# The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

Authors: Xiaoyao Jia; Zhenluo Lou
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this paper, we study the following quasilinear equation:−divϕ∇u∇u+ϕuu=fuinℝN, where ϕ∈C1ℝ+,ℝ+ and Φt=∫0tsϕ∣s∣ds. In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.


---

## Body

## 1. Introduction

In this paper, we consider the following quasilinear elliptic equation:(1)−ΔΦu+ϕuu=fuinℝN,where N≥3 is the dimension of space and ϕ∈C1ℝ+,ℝ+ satisfies the following conditions:(ϕ1) limt⟶0ϕtt=0, limt⟶+∞ϕtt=+∞, ϕtt′>0, for t>0(ϕ2) There exist p,q∈1,N,q<p<q∗≡Nq/N−q, such that q−2≤ϕ′tt/ϕt≤p−2, for t>0Φ is a continuous function on ℝ defined by
(2)Φt=∫0tϕssds,fort>0,and Φ-Laplace is defined by
(3)ΔΦu=divϕ∇u∇u.Let(4)Φ∗−1t=∫0tΦ−1ssN+1/Nds.The quasilinear elliptic equation is an important partial differential equation; in recent years, many researchers have studied the following equation:(5)−Δpu=fx,uinΩ,where Ω⊂ℝN is an open set, 1<p<N. Under some growth conditions, many people proved the existence and multiplicity of solutions to (5) and several mathematicians also obtained the bifurcation and asymptotic properties. By variational method and maximal principle, Guo [1] and Guo-Webb [2] obtained the existence and uniqueness of the solution to (5) and they also considered the partial symmetric properties of the solutions. Guedda-Veron [3] used topology and spectrum analysis to study the bifurcation and multiplicity of the solutions. We point out that by constructing the pseudogradient vectors of p-Laplace operator, Zhang-Li [4] firstly obtained the sign-changing solutions to (5) and see also Zhang-Chen-Li [5], for more results of p-Laplace equations, one can see [6–8] and the references therein. By variational method and minimax theorem, Li-Guo [9] and Li-Liang [10] studied the p−q-Laplace equation and they obtained the existence and multiple solutions.In [11], Franchi-Lanconelli-Serrin studied the quasilinear equation (1), and in weighted Sobolev space, they considered the existence and uniqueness of the solution to (1). In fact, the function ϕ is a general elliptic equation. For example, if ϕu≡1, then (1) is the Laplace equation; if ϕu=up−2, then (1) is the p-Laplace equation; ϕu=1/1+p2, then (1) is the curvature equation. The Orlicz-Sobolev space is a kind of general norm space and one can study the quasilinear equations in this space. Carvalho-Silva-Goulart [12] and Carvalho-Silva-Goncalves-Goulart [13] considered (1) in the Orlicz-Sobolev space, by variational method and concentration-compactness theorem. They obtained the existence of (1), and they also studied the famous problem, i.e., concave and convex terms.In this paper, letft satisfy the following conditions:(f1) f∈C1ℝ,ℝ. For t≥0,ft≥0, and for t<0,ft=0(f2) limt⟶0+ft/tϕt=0, limt⟶+∞ft/tϕt=+∞(f3) There exists a constant λ∈0,1, such that limt⟶+∞ft/gt=0, with gt≡tϕtλΦ∗′t1−λ for t>0(f4) p−1ft<f′tt for all t>0(f5) ft satisfies the Ambrosetti-Rabinowitz condition: there exists a θ>0 such that
(6)0≤Ft≡∫0tfsds≤1p+θftt,fort>0Under the preceding conditions, by a variational method, we obtain the existence of a nontrivial solution to (1). We follow the idea in [10] to obtain the existence of nontrivial solution. Our conditions f1-f5 are slightly different from what is in [10]. Conditions f1 and f2 are standard. Our condition f3 is the improvement of f3 in [10], which is used to obtain the superlinear growth of f at t⟶+∞. The condition f4 is to guarantee that A is a Finsler manifold which is used to obtain a special minimizing sequence (see Lemma 14–Lemma 16). Condition f5 is the compactness condition.Remark 1.
The conditionϕ2 implies that for t>0,
(7)q≤ϕtt2Φt≤p.Remark 2.
Byf2 and (7), we have that
(8)limt⟶0+FtΦt=limt⟶0+fttϕt=0;(9)limt⟶0+tftΦt=limt⟶0+tftt2ϕtt2ϕtΦt=0.
This paper is organized as follows. In Section2, we recall some results of the Orlicz-Sobolev space; in Section 3, we list and prove some preliminary results; and in Section 4, we prove our main theorem.


## 2. Orlicz-Sobolev Spaces

In this section, we recall some useful knowledge for the Orlicz space and Orlicz-Sobolev space and give some inequalities onΦ. The reader can refer to [14, 15] for more details.By conditionϕ1 and the definition of Φ, the function Φ is a N-function (see [14] for the definition of N-function). The complementary of Φ is defined by
(10)Φ~s=maxt≥0st−Φt,for s≥0. It is easy to see that Φ and Φ~ are complementary N-functions. By (7) (Chapter 8 [14]), the function Φ and Φ~ satisfies the following inequality:
(11)Φ~Φtt<Φt,fort>0.By Lemma4, Lemma 5, and Lemma 8 below, one can see that Φ, Φ~, and Φ∗ satisfy Δ2-condition (see [14] for the definition of Δ2-condition).For anyΩ⊂ℝN, under the assumptions ϕ1 and ϕ2, the Orlicz space LΦΩ contains all measurable functions u:Ω⟶ℝ which satisfy
(12)∫ΩΦ∣ux∣dx<∞,and the Luxemburg norm on LΦΩ is defined by
(13)uΦ,Ω=infλ>0∣∫ΩΦuxλdx≤1.For any integerm≥1, the corresponding Orlicz-Sobolev space WmLΦΩ is defined by
(14)WmLΦΩ=u∈LΦΩ∣Dαu∈LΦΩ,α≤m,and the norm on WmLΦΩ is defined by
(15)um,Φ,Ω=uΦ,Ω+∑α≤mDαuΦ,Ω.LetEΦΩ denote the closure in LΦΩ of function u which are bounded on Ω and have bounded support in Ω¯. The space WmEΦΩm∈ℕ,m≥1 is defined by
(16)WmEΦΩ=u∈EΦΩ∣Dαu∈EΦΩ,α≤m.In the following, for simplicity, we denoteuΦ,ℝN and um,Φ,ℝN by uΦ and um,Φ. Then, the Orlicz-Sobolev space has the following properties.Theorem 3 (Theorem 8.20 and Theorem 8.31 [14]).
(a)
The spacesLΦΩ is reflexive if and only if both Φ,Ω and Φ~,Ω are Δ-regular(b)
IfΦ,Ω and Φ~,Ω are Δ-regular, then WmEΦΩ=WmLΦΩ is reflexive(c)
C0∞ℝN is dense in WmEΦℝN. Thus, WmEΦℝN=W0mLΦℝN
SinceΦ and Φ~ are complementary N-functions, the following generalize Hölder inequality (see [14]) holds:
(17)∫Ωuxvxdx≤2uΦ,ΩvΦ~,Ω,foranyu∈LΦΩ,v∈LΦ~Ω.
Next, we recall some inequalities onΦ.Lemma 4 (Lemma 2.1 [15]).
Letζ0t=mintq,tp, ζ1t=maxtq,tp, t≥0. Then,
(18)ζ0tΦρ≤Φρt≤ζ1tΦρ,forρ,t≥0,ζ0uΦ≤∫ℝNΦudx≤ζ1uΦ,foru∈LΦℝN.Lemma 5 (Lemma 2.2 [15]).
Letζ2t=mintq∗,tp∗, ζ3t=maxtq∗,tp∗, t≥0. Then,
(19)ζ2tΦ∗ρ≤Φ∗ρt≤ζ3tΦ∗ρ,forρ,t≥0,(20)ζ2uΦ∗≤∫ℝNΦ∗udx≤ζ3uΦ∗,foru∈LΦ∗ℝN.Lemma 6 (Lemma 2.3 [15]).
Φt increases essentially more slowly than Φ∗t near infinity, i.e.,
(21)limt⟶+∞ΦktΦ∗t=0,for every constant k>0.Lemma 7 (Lemma 2.4 [15]).
(19) is equivalent to
(22)q∗≤Φ∗′ttΦ∗t≤p∗.Lemma 8 (Lemma 2.5 [15]).
Letζ4t=mintq/q−1,tp/p−1, ζ5t=maxtq/q−1,tp/p−1, t≥0. Then,
(23)ζ4tΦ~ρ≤Φ~ρt≤ζ5tΦ~ρ,forρ,t≥0,ζ4uΦ~≤∫ℝNΦ~udx≤ζ5uΦ~,foru∈LΦ~ℝN.
We recall some imbedding results in the Orlicz-Sobolev space.Theorem 9 (Theorem 8.12 [14]).
The imbeddingLBℝN⊂LAℝN holds if and only if B dominates A globally, i.e., there exists a constant k>0 such that At≤Bkt holds for all t≥0.Theorem 10 (Theorem 8.36 [14]).
LetΩ be an arbitrary domain in ℝN. If the N-function Φt satisfies
(24)∫01Φ−1ttN+1/Ndt<∞,∫1∞Φ−1ttN+1/Ndt=∞,then W0mLΦΩ⊂LΦ∗Ω, for any integer m≥1. Moreover, if Ω0 is a bounded subdomain of Ω, then the imbedding W0mLΦΩ⊂LBΩ0 exists and is compact for any N-function B increasing essentially more slowly than Φ∗ near infinity.SinceΦ and Φ~ satisfy Δ2-condition, so by Theorem 3, WmLΦℝN=W0mLΦℝN is reflexive. Using Lemma 4 and Theorem 10, one can see that
(25)WmLΦℝN=W0mLΦℝN⊂LΦ∗ℝN.By (7), (11), Lemma 4, and Lemma 8, we can see that for u∈W1LΦℝN and Ω⊂ℝN, there exists some C>0 such that
(26)ϕuuΦ~,Ω≤ζ4−1∫ΩΦ~ϕuudx≤ζ4−1∫ΩΦ~pΦ∣u∣∣u∣dx≤Cζ4−1∫ΩΦ∣u∣dx≤Cζ4−1ζ1uΦ,Ω.


## 3. Preliminary Results

In this section, we prove some preliminary results for future use.To study the existence of solution to (1), we first study its energy functional. It is clear that the functional defining on W1LΦℝN is given by
(27)Iu=∫ℝNΦ∇u+Φudx−∫ℝNFudx.It is easy to see that under the assumptionsf1-f5 and (ϕ1)-(ϕ3), the functionals I is of C1. For any u,v∈W1LΦℝN, we have
(28)I′u,v=∫ℝNϕ∇u∇u·∇v+ϕuuvdx−∫ℝNfuvdx.The·,· is the dual pairing between W1LΦℝN and its dual space W1LΦℝN∗.By conditionf2 and f3, we have that for any ε>0, there exists a constant Cε>0 such that
(29)ft≤εtϕt+Cεgt,fort>0.Hence, it follows that for anyε>0, there exists a constant Cε>0 such that
(30)Ft≤εΦt+Cεtgt,fort>0.Lemma 11.
Supposef5 holds. Then, there exists a positive constant C>0 such that for any u∈ℝ+, t∈ℝ+,
(31)Ftu≤Futp+θ,for0<t<1,≥Futp+θ,fort≥1.Proof.
Byf5, one has that, for s∈ℝ+,
(32)p+θs≤F′sFs.
Integrating the last inequality fromu to tu, we get the result in the lemma. This ends the proof.
Set(33)A=u∈W1LΦℝN∣I′u,u=0,u≡0.
Then, we show thatA is not an empty set.Lemma 12.
Suppose thatϕ1-ϕ3 and f1-f5 hold. Then, A≠∅.Proof.
Byf1 and f2, f≡0. Hence, there exists a constant a0>0 such that fa0>0. Notice that C0∞ℝN⊂W1LΦℝN. We choose u0∈C0∞ℝN such that u0≥0, u0≡0, and supu0x∣x∈ℝN>a0. Let
(34)ht=I′tu0,tu0.It is easy to show that ht is continuous. For t>1, by (7), f5, and Lemma 11, we have
(35)ht=∫ℝNϕt∇u0t2∇u02+ϕtu0t2u02dx−∫ℝNftu0tu0dx≤p∫ℝNΦt∇u0+Φtu0dx−p+θ∫ℝNFtu0dx≤ptp∫ℝNΦ∇u0+Φu0dx−p+θtp+θ∫ℝNFu0dx.
Using the definition ofu0, one has that ∫ℝNFu0dx>0. It follows that
(36)limt⟶+∞ht=−∞.
For0<t<1, by (7), we have
(37)ht≥q∫ℝNΦt∇u0+Φtu0dx−∫ℝNftu0tu0Φtu0Φtu0dx.
By (9), we can see
(38)limt⟶0+ftu0tu0Φtu0=0.
It follows that, for any0<ε<1, there exists some t~>0 such that, for any t∈0,t~,
(39)ftu0tu0Φtu0<ε.
Hence,(40)ht≥q−ϵ∫ℝNΦt∣∇u0∣+Φtu0dx>0.
Therefore, there exists somet0>0 such that ht0=0. This ends the proof.Lemma 13.
For anyϵ>0, there exists some Cϵ>0 such that
(41)∫ℝNugudx≤ϵ∫ℝNΦudx+Cϵ∫ℝNΦ∗udx,for any u∈W1LΦℝN.Proof.
By Lemma7 and (7), for t>0(42)tgt=t2ϕtλtΦ∗′t1−λ≤pλp∗1−λΦtλΦ∗t1−λ.It follows from Young’s inequality that, for any ϵ>0, there exists Cϵ>0 such that
(43)tgt≤ϵΦt+CϵΦ∗t.
Hence, foru∈W1LΦℝN,
(44)∫ℝN∣u∣g∣u∣dx≤ϵ∫ℝNΦudx+Cϵ∫ℝNΦ∗udx.
This ends the proof.
Set(45)I0=infIu∣u∈A.Lemma 14.
Supposeϕ1-ϕ3 and f1-f5 hold. Then, I0>0.Proof.
For anyu∈A, let Tu=∫ℝNΦ∣∇u∣+Φ∣u∣dx. Using (7) and f5, we obtain that
(46)Iu=Tu−∫ℝNFudx≥Tu−1p+θ∫ℝNfuudx=Tu−1p+θ∫ℝNϕ∇u∇u2+ϕuu2dx≥1−pp+θTu≥0.
Hence,(47)I0=infIu∣u∈A≥0.
By (7), (29), (30), and Lemma 13, for any 0<ϵ1<ϵ2, one has that there exist two positive constants Cϵ1 and Cϵ2 such that
(48)Iu≤1q∫ℝNϕ∣∇u∣∇u2+ϕ∣u∣u2dx−∫ℝNFudx=1q∫ℝNfuudx−∫ℝNFudx≤ϵ1∫ℝNΦ∣u∣dx+Cϵ1∫ℝN∣u∣g∣u∣dx≤ϵ2∫ℝNΦ∣u∣dx+Cϵ2∫ℝNΦ∗∣u∣dx.
Chooseϵ2 small enough such that 1−p/p+θ−ϵ2>0. Then, we obtain that
(49)1−pp+θ−ϵ2Tu≤Cϵ2∫ℝNΦ∗∣u∣dx.
By (25), Lemma 4, and Lemma 5, we have that
(50)∫ℝNΦ∗∣u∣dx≤ζ3uΦ∗≤Cζ3u1,Φ≤Cζ3ζ0−1∫ℝNΦ∣u∣dx+ζ0−1∫ℝNΦ∣∇u∣dx.
By the definition ofζ0 and ζ3, we have
(51)ζ3ζ0−1∫ℝNΦ∣u∣dx+ζ0−1∫ℝNΦ∣∇u∣dx≤ζ3∫ℝNΦ∣u∣dx1/p+∫ℝNΦ∣u∣dx1/q+∫ℝNΦ∣∇u∣dx1/p+∫ℝNΦ∣∇u∣dx1/q≤Cζ3Tu1/p+Tu1/q≤CmaxTup∗/p+Tup∗/q,Tuq∗/p+Tuq∗/q.
Combining (49)–(51), there exists a positive constant Cϵ2′>0, such that
(52)maxTup∗/p−1+Tup∗/q−1,Tuq∗/p−1+Tuq∗/q−1≥Cϵ2′1−pp+θ−ϵ2,which means Tu>C for some C>0. It follows that I0>1−p/p+θC. This completes the proof.Foru∈W1LΦℝN, set
(53)Hu=I′u,u=∫ℝNϕ∇u∇u2+ϕ∣u∣u2dx−∫ℝNfuudx.It is easy to see thatH is a C1-functional. For any u∈A, one has that u+≡0. If u+≡0, then fu≡0 by f1. Noting that u≡0, one has that Hu>0, which contradicts with u∈A. By ϕ2 and f4(54)H′u,u=∫ℝNϕ′∇u∇u3+2ϕ∇u∇u2+ϕ′uu3+2ϕuu2dx−∫ℝNfuu+f′uu2dx≤p∫ℝNϕ∇u∇u2+ϕuu2dx−∫ℝNfuu+f′uu2dx=∫ℝNp−1fu+−f′u+u+u+dx<0.Hence,A is a closed and a complete submanifold of W1LΦℝN with the natural Finsler structure (see [16]). Using Lemma 2.14 [10] with n=1,e1=u/G′u,u, one has the following.Lemma 15.
For anyu∈A, v∈W1LΦℝN, we have
(55)dIAu,πv=Iu,v,with π is the projection from W1LΦℝN to TuA.By Lemma12, A is not empty, and by Lemma 14, I is bounded from below. Hence, by Lemma 2.15 of [10] and Lemma 15, we can get the following result.Lemma 16.
There exists a sequenceun⊂A, such that,
(56)limn⟶∞Iun=I0andlimn⟶∞I′un=0inW1LΦℝN∗.Lemma 17.
Letρn⊂L1ℝN be a bounded sequence and ρn≥0, then there exists a subsequence, still denoted by ρn, such that one of the following two possibilities occurs:
(1)
(Vanishing):limn⟶∞supy∈ℝN∫y+BRρnxdx=0 for all 0<R<+∞(2)
(Nonvanishing): there existsα>0, 0<R<+∞ and yn⊂ℝN, such that limn⟶∞∫yn+BRρnxdx≥α>0Lemma 18.
Supposeun is a bounded sequence in W1LΦℝN, and for some r>0,
(57)limn⟶+∞supy∈ℝN∫y+BrΦundx=0.
Then,limn⟶∞∫ℝNgunundx=0.Proof.
By Lemma7, (7), and Hölder inequality, for any τ∈0,1,
(58)∫By,rϕ∣un∣un2τΦ∗′∣un∣∣un∣1−τdx≤∫By,rϕunun2dxτ∫By,rΦ∗′∣un∣∣un∣dx1−τ≤pτp∗1−τ∫By,rΦ∣un∣dxτ∫By,rΦ∗∣un∣dx1−τ.
Noting thatun is bounded in W1LΦℝN, there exists a constant M>0 such that
(59)∫ℝNΦunMdx≤1,∫ℝNΦ∣∇un∣Mdx≤1.
Then, it follows from imbedding theorem (25) and Lemma 4 that
(60)uΦ∗,By,r≤C1uΦ,By,r+∇uΦ,By,r≤C2uMΦ,By,r+∣∇u∣MΦ,By,r≤C2∫By,rΦ∣u∣Mdx1/p+∫By,rΦ∣∇u∣Mdx1/p,for some positive constants C1 and C2. Thus, by Lemma 5,
(61)∫By,rΦ∗∣un∣dx1−τ≤ζ31−τuΦ∗,By,r≤C3ζ31−τ∫By,rΦ∣u∣Mdx1/p+ζ31−τ∫By,rΦ∣∇u∣Mdx1/p≤C3∫By,rΦ∣u∣Mdxq∗1−τ/p+∫By,rΦ∣∇u∣Mdxq∗1−τ/p≤C4∫By,rΦ∣u∣dxq∗1−τ/p+∫By,rΦ∣∇u∣dxq∗1−τ/p,for some positive constants C3 and C4. Letting 1−τ=p/q∗, it follows from (58) and (61) that
(62)∫By,rϕ∣un∣un2τΦ∗′∣un∣∣un∣1−τdx≤C∫By,rΦ∣un∣dxτ∫By,rΦ∣u∣dx+∫By,rΦ∣∇u∣dx,for some C>0. Then, coving ℝN by balls of radius r, in such a way that each point of ℝN is contained in at most N+1 balls, we have
(63)∫ℝNϕ∣un∣un2τΦ∗′∣un∣∣un∣1−τdx≤N+1Csupy∈ℝN∫By,rΦundxτ∫ℝNΦudx+∫ℝNΦ∇udx⟶0,as n⟶∞. Noting that un is bounded in W1LΦℝN, by (2.3), we get that ∫ℝNϕ∣un∣un2dx is a bounded sequence. If λ>τ, letting μ=λ−τ/1−τ, then
(64)∫ℝNgunundx≤∫ℝNϕunun2dxμ∫ℝNϕ∣un∣un2τΦ∗′∣un∣∣un∣1−τdx1−μ⟶0,as n⟶∞. By imbedding theorem (25) and Lemma 7, one sees that ∫ℝNΦ∗′∣un∣∣un∣dx is a bounded sequence. If λ<τ, setting μ=τ−λ/τ, then
(65)∫ℝNg∣un∣∣un∣dx≤∫ℝNΦ∗′∣un∣∣un∣dxμ∫ℝNϕ∣un∣un2τΦ∗′∣un∣∣un∣1−τdx1−μ⟶0,as n⟶∞. This ends the proof.Theorem 19 (Vitali’s convergence theorem [17]).
LetX,A,μ be a measure space and let 1≤p<∞. Let fnn=1∞ be a sequence in LpX,A,μ and let f be an A-measurable function such that f is finite μ-a.e. and fn⟶fμ-a.e. Then, f∈LpX,A,μ and f−fnp⟶0 if and only if
(i)
For eachε>0, there exists a set Aε∈A such that μAε<∞ and ∫Aεcfnpdμ<ε for all n∈ℝN(ii)
limμE⟶0∫Efnpdμ=0
uniformly inn, i.e., for each ε>0, there is a δ>0 such that E∈A and μE<δ imply ∫Efnpdμ<ε for all n∈ℕ.


## 4. Main Result

Theorem 20.
Suppose conditions (ϕ1)-(ϕ3) and (f1)-(f5) hold. Then, (1) has a nonnegative weak solution.Proof.
By Lemma16, there exists a sequence un⊂A, such that
(66)Iun⟶I0,I′un⟶0inW1LΦℝN∗.Step 1.
There exists some constantC>0 such that un1,Φ≤C. By (66) and (7), one has
(67)I0+on1=∫ℝNΦ∣un∣+Φ∣∇un∣dx−1p+θ∫ℝNϕ∣un∣un2+ϕ∣∇un∣∇un2dx+∫ℝN1p+θfunun−Fundx≥1−pp+θ∫ℝNΦ∣un∣+Φ∣∇un∣dx+∫ℝN1p+θf∣un∣∣un∣−F∣un∣dx≥1−pp+θ∫ℝNΦ∣un∣+Φ∣∇un∣dx.
In the last step, we use condition (f5). Hence, by Lemma 4, there exists some constant C>0 such that un1,Φ≤C.Step 2.
There existR>0, α>0, and yn⊂ℝN, such that
(68)∫yn+BRΦ∣un∣dx≥α>0.By Step1, Φ∣un∣ is bounded in L1ℝN. Lemma 17 shows that one of two situations (vanishing and nonvanishing) occur. We need only to show that the vanishing case is impossible. If vanishing occurs, by Lemma 18, limn⟶∞∫ℝNgunundx=0. It follows from (29) and (30) that
(69)limn⟶∞∫ℝNfunundx=limn⟶∞∫ℝNFundx=0.Noticeun∈A, we obtain
(70)0<I0≤limsupn⟶∞1q∫ℝNϕunun2+ϕ∇un∇un2dx−∫ℝNFundx=limsupn⟶∞∫ℝN1qfunun−Fundx=0.This is a contradiction. Hence, only nonvanishing is possible, i.e., there existR>0, α>0, and yn⊂ℝN, such that
(71)∫yn+BRΦundx≥α>0.Step 3.
There existsu¯∈W1LΦℝN, such that ∇u¯n⟶∇u¯ a.e. in ℝN, where u¯nx=unx+yn.For anyψ∈W1LΦℝN, set ψnx=ψx−yn. Then, ψ1,Φ=ψn1,Φ, and
(72)I′u¯n,ψ=I′un,ψn≤I′unW1LΦℝN∗ψn1,Φ=I′unW1LΦℝN∗ψ1,Φ⟶0,as n⟶∞. Since u¯n1,Φ=u¯n1,Φ, so u¯n is a bounded sequence in W1LΦℝN. Hence, there exists u¯∈W1LΦℝN and u¯≡0 such that some sequence of u¯n, still denoted by u¯n,
(73)u¯n⇀u¯inW1LΦℝN,u¯n⟶u¯a.e.inℝN,∫Ωgu¯n−u¯u¯n−u¯dx⟶0foranyboundedsubsetΩ⊂ℝN.Letηr∈C0∞ℝN such that 0≤η≤1, with ηr=1 in Br=x∈ℝN∣x≤r, ηr=0 in B2rc, and ∇ηr≤C/r for some C>0. Then, it is easy to see that
(74)I′u¯n−I′u¯,u¯n−u¯ηr=on1.On the other hand,(75)I′u¯n−I′u¯,u¯n−u¯ηr=∫B2rηrϕu¯nu¯n−ϕu¯u¯u¯n−u¯dx+∫B2rηrϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇u¯n−u¯dx+∫B2r\Brϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇ηru¯n−u¯dx−∫B2rηrfu¯n−fu¯u¯n−u¯dx.By (17), (26), and Lemma 4,
(76)∫B2r\Brϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇ηru¯n−u¯dx≤Crϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯Φ~,B2r\Bru¯n−u¯Φ,B2r\Br≤Crζ4−1∫B2r\BrΦ∇u¯ndx+ζ4−1∫B2r\BrΦ∇u¯dxζ0−1∫B2r\BrΦu¯ndx+ζ0−1∫B2r\BrΦu¯dx.Noting thatu¯n is bounded in W1LΦℝN, for any ε>0, there exists r0>0 such that for r>r0,
(77)∫B2r\Brϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇ηru¯n−u¯dx≤ε.For fixr and δ>0, by the Egoroff theorem, there exists Eδ⊂B2r such that mEδ<δ, and fu¯n−fu¯u¯n−u¯⟶0 uniformly on B2r\Eδ as n⟶+∞. Thus, for any ε>0, δ>0, there exists n0>0 such that, for n≥n0,
(78)∫B2r\Eδηrfu¯n−fu¯u¯n−u¯dx≤ε.By (f2) and (f3), there exists some C>0 such that
(79)ft≤Ctϕtfor0<t<1,Cgtfort≥1.Let1<μ<q∗/pλ+1−λq∗ be a fixed constant. Then, by Lemma 4, Lemma 5, and Lemma 7, for t≥1,
(80)tftμ≤Ct2ϕtλμtΦ∗′t1−λμ≤CΦtλμΦ∗t1−λμ≤CtλμpΦ∗t1−λμ≤CΦ∗tλμp/q∗+1−λμ≤CΦ∗t,for some C>0. Set Eδ,1=x∈Eδ∣u¯x<1, and Eδ,2=Eδ\Eδ,1. Using Hölder inequality, we obtain
(81)∫Eδηrfu¯n−fu¯u¯n−u¯dx≤C∫Eδfu¯n−fu¯u¯n−u¯μdx1/μmEδ1−1/μ≤C∫Eδfu¯nu¯nμ+fu¯u¯μdx1/μδ1−1/μ.Sinceun and u is bounded in W1LΦℝN, so there exists a constant C>0 such that
(82)∫ℝNΦ∗∣un∣dx≤C,∫ℝNΦ∗∣u∣dx≤C.Therefore, for someC>0,
(83)∫Eδfu¯u¯μdx≤C∫Eδ,1u¯2ϕ∣u¯∣μdx+C∫Eδ,2Φ∗∣u¯∣dx≤C,∫Eδfu¯nu¯nμdx≤C∫Eδ,1u¯n2ϕ∣u¯n∣μdx+C∫Eδ,2Φ∗∣u¯n∣dx≤C.Choosingδ>0 small enough, one has
(84)∫Eδηrfu¯n−fu¯u¯n−u¯dx≤ε.Hence, such that forn>n0,
(85)∫B2rηrfu¯n−fu¯u¯n−u¯dx≤2ϵ.Sinceϕu¯nu¯n−ϕu¯u¯u¯n−u¯≥0 and ϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇u¯n−u¯≥0, it follows from (74)–(85) that for any ϵ>0, there exist r0 and n1≥n0 such that for n>n1 and r≥r0,
(86)∫Brϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇u¯n−u¯dx≤Cϵ.It follows that, passing to a subsequence if necessary,ϕ∇u¯n∇u¯n−ϕ∇u¯∇u¯·∇u¯n−u¯⟶0 a.e. in Br. Using Lemma 6 [18], we get that
(87)∇u¯n⟶∇u¯a.e.inBr.Sincer is arbitrary, one can see that ∇u¯n⟶∇u¯ a.e. in ℝN.Step 4.
I′u¯=0, u¯∈A.
We first show thatϕ∇u¯n∇u¯n⇀ϕ∇u¯∇u¯ in LΦ~ℝN, i.e., for any ψ∈LΦℝN,
(88)limn⟶∞∫ℝNϕ∇u¯n∇u¯nψdx=∫ℝNϕ∇u¯∇u¯ψdx.
Since∇u¯n is bounded in W1LΦℝN, by (26), there exists some M>0 such that
(89)ϕ∇u¯n∇u¯nΦ~≤M.
Then, for anyϵ>0, there exist r>0 and δ>0 such that for any mE≤δ,
(90)∫Brcϕ∇u¯n∇u¯nψdx≤2ϕ∇u¯n∇u¯nΦ~ψΦ,Brc≤ϵ,∫Eϕ∇u¯n∇u¯nψdx≤2ϕ∇u¯n∇u¯nΦ~ψΦ,E≤ϵ.
Then, by Theorem19,
(91)ϕ∇u¯n∇u¯n⇀ϕ∇u¯∇u¯inW1LΦℝN∗.
Similarly, we can have(92)ϕu¯nu¯n⇀ϕu¯u¯,fu¯n⇀fu¯,inW1LΦℝN∗.
It follows from (91) and (92) that for any ψ∈W1LΦℝN,
(93)0=limn⟶∞I′u¯n,ψ=limn⟶∞I′u¯,ψ.
In particular, takingψ=u¯, we get that I′u¯,u¯=0. Since u¯≡0, so u¯∈A.Step 5.
Iu¯=I0.
By the definition ofu¯k, we obtain that
(94)Iu¯n=Iun,I′u¯n,u¯n=I′un,un=on1.
Hence, by (7), condition (f5), and Fatou’s lemma,
(95)Iu¯≥I0=limn⟶∞Iu¯n=limn⟶∞∫ℝNΦu¯n+Φ∇u¯ndx−∫ℝNFu¯ndx=limn⟶∞∫ℝNΦu¯n+Φ∇u¯n−1pϕu¯nu¯n2−1pϕ∇u¯n∇u¯n2dx+∫ℝN1pfu¯n−Fu¯ndx≥∫ℝNΦu¯+Φ∇u¯−1pϕu¯u¯2−1pϕ∇u¯∇u¯2dx+∫ℝN1pfu¯u¯−Fu¯dx=∫ℝNΦu¯+Φ∇u¯dx−∫ℝN1pFu¯dx=Iu¯.
It follows thatIu¯=I0. By (f1), we get that u¯−=max−u¯,0=0. Hence, u¯≥0. This ends the proof.



---

*Source: 9986047-2021-04-26.xml*