# Strauss’s Radial Compactness and Nonlinear Elliptic Equation Involving a Variable Critical Exponent

Authors: Masato Hashizume; Megumi Sano
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study existence of a nontrivial solution of-Δpu(x)+u(x)p-1=u(x)q(x)-1,u(x)≥0,x∈RN,u∈Wrad1,p(RN), under some conditions on q(x), especially, liminf|x|→∞q(x)=p. Concerning this problem, we firstly consider compactness and noncompactness for the embedding from Wrad1,p(RN) to Lq(x)(RN). We point out that the decaying speed of q(x) at infinity plays an essential role on the compactness. Secondly, by applying the compactness result, we show the existence of a nontrivial solution of the elliptic equation.


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## Body

## 1. Introduction and Main Results

In this article, we consider the following nonlinear elliptic equation:(1)-Δpu+up-1=uq·-1,u≥0inRN,u∈Wrad1,pRN,for 1<p<N, where Δpu=div(|∇u|p-2∇u) is p-Laplacian and variable exponent q(x) is a measurable function satisfying q(x)>p,  liminf|x|→∞q(x)=p. p(x)-Laplacian type elliptic equation is one of the problems with variable exponent and this type equation on is studied by many researchers in both bounded domain case and unbounded domain case. In this paper, we refer the paper studying on RN, in several subjects: multiplicity of solutions (see, e.g., [1, 2]), existence of solutions of equations involving several nonlinearities (see, e.g., [3, 4]), equations under periodic assumptions (see e.g. [5, 6]), and so on. Moreover, existence of solutions of (1) involving variable exponent touching the critical exponent, that is, esssupRNq(x)=p∗≔Np/N-p, is studied by [7, 8].Concerning the classical Sobolev embedding inRN, it is well known that the embedding from Wrad1,p(RN) to Lp∗(RN) is not compact. And the embedding from Wrad1,p(RN) to Lp(RN) is not compact, either. In the viewpoint of the lack of compactness, consider the case where essinfRNq(x)=p is natural as another critical case.However, even forp-Laplace equation there are no results in this case. Thus we study problem (1) at the opening of this article. In this case, unlike the subcritical case, we need to overcome some difficulties to show the existence of a nontrivial solution of (1). We will explain them more precisely after Remark 5. Thus in advance of study of (1), we consider the related embedding to the equation. Namely, we study the embedding from Wrad1,p(RN) to Lq(x)(RN).We define the generalized Sobolev spacesWk,p(x)(Ω) with variable exponents p(x) according to [9]. For a domain Ω⊂RN and a function p∈L∞(Ω) with p(x)≥1 we set (2)LpxΩ=uis  a  real  measurable  function  onΩ∣∫Ωuxpxdx<∞,Wk,pxΩ=u∈LpxΩ∣Dαu∈LpxΩ,α≤k. These Lp(x)(Ω) and Wk,p(x)(Ω) are Banach spaces with the following norms: (3)upx=infλ>0∣∫Ωuλpxdx≤1,uWk,px=upx+∑α≤kDαupx.The Sobolev embedding theorem ofWk,p(x) and related subjects have been well studied so far; see, e.g., [10–19], and we refer to the book [20]. For example, in bounded domain case, the Sobolev best constant with a variable critical exponent and the existence of extremals were studied in [21]; see also [22] in the Sobolev trace embedding case. One of the results in [18] is the existence of the compact embedding. They consider the situation when p(x) is uniformly continuous on Ω¯ and 1<essinfΩ¯p(x)≤esssupΩ¯p(x)<N/k. Under this situation there exists a compact embedding from Wk,p(x)(Ω) to Lq(x)(Ω) for q(x) satisfying p(x)≤q(x) a.e. in Ω and essinfΩ¯(p∗(x)-q(x))>0, where p∗(x)=Np(x)/(N-kp(x)). On the other hand, for W1,p(Ω) Kurata and Shioji [17] consider the critical case, that is, esssupΩ¯q(x)=p∗. They showed that if there exist x0∈Ω,C0>0,η>0 and 0<l<1 such that esssupΩ∖Bη(x0)q(x)<p∗ and (4)qx≤p∗-C0log⁡x-x0lfor  a.e.x∈Ω∩Bηx0, then the embedding from W1,p(Ω) to Lq(x)(Ω) is compact. Conversely, if (5)qx≥p∗-C0log⁡x-x0for  a.e.x∈Ω∩Bηx0, then the embedding from W1,p(Ω) to Lq(x)(Ω) is not compact.WhenΩ=RN, Strauss [23] and Lions [24] showed that the radial Sobolev space Wrad1,p(RN) can be embedded to Lq(RN) compactly for q∈(p,p∗). In addition, related results are in [25, 26] and so on. In p(x) case, under the same conditions as those of bounded domain case the compact embedding from Wrad1,p(x)(RN) to Lq(x)(RN) is obtained for q(x) satisfying essinfRN(q(x)-p(x))>0 and essinfRN(p∗(x)-q(x))>0 by [27]. On the other hand, the critical case, that is, essinfRN(q(x)-p(x))=0 or essinfRN(p∗(x)-q(x))=0, has not been treated so far even if p(x)≡p. In this paper, we fix p(x)≡p. Our first study is to obtain a sufficiently condition of compactness and noncompactness of the embedding from Wrad1,p(RN) to Lq(x)(RN) for variable exponent q(x) which also includes the excluded case in [27]. Based on these results, as the second study we obtain a nontrivial solution of (1) under the compactness conditions with liminf|x|→∞q(x)=p.Before introducing main results, we fix several notations.BR denote an open ball centered 0 with radius R. ωN-1 is an area of the unit sphere SN-1 in RN. Throughout this paper we assume that q(x)∈L∞(RN) and q(x)≥1 for a.e. x∈RN. A letter C denotes various positive constant. If u is a radial function in RN, then we can write as u(x)=u~(|x|) by some function u~=u~(r) in R+. For simplicity we write u(x)=u(|x|) with admitting some ambiguity.Theorem 1 (noncompactness).
If there exist positive constantsR and C0 and an open set Γ in SN-1 such that(6)qx≤p+C0log⁡xforx∈R,+∞×Γ;then the embedding from Wrad1,p(RN) to Lq(x)(RN) is not compact.Theorem 2 (compactness).
If there exist positive constantsr,R,C0,C1, and k,l∈(0,1) such that(7)qx≤p∗-C0log⁡xkforx∈Br,(8)qx≥p+C1log⁡xlforx∈RN∖BR,then the embedding from Wrad1,p(RN) to Lq(x)(RN) is compact.Remark 3.
In Theorem2, we do not need the constraint p≤q(x)≤p∗. Wrad1,p(RN)⊂Lq(x)(RN) holds whenever q(x) satisfies q(x)≤p∗ in Br and q(x)≥p in RN∖BR.Theorem 4.
Assume thatq(x) satisfies the hypotheses (7) and (8) in Theorem 2 and essinfx∈BR⁡q(x)>p. Then there exists a nontrivial weak solution u∈Wrad1,p(RN) of (1) in the sense of(9)∫RN∇up-2∇u∇ϕ+up-1ϕ-uqx-1ϕdx=0for any ϕ∈Wrad1,p(RN).Remark 5.
Ifq(x) is a radially symmetric function satisfying the hypotheses of Theorem 4, then we can show that the weak solution u obtained in Theorem 4 satisfies u∈Cloc⁡1,α(RN∖{0}) and u(x)>0 for all x∈RN∖{0}. Indeed, since u and q(x) are radially symmetric, it follows that, for all ϕ∈Wrad1,p(RN), (10)∫0∞u′rp-2u′rϕ′r+up-1ϕ-uqr-1ϕrN-1dr=0, where r=|x|. If for any ψ∈Cc∞(RN) we consider the radial function Ψ(r)=∫ω∈SN-1ψ(rω)dSω, then we have (11)∫RN∇up-2∇u∇ψ+up-1ψ-uqx-1ψdx=∫0∞u′rp-2u′rΨ′r+up-1Ψ-uqr-1ΨrN-1dr=0. Therefore we see that u satisfies (9) even for nonradial functions ϕ. Finally, by Corollary of Theorem 2 in [28] we have u∈Cloc⁡1,α(RN∖{0}). And also, by Theorem 2.5.1 in [29] we have u(x)>0 for all x∈RN∖{0}.We note the difficulties to obtain Theorem4 caused by the condition essinfx∈RN⁡q(x)=p.Ambrosetti-Rabinowitz condition (AR) is well known in order to obtain a nontrivial weak solution to the following problem by mountain pass method:(12)-Δpu+up-2u=fx,uinRN.ARThere  areμ>pandM>0such  that  foru≥M,0<μFx,u≤ufx,u, where F(x,u)≔∫0uf(x,s)ds. Especially, condition (AR) has been used to establish not only the mountain pass structure of the energy functional but also the Palais-Smale condition. A weaker condition has also been considered, for instance, Liu-Wang [30] studied (SQ) which is calledsuperquadratic condition. SQlimu→∞⁡Fx,uup=∞uniformly  inx∈RN. However, assuming that the nonlinear term u(x)q(x)-1 in (1) is a special case of the general nonlinear term f(x,u), this does not satisfy even condition (SQ) when essinfx∈RNq(x)=p. From these facts, it seems to be difficult to confirm whether the energy functional J (see Section 4) corresponding to (1) satisfies the Palais-Smale condition or not. In more detail, while the fact that bounded Palais-Smale sequence has a convergent subsequence is straightforward from Theorem 2, boundedness of all Palais-Smale sequence is nontrivial. Besides that, satisfying the mountain pass structure for J is not trivial since we cannot apply the fibering map method directly.To overcome these difficulties, in Section3, we construct a solution of (1) as a limit of mountain pass solutions of some elliptic equations approaching (1) in the sense of energy functional. In Section 4, we show another proof by using the variant of the mountain pass theorem. More precisely, by introducing the condition (C) (see Section 4) defined in [31] or [32] instead of the Palais-Smale condition, we obtain a solution of (1) in a different way from Section 3.


## 2. Compactness and Noncompactness of the Embedding

We prove Theorems1 and 2. Before beginning the proof we recall the point-wise estimate and the compactness theorem introduced in [23, 24] (p=2). For the reader’s convenience, the proofs are in Appendix.Proposition 6.
For anyu∈Wrad1,p(RN) we have(13)ux≤pωN-11/px-N-1/puLpRNp-1/p∇uLpRN1/p.Proposition 7.
The embedding fromWrad1,p(RN) to Lq(RN) is compact for q∈(p,p∗).Proof of Theorem1.
We shall show Theorem1 in the same way as [17]. Set r(x)=q(x)-p for x∈RN. Let ϕ∈Cc∞(RN) be a radial function satisfying ϕ≡1 on B1/2 and suppϕ⊂B1. For m∈N, we define ϕm(x)=m-N/pϕ(x/m). Then for any m∈N we obtain (14)ϕmLpRN=ϕLpB1,∇ϕmLpRN=m-1∇ϕLpB1. Since {ϕm}m=1∞ is a bounded sequence in Wrad1,p(RN) and Wrad1,p(RN) is reflexive (see, e.g., Proposition 3.20. in [33]), there exist a weakly convergent subsequence {ϕmj}j=1∞ and ϕ∞∈Wrad1,p(RN) such that ϕmj⇀ϕ∞ in Wrad1,p(RN) as j→∞. By compactness of the embedding from Wrad1,p(RN) to Lr(RN) for p<r<p∗, we have ϕmj→ϕ∞ in Lr(RN) and ϕmj→ϕ∞ a.e. in RN which yields that ϕ∞≡0. On the other hand, we have (15)∫RNϕmxqxdx=∫Bmm-N/pp+rxϕxmqxdx=∫B1m-N/prmyϕyqmydy≥∫B1/2∖B1/4m-N/prmydy. Since Γ is open in SN-1, there exists a smooth subset D⊂SN-1 such that D⊂Γ. By using the polar coordinates as y=sω(s>0,ω∈SN-1) we obtain (16)∫RNϕmxqxdx≥∫1/41/2∫ω∈Dm-N/prmsωsN-1dsdSω. By the assumption (6), we obtain r(msω)≤C0log⁡ms-1 for large m, s∈(1/4,1/2), and ω∈D⊂Γ. Moreover for s∈(1/4,1/2) and large m, it holds that log⁡ms=log⁡m+log⁡s≥1/2log⁡m which yields (17)rmsω≤2C0log⁡m. Therefore we obtain (18)∫RNϕmxqxdx≥∫1/41/2∫ω∈De-N/plog⁡m2C0/log⁡msN-1dsdSω=HN-1De-2C0N/p2-N-4-NN>0 for large m, where Hd is the d-dimensional Hausdorff measure. Thus, if we assume that the embedding from Wrad1,p(RN) to Lq(x)(RN) is compact, then we have ∫RN|ϕ∞|q(x)dx>0 which contradicts ϕ∞≡0. Hence the embedding from Wrad1,p(RN) to Lq(x)(RN) is not compact.Proof of Theorem2.
We assume thatr<R without loss of generality. Let {um}m=1∞ be a bounded sequence in Wrad1,p(RN). We shall show the existence of a strongly convergence subsequence of {um}m=1∞ in Lq(x)(RN). By the reflexivity of Wrad1,p(RN), there exist a subsequence {umj}j=1∞ and u0∈Wrad1,p(RN) such that umj⇀u0 in Wrad1,p(RN) as j→∞. Especially it also holds that umj⇀u0 in W1,p(RN) as j→∞. And also, by Proposition 7 we have umj→u0 in Lq(RN) for any q∈(p,p∗) and(19)umj→u0a.e.  inRNasj→∞.Furthermore, {umj|Br}j=1∞⊂W1,p(Br) is a bounded sequence and the embedding from W1,p(Br) to Lq(x)(Br) is compact by the assumption (7) (see Remark 2 in [17]). Thus there exist a subsequence of {umj|Br}j=1∞ (we use {umj|Br}j=1∞ again for simplicity) and v0∈Lq(x)(Br) such that the following hold true:(20)umjBr⇀v0inW1,pBr,umjBr→v0inLqxBr,umjBr→v0inLpBr,umjBr→v0a.e.  inBr(21)asj→∞. By (19) and (20), we can check that u0|Br=v0 a.e. in Br which yields (22)umjBr→u0BrinLqxBrasj→∞.
In the similar way as above, we also obtain the following:(23)umjBK∖Br⇀u0BK∖BrinWrad1,pBK∖Br,umjBK∖Br→u0BK∖BrinLsBK∖Br,umjBK∖Br→u0BK∖Bra.e.  inBK∖Br for any K>0 and any s≥1 as j→∞ since the embedding from Wrad1,p(BK∖Br) to Ls(BK∖Br) is compact for any K,s.
Setvmj≔umj-u0. In order to make good use of (22) and (23) we divide ∫RN|vmj(x)|q(x)dx into three terms as follows:(24)∫RNvmjxqxdx=∫Brvmjxqxdx+∫BK∖Brvmjxqxdx+∫RN∖BKvmjxqxdx≕I1j+I2j,K+I3j,K,where K is sufficiently large.
Firstly, by (22) we have(25)I1j=o1asj→∞.
Next, forI2(j,K) we have (26)I2j,K=∫BK∖Brvmjxqxdx≤∫BK∖Brvmjxdx+∫BK∖BrvmjxqL∞RNdx.Thus, by (23) we obtain(27)I2j,K=o1asj→∞for  fixedK>0.
Finally we shall estimateI3(j,K). Since (28)vmjx≤pωN-11/pvmjW1,pRNx-N-1/p≤Cx-N-1/p by Proposition 6 and the boundedness of {vmj}j=1∞, we can assume |vmj(x)|≤1 for x∈RN∖BK with large K. Therefore by assumption (8) we obtain (29)I3j,K=∫RN∖BKvmjqxdx≤∫RN∖BKvmjp+C1log⁡x-ldx≤∑n=2∞∫BKn∖BKn-1vmjp+C1nlog⁡K-ldx≤∑n=2∞∫BKn∖BKn-1vmjpCx-N-1/pC1nlog⁡K-ldx≤CC12log⁡K-lvmjW1,pRNp∑n=2∞K-N-1/pn-1C1nlog⁡K-l≤C∑n=2∞δ1n-11-l=C∑n=1∞δ1n1-l, where δ1=δ1(K)≔K-N-1/pC1(log⁡K)-l→0 as K→∞. Since ∑n=1∞δ1n1-l=δ1+∫1∞δ1x1-ldx<∞ for each δ1∈(0,1), we have (30)∑n=1∞δ1n1-l→0asK→∞. Hence we have(31)I3j,K=o1uniformly  injasK→∞.We go back to (24) and by (25), (27), and (31) we have (32)limj→∞⁡∫RNvmjxqxdx=0. As a consequence we obtain umj→u0 in Lq(x)(RN).


## 3. Approximation Method: Proof of Theorem4

In this section, we show Theorem4 by using Theorem 2. First, we prepare the mountain pass theorem (Theorem 8) introduced in [34, 35] and so on which are based on [36]. Let V be a Banach space and E∈C1(V,R). We define a Palais-Smale sequence for E as {um}⊂V satisfying |E(um)|≤c uniformly in m, and E′(um)→0 in V∗, where E′(·) is Fréchet derivative and V∗ is the dual space of V. We say that E satisfies (P.-S.) condition if any Palais-Smale sequence has a strongly convergent subsequence.Theorem 8 (see [34, 35]).
SupposeE∈C1(V,R) satisfies (P.-S.) condition. Assume that (i)
E(0)=0(ii)
There existρ>0,α>0 such that E(u)≥α for any u∈V with ‖u‖=ρ.(iii)
There existsu1∈V such that ‖u1‖≥ρ and E(u1)<α.
Define(33)P=p∈C0,1,V∣p0=0,p1=u1. Then (34)β=infp∈P⁡sup0≤t≤1⁡Ept is a critical value.Proof of Theorem4.
⁡
Step 1. We may assume that R in the hypotheses of Theorem 2 is sufficiently large such that essinfx∈BRq(x)≥p+C1(log⁡R)-l. Because, if not, we take a sufficiently large R~ which satisfies it instead of R. For m∈N let {Rm} be a sequence such that R1=R and Rm→∞ as m→∞. Then we set functions as (35)qmx=qxifqx≥p+C1log⁡Rm-l,p+C1log⁡Rm-lifqx<p+C1log⁡Rm-l. Define a functional Jm from Wrad1,p(RN) to R by (36)Jmu=1p∫RN∇up+updx-∫RN1qmxu+qmxdx. We can check that Jm∈C1(Wrad1,p(RN),R). Indeed, for fixed m and any u,ϕ∈Wrad1,p(RN)(37)DGJmuϕ≔ddtt=0Jmu+tϕ=∫RN∇up-2∇u·∇ϕ+up-2uϕdx-∫RNu+qmx-1ϕdx. Then we see that Jm is Gâteaux differentiable in Wrad1,p(RN). By the Vitali convergence theorem, we see that DGJm is continuous from Wrad1,p(RN) to its dual space Wrad1,p(RN)∗. Hence Jm∈C1(Wrad1,p(RN),R). Moreover, for each m, Jm satisfies as follows: (i)
Jm satisfies (P.-S.) condition.(ii)
Jm(0)=0,(iii)
There exist positive constantsα,ρ such that Jm(u)≥α for any u∈Wrad1,p(RN) with ‖u‖W1,p(RN)=ρ,(iv)
There existsv∈Wrad1,p(RN) such that ‖v‖W1,p(RN)≥ρ,  Jm(v)<α.
By Theorem8 there exists a critical point um∈Wrad1,p(RN) of Jm such that (38)Jmum=βm, where βm is defined in the same way as β in Theorem 8. Thus um is a nontrivial weak solution of(39)-Δpw+wp-2w=w+qmx-1inRN.We can also see that um≥0 by multiplying both sides of (39) by (um)-.
Proposition 9. {um}is bounded inWrad1,p(RN).
We will prove this proposition at the end of this section.
Step 2. Since {um} is a bounded sequence, there exists u0∈Wrad1,p(RN) such that um⇀u0 weakly in Wrad1,p(RN). Put (40)Gm=∇ump-2∇um-∇u0p-2∇u0,∇um-∇u0RN+ump-1-u0p-1um-u0. Then we have (41)∫RNGmdx=∫RN∇ump+umpdx-∫RN∇ump-2∇um∇u0+ump-1u0dx+hm, where hm=∫RN|∇u0|p-2∇u0(∇u0-∇um)+u0p-1(u0-um)dx=o(1) as m→∞. Moreover, from (56) and (57) in the proof of Proposition 9 it follows that (42)∫RN∇ump+umpdx-∫RN∇ump-2∇um∇u0+ump-1u0dx=∫RNum+qmx-1um+-u0dx≤2umqmx-1qx/qx-1um-u0qx=2umqxum-u0qx by the generalized Hölder inequality (see, e.g., [9] Theorem 2.1). By the boundedness of {um} in Wrad1,p(RN) and Theorem 2 we have ‖um‖q(x)‖um-u0‖q(x)=o(1) as m→∞. Hence(43)∫RNGmdx=o1as m→∞. Recall that, for p≥1, a,b∈Rd, we have (44)bp-2b-ap-2a,b-a≥22-pb-apifp≥2,p-1b-a21+a2+b2p-2/2if1≤p≤2. From this inequality and (43) it follows that (45)∫RN∇um-∇u0p+um-u0pdx=o1 which is equivalent to um→u0 strongly in W1,p(RN). Thus u0 satisfies (46)-Δpu0+u0p-1=u0qx-1,u0≥0inRN.Step3. Finally, we have to show u0≢0. From the boundedness of {um} and Proposition 6, we see that um≤1 in RN∖BL for large L. Therefore we have(47)∫RN∇ump+umpdx=∫RNumqmxdx≤∫RNumpdx+∫Brump∗dx+∫BL∖BrumqL∞RNdx.By the Sobolev inequality it follows that(48)∫Brump∗dx≤∫RNump∗dx≤S-p∗/p∫RN∇umpdxp∗/p.Moreover, we have(49)∫BL∖BrumqL∞RNdx≤C∫BL∖Br∇ump+umpdxqL∞RN/p≤C∫BL∖Br∇umpdx+∫BL∖Brump∗dxp/p∗BL∖Br1-p/p∗qL∞RN/p≤C∫RN∇umpdxqL∞RN/p.Put q∗≔min⁡{p∗,‖q‖L∞(RN)}. From (47), (48), and (49), we obtain (50)C≤∫RN∇umpdxq∗-p/p, where we used that um≢0. By Theorem 2 we have (51)C≤limm→∞⁡∫RN∇umpdx=limm→∞⁡∫RN-ump+umqmxdx≤∫RNu0qxdx. Consequently we have u0≢0.Proof of Proposition 9.
We take a smooth radial functionu^>0 on RN. Since (52)JmKu^≤Kpp∫RN∇u^p+u^pdx-∫BRKqxqxu^+qxdx≤Kpp∫RN∇u^p+u^pdx-Kp+C1log⁡RlesssupBR⁡qx∫BRu^+qxdx→-∞ as K→+∞, there exists K^>0 independent of m such that Jm(K^u^)<0. If we set p^(t)=tK^u^ for t∈[0,1], then we see that (53)p^∈P^=p∈C0,1,Wrad1,pRN∣p0=0,p1=K^u^. Moreover, we have(54)βm=infp∈P^⁡max0≤t≤1⁡Jmpt≤max0≤t≤1⁡Jmp^t=max0≤t≤K^⁡tpp∫RN∇u^p+u^pdx-∫BRtqxqxu^+qxdx≤C.On the other hand, since um is a critical point of Jm at βm we have(55)βm=1p∫RN∇ump+umpdx-∫RN1qmxum+qmxdxand, for any ϕ∈Wrad1,p(RN),(56)∫RN∇ump-2∇um∇ϕ+ump-2umϕdx-∫RNum+qmx-1ϕdx=0.In particular,(57)∫RN∇ump+umpdx-∫RNum+qmxdx=0.From (54), (55), and (57), it follows that (58)∫RN1p-1qmxum+qmxdx≤C. Furthermore, by q(x)≤qm(x) we have(59)∫RN1p-1qxum+qmxdx≤C.Thus for any L>0 there exists a positive constant C(L) such that(60)∫BLum+qmxdx≤CL.
Here, we take a constantR0>R sufficiently large (this R0 will be chosen again later) and we have(61)umW1,pRNp≤CR0+∫RN∖BR0um+qmxdxby (57) and (60). Set δ=C1(log⁡R0)-l and An,m≔{x∈BR0n∖BR0n-1∣qm(x)≤p+δ}. Then we obtain (62)∫RN∖BR0um+qmxdx≤∫x∈RN∣qmx>p+δum+qmxdx+∫x∈RN∖BR0∣qmx≤p+δum+qmxdx≤∫x∈RN∣qmx>p+δum+qmxdx+∑n=2∞∫An,mum+p+C1nlog⁡R0-ldx+∑n=2∞∫An,mum+p+δdx≕L1+L2+L3, where the third inequality comes from the assumption (8). We shall estimate L1, L2, and L3. For L1, by (59) we have(63)L1≤1p-1p+δ-1∫RN1p-1qmxum+qmxdx=C.In order to estimate L2 and L3, we prepare an estimate of ‖um‖Lp(An,m). For each n,m∈N we have(64)∫An,mumpdx≤2umLqmxAn,mp1LrmxAn,mby the generalized Hölder inequality, where rm(x)≔qm(x)/qm(x)-p. Now we assume ‖um‖Lqm(x)(An,m)>1 and ‖1‖Lrm(x)(An,m)>1 (if not, the proof is much simpler). By Proposition 2.2. in [27] we have (65)umLqmxAn,m≤∫An,mumqmxdxess.inf⁡x∈An,mqmx-1≤∫An,mumqmxdx1/p+nlog⁡R0-l. Since (66)∫An,mumqmxdx≤1p-1p+nlog⁡R0-l-1∫RN1p-1qmxumqmxdx≤Cnlog⁡R0l, we obtain(67)umLqmxAn,m≤Cnlog⁡R0l/p+nlog⁡R0-l.In the same way as above, we have(68)1LrmxAn,m≤∫An,mdxess.inf⁡x∈An,mrmx-1≤An,m1/1+pC1-1log⁡R0l≤CR0n/1+pC1-1log⁡R0l,where the second inequality comes from (69)essinfx∈An,m⁡rmx=1+pesssupx∈An,mqmx-p≥1+pδ=1+pC1-1log⁡R0l. From (67) and (68) we obtain(70)∫An,mumpdx≤CR0n/1+pC1-1log⁡R0lnlog⁡R0pl/p+nlog⁡R0-l.For L2, by using (70) and Proposition 6, we have (71)L2≤C∑n=2∞umW1,pRNC1nlog⁡R0-lR0-N-1/pn-1C1nlog⁡R0-l∫An,mumpdx≤CumW1,pRNp∑n=2∞R0-N-1C1/2lpn-11-llog⁡R0-l∫An,mumpdxC1nlog⁡R0-l/p≤CumW1,pRNp∑n=2∞R0-N-1C1/2lpn-11-llog⁡R0-l+C1n1-llog⁡R0-l/p+p2C1-1log⁡R0-lnlog⁡R0lC1/p+nlog⁡R0-lnlog⁡R0-l=CumW1,pRNp∑n=2∞δ1n,R0n-11-lδ2n,R0. Since (72)δ2n,R0=nlog⁡R0lC1/p+nlog⁡R0-lnlog⁡R0-l→1asn→∞orR0→∞, there exists a positive constant C~ which is independent of n and R0 such that(73)δ2n,R0≤C~.On the other hand, for large R0 we obtain (74)δ1n,R0=R0-C1/plog⁡R0-lN-1/2l-1/1+pC1-1log⁡R0ln/n-11-l≤R0-C1N-1/2l+1plogR0-l which yields (75)δ1=δ1n,R0→0uniformly  innasR0→∞.From (73) and (75) we have (76)L2≤CumW1,pRNp∑n=2∞δ1n-11-l=o1umW1,pRNpasR0→∞ in the same way as the proof of Theorem 2. Thus for sufficiently large R0 we have(77)L2≤13umW1,pRNp.In the same way as L2, we obtain the estimate of L3 for large R0 as follows:(78)L3≤CumW1,pRNδ∑n=2∞R0-N-1/pn-1δ∫An,mumpdx≤CumW1,pRNp∑n=2∞R0-N-1/pC1n-1log⁡R0-l∫An,mumpdxC1/plog⁡R0-l≤CumW1,pRNp∑n=2∞R0-N-1/pC1n-1log⁡R0-l+n/1+pC1-1log⁡R0-lC1/plog⁡R0-lnlog⁡R0lC1/plog⁡R0-l+n-l≤CumW1,pRNp∑n=2∞R0-N-1/2pC1n-1log⁡R0-lnlog⁡R0lC1/plog⁡R0-l+n-l≤CumW1,pRNp∑n=2∞R0-N-1/4pC1n-1log⁡R0-l, where the last inequality comes from (79)nlog⁡R0lC1/plog⁡R0l+n-l=oR0N-1/4pC1n-1log⁡R0-lasn→∞orR0→∞.Therefore for sufficiently large R0 we have(80)L3≤13umW1,pRNp.From (61), (63), (77), and (80) we have (81)umW1,pRNp≤C+23umW1,pRNp. As a consequence {um} is bounded.


## 4. Mountain Pass Theorem under the Condition (C): Proof of Theorem4

In this section, we show Theorem4 by a different method from Section 3.Cerami [31] and Bartolo-Benci-Fortunato [32] have proposed a variant of (P.-S.) condition. In this paper, we use the condition (C) introduced by [31, 32] and the mountain pass theorem under the condition (C) (Theorem 11). Let V be a real Banach space and E∈C1(V,R). First, we define the condition (C) based on [31, 32].Definition 10 ([31, 32] Definition 1.1.).
We say thatE satisfies the condition (C) in (c1,c2),(-∞≤c1<c2≤+∞), if (i)
every bounded sequence{uk}⊂E-1((c1,c2)), for which {E(uk)} is bounded and E′(uk)→0, possesses a convergent subsequence, and(ii)
for anyc∈(c1,c2) there exist σ,ρ,α>0 such that [c-σ,c+σ]⊂(c1,c2) and for any u∈E-1([c-σ,c+σ]) with ‖u‖≥ρ,‖E′(u)‖∗‖u‖≥α.Theorem 11 (mountain pass theorem under the condition (C)).
LetE satisfy the condition (C) in (0,+∞). Assume that (i)
E(0)=0(ii)
There existρ>0,α>0 such that E(u)≥α for any u∈V with ‖u‖=ρ.(iii)
There existsu1∈V such that ‖u1‖≥ρ and E(u1)<α.
Define(82)P=p∈C0,1,V∣p0=0,p1=u1. Then (83)β=infp∈P⁡sup0≤t≤1⁡Ept≥α is a critical value.Forc∈R, we set (84)Ec=u∈V∣Eu<c,Kc=u∈V∣E′u=0,Eu=c.Note that Theorem 11 can be shown in the same way as the proof of Theorem 6.1 in p.109 in [35] by substituting the following deformation theorem under the condition (C) for Theorem 3.4 in p.83 in [35].Theorem 12 ([32] Theorem 1.3.).
LetE satisfy the condition (C) in (c1,c2). If β∈(c1,c2) and N is any neighborhood of Kβ, there exist a bounded homeomorphism η of V onto V and constants ε¯>ε>0 such that [β-ε¯,β+ε¯]⊂(c1,c2), satisfying the following properties:(I)
η(Eβ+ε∖N)⊂Eβ-ε(II)
η(Eβ+ε)⊂Eβ-ε if Kβ=∅(III)
η(u)=u if |E(u)-β|≥ε¯.We set an energy functional fromWrad1,p(RN) to R as (85)Ju=1p∫RN∇up+updx-∫RN1qxu+qxdx. We can check that J∈C1(Wrad1,p(RN),R) in the same way as the proof of Theorem 4.Proposition 13.
Assume thatq(x) satisfies the hypotheses (7) and (8) in Theorem 2 and essinfx∈BRq(x)>p. Then J satisfies the condition (C) on R.Proof.
We takec1,c2∈R with c1<c2 arbitrary. First, we shall show that J satisfies (i) in Definition 10. Let {um}⊂Wrad1,p(RN) be a bounded sequence satisfying J(um)∈(c1,c2) and ‖J′(um)‖∗→0 as m→+∞. Then the following holds true for any ϕ∈Wrad1,p(RN):(86)∫RN∇ump-2∇um∇ϕ+ump-2umϕdx-∫RNum+qx-1ϕdx=o1.In particular, since {um} is bounded it follows that(87)∫RN∇ump+umpdx-∫RNum+qxdx=o1.Likewise since {um} is bounded, there exists a subsequence written as {um} for simplicity and u0∈Wrad1,p(RN) such that um⇀u0 weakly in W1,p(RN). Put (88)Gm=∇ump-2∇um-∇u0p-2∇u0,∇um-∇u0RN+ump-1-u0p-1um-u0as in Section 3. In the same way as Step 2 in the proof of Theorem 4 in Section 3 by substituting (86) and (87) for (56) and (57), respectively, we have (89)∫RNGmdx=o1 as m→∞ by Theorem 2. Recalling that (90)bp-2b-ap-2a,b-a≥22-pb-apifp≥2,p-1b-a21+a2+b2p-2/2if1≤p≤2, consequently we have (91)limm→∞⁡∫RN∇um-u0p+um-u0pdx≤Climm→∞⁡∫RNGmdx=0. This implies that um→u0 strongly in W1,p(RN).
Next, we shall show (ii). For anyc∈(c1,c2), we take some σ with [c-σ,c+σ]⊂(c1,c2). We will choose suitable ρ>0 again later. By deriving a contradiction, we show that there exists α>0 such that for any u∈J-1([c-σ,c+σ]) with ‖u‖≥ρ,‖J′(u)‖∗‖u‖≥α. We assume that there exists {um}⊂Wrad1,p(RN) such that um∈J-1([c-σ,c+σ]) with ‖um‖W1,p(RN)≥ρ and ‖J′(um)‖∗‖um‖W1,p(RN)=:αm→0 as m→+∞. Since J′(um)um→0 as m→+∞, we have (92)umW1,pRNp-∫RNum+qxdx≤αm which yields (93)c+σ≥Jum≥∫RN1p-1qxum+qxdx-αm.Moreover, in the same way as the proof of Proposition 9, for large m we have(94)∫Anumpdx≤CR0n/1+pC1-1log⁡R0lnlog⁡R0pl/p+nlog⁡R0-l,where An:={x∈BR0n∖BR0n-1∣q(x)≤p+δ} for n≥2 and R0 is the same as the proof of Proposition 9. By substituting (93) and (94) for (59) and (70), we obtain the following estimates: (95)umW1,pRNp-αm≤∫BR0um+qxdx+∫RN∖BR0um+qxdx≤CR0c+σ+αm+23umW1,pRNp, where C(R0) is a positive constant independent of ρ. Therefore we have(96)umW1,pRNp≤3αm+CR0c+σ+αm≤31+CR0c2+1for large m. If we choose sufficiently large ρ satisfying ρ>31/p{1+C(R0)(c2+1)}1/p, then we see that (96) contradicts ‖um‖W1,p(RN)≥ρ.
The proof of Proposition13 is now complete.Proposition 14.
Assume thatq(x) satisfies the hypotheses (7) and (8) in Theorem 2 and essinf⁡x∈BRq(x)>p. Then J has the mountain pass geometry, that is, J satisfies (i), (ii), and (iii) in Theorem 11.Proof.
(i) is obvious. We prove (ii). LetS be the best constant of the Sobolev inequality: S‖v‖Lp∗(RN)p≤‖∇v‖Lp(RN)p for v∈Cc∞(RN). Set q∗=max⁡{p∗,p2,‖q‖L∞(RN)}. Note that q∗≥p∗>pN/(N-1). For u∈Wrad1,p(RN) with ‖u‖W1,p(RN)=γ, it follows that (97)∫RN1qxu+qxdx≤1p∫RNupdx+1p∫Brup∗dx+∫RN∖Bruq∗dx≤1p∫RNupdx+1pS-1∫RN∇updxp∗/p+uLpRNq∗p-1/p∇uLpRNq∗/pKr≤1p∫RNupdx+1p∫RN∇updxS-p∗/pγp∗-p+Krγq∗-p, where K(r)=p/ωN-1q∗/p∫RN∖Br|x|-q∗(N-1)/pdx<∞ and the second inequality comes from Proposition 6. From this if γ is sufficiently small, we have(98)Ju≥1p∫RN∇updx1-S-p∗/pγp∗-p-Krγq∗-p>0.For {um}⊂Wrad1,p(RN) and γ satisfying ‖um‖W1,p(RN)=γ and (98), we assume that J(um)→0 and derive a contradiction. From (98) it follows that ∫RN|∇um|pdx→0. In addition, for sufficiently large R we have (99)∫RN1qxum+qxdx≤1p∫Brumqxdx+∫BR∖Brumqxdx+∫RN∖BRumqxdx≤1p∫Brump∗dx+∫BRumpdx+∫RN∖Brumq∗dx+∫RN∖BRump+C1log⁡x-ldx=1pH1+H2+H3+H4. By using the estimates in the calculation of ∫RN(u)+q(x)/q(x)dx to show (98) we have H1=o(1) and H3=o(1) as m→∞. For H2 we have (100)H2≤BR1-p/p∗S-1∫RN∇umpdx=o1. We can show that H4 is bounded uniformly for m and H4→0 as R→∞ in the same way as the estimate of I3(j,K) in the proof of Theorem 2. Therefore (101)∫RN1qxumqxdx→0 as m→∞, which implies that ‖um‖W1,p(RN)→0 since J(um)→0 as m→∞. This contradicts ‖um‖W1,p(RN)=γ.
Finally, we prove (iii). We take a smooth radial functionv such that ‖v‖W1,p(RN)=γ, v>0 in BR, where R is in the hypothesis (8). Recall that q_≔essinfx∈BRq(x)>p. By taking sufficiently large t we have (102)Jtv=tpp∫RN∇vp+vpdx-∫RNtqxqxv+qxdx≤tpp∫RN∇vp+vpdx-tq_∫BR1qxv+qxdx<0. Since ‖tv‖W1,p(RN)>γ we prove (iii).Proof of Theorem4.
From Propositions13 and 14 and Theorem 11, we can show the existence of a nontrivial critical point u∈Wrad1,p(RN) which is a weak solution to -Δpu+|u|p-2u=u+q(x)-1 in RN. Then we also see that u≥0 in RN.



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