ)>0 such that ∫|x|≥R‍(|∇un|2+|un|2)dx≤ɛ holds.Lemma 8.
Suppose thath∈L2(R3), h≥0, (f1)–(f3), and (k1)-(k2) hold. Then the sequence {un} in (27) has a convergent subsequence. Moreover, I possesses a nonzero critical point u in H1(R3) and I(u)>0.Proof.
By Lemma6, the sequence {un} in (27) is bounded in H1(R3). We may assume that up to a subsequence un⇀u weakly in H1(R3) for some u∈H1(R3). Now, we shall show that ∥un∥→∥u∥ as n→∞.
By (11), we have(59)〈I′(un),un〉=(a+b∫R3‍|∇un|2dx)∫R3‍|∇un|2dx+∫R3‍un2dx-∫R3‍k(x)f(un)un-∫R3‍h(x)undx,〈I′(un),u〉=(a+b∫R3‍|∇un|2dx)∫R3‍∇un·∇udx+∫R3‍unudx-∫R3‍k(x)f(un)u-∫R3‍h(x)udx.
By (59), b>0, and the boundedness of {un} in H1(R3), we easily get
(60)〈I′(un),un-u〉=(a+b∫R3‍|∇un|2dx)∫R3‍|∇un|2dx+∫R3‍un2dx-∫R3‍k(x)f(un)undx-∫R3‍h(x)undx-(a+b∫R3‍|∇un|2dx)∫R3‍∇u·∇undx-∫R3‍uundx+∫R3‍k(x)f(un)udx+∫R3‍h(x)udx=(a+b∫R3‍|∇un|2dx)×∫R3‍|∇(un-u)|2dx+∫R3‍|un-u|2dx+(a+b∫R3‍|∇un|2dx)×∫R3‍∇u∇(un-u)dx+∫R3‍u(un-u)dx-∫R3‍k(x)f(un)(un-u)dx-∫R3‍h(x)(un-u)dx≥a∫R3‍∇|(un-u)|2dx+∫R3‍|un-u|2dx+(a+b∫R3‍|∇un|2dx)×∫R3‍∇u∇(un-u)dx+∫R3‍u(un-u)dx-∫R3‍k(x)f(un)(un-u)dx-∫R3‍h(x)(un-u)dx≥∥un-u∥2+(a+b∫R3‍|∇un|2dx)×∫R3‍∇u∇(un-u)dx+∫R3‍u(un-u)dx-∫R3‍k(x)f(un)(un-u)dx-∫R3‍h(x)(un-u)dx.
One has
(61)∥un-u∥2≤〈I′(un),un-u〉-(a+b∫R3‍|∇un|2dx)∫R3‍∇u∇(un-u)dx-∫R3‍u(un-u)dx+∫R3‍k(x)f(un)(un-u)dx+∫R3‍h(x)(un-u)dx.
It is clear that
(62)〈I′(un),un-u〉⟶0,asn⟶∞.
Sinceun⇀u weakly in H1(R3), we obtain
(63)∫R3‍(∇un·∇u+unu)dx=∫R3‍(|∇u|2+|u|2)dx+o(1),asn⟶∞.
By the continuity of imbedding H1(R3)↪L2(R3), we have that un⇀u weakly in L2(R3); that is,
(64)∫R3‍unudx=∫R3‍u2dx+o(1),asn⟶∞.
By (63) and (64), we deduce
(65)∫R3‍∇un·∇udx=∫R3‍|∇u|2dx+o(1),asn⟶∞.
Combining the boundedness of {un} in H1(R3), (64), and (65), we obtain
(66)(a+b∫R3‍|∇un|2dx)∫R3‍∇u∇(un-u)dx+∫R3‍u(un-u)dx=o(1),asn⟶∞.
Moreover, by (32), Lemma 7, and un⇀u in H1(R3), for any ɛ>0 and n large enough, one has
(67)∫|x|≥R(ɛ)‍k(x)f(un)undx-∫|x|≥R(ɛ)‍k(x)f(un)udx=∫|x|≥R(ɛ)‍k(x)f(un)(un-u)dx≤∫|x|≥R(ɛ)‍|k(x)f(un)||un-u|dx≤(∫|x|≥R(ɛ)‍|k2(x)f2(un)|dx)1/2×(∫|x|≥R(ɛ)‍|un-u|2dx)1/2≤θ(∫|x|≥R(ɛ)‍|un2|dx)1/2(∫|x|≥R(ɛ)‍|un-u|2dx)1/2≤θ(∫|x|≥R(ɛ)‍(a|∇un|2+|un|2)dx)1/2×(∫|x|≥R(ɛ)‍|un-u|2dx)1/2≤θɛ.
This and the compactness of embedding H1(R3)↪Lloc⁡2(R3) imply that
(68)∫R3‍k(x)f(un)undx=∫R3‍k(x)f(un)udx+o(1).
Sinceun is bounded in H1(R3) and the continuity of the Sobolev embedding of H1(R3) imbedding in L2(R3), for any choice of ɛ>0 and ρ>0, the relation
(69)∥un-u∥2,Bρ(0)<ɛ
holds for large n. By h∈L2(R3), for any ɛ>0 there exists ρ¯=ρ¯(ɛ) such that
(70)∥h∥2,R3∖Bρ(0)<ɛ,∀ρ≥ρ¯.
By (70) and (69), we have
(71)∫R3‍h(x)undx-∫R3‍h(x)udx≤∫R3∖Bρ(0)‍|h(x)(un-u)|dx+∫Bρ(0)‍|h(x)(un-u)|dx≤∥h(x)∥2,R3∖Bρ(0)∥un-u∥2,R3∖Bρ(0)+∥h(x)∥2,Bρ(0)∥un-u∥2,Bρ(0)≤ɛ∥un-u∥2,R3∖Bρ(0)+ɛ∥h(x)∥2,Bρ(0)≤C7ɛ.
This yields
(72)∫R3‍h(x)undx=∫R3‍h(x)udx+o(1).
By (61), (62), (66), (68), and (72), we have
(73)∥un-u∥2=o(1),asn⟶∞,
by a>0. This yields that ∥un∥→∥u∥ as n→∞ and u is a nonzero critical point of I in H1(R3) and I(u)>0 by Mountain Pass Theorem in [35].Now, we give local properties of the variational functionalI, which is required by using Ekeland’s variational principle.Lemma 9.
Suppose thath∈L2(R3), h≥(≢)0, (f1)–(f3), and (k1) hold. If ∥h∥2<m, then there exists u0∈H1(R3) such that
(74)I(u0)=inf⁡{I(u):u∈B¯ρ}<0,whereBρ={u∈H1(R3):∥u∥<ρ},m,ρ are given by Lemma 4 and u0 is a positive solution of system (1).Proof.
Becauseh∈L2(R3), h≥(≢)0, we can choose a function φ∈H1(R3) such that
(75)∫R3‍h(x)φdx>0.
Together with (f1), (k1), and (75), for t>0, we have
(76)I(tφ)=at22∫R3‍|∇φ|2dx+bt44(∫R3‍|∇φ|2dx)2+t22∫R3‍|φ|2dx-∫R3‍k(x)F(tφ)dx-∫R3‍h(x)tφdx≤t22∥φ∥2+bt44∥φ∥4-t∫R3‍h(x)φdx≤0
for t>0 small enough. Thus there exists u small enough such that I(u)<0. By Lemma 4, we deduce that
(77)c0≔inf⁡u∈B¯ρI(u)<0<inf⁡u∈∂B¯ρI(u).
By applying Ekeland’s variational principle [36, Theorem 4.1] in B¯ρ, there is a minimizing sequence {un}⊂B¯ρ such that
(78)(i)c0≤I(un)<c0+1n,(ii)I(w)≥I(un)-1n∥w-un∥∀w∈B¯ρ.
Then, by a standard procedure, we can show that {un} is a bounded (PS) sequence of I. Lemmas 7 and 8 imply that there exists u0∈H1(R3) such that I′(u0)=0 and I(u0)=c0<0. So this lemma is proved.Proof of Theorem 1.2.
By Lemmas4–8, we obtain the existence of a mountain pass solution u for problem (1) and I(u)>0. By Lemma 9, we know that problem (1) has a local minimum solution u0 and I(u0)<0. Thus, u≠u0 and u,u0 are positive. Thus this theorem is proved.



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