# Non-Nehari Manifold Method for Fractional p-Laplacian Equation with a Sign-Changing Nonlinearity

Authors: Huxiao Luo; Shengjun Li; Wenfeng He
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We consider the following fractional p-Laplacian equation:-Δpαu+V(x)up-2u=f(x,u)-Γ(x)uq-2u,x∈RN, where N≥2, pα⁎>q>p≥2, α∈(0,1), -Δpα is the fractional p-Laplacian, and Γ∈L∞(RN) and Γ(x)≥0 for a.e. x∈RN. f has the subcritical growth but higher than Γ(x)uq-2u; however, the nonlinearity f(x,u)-Γ(x)uq-2u may change sign. If V is coercive, we investigate the existence of ground state solutions for p-Laplacian equation.


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## Body

## 1. Introduction

Consider the following nonlinear Schrödinger equation with fractionalp-Laplacian:(1)-Δpαu+Vxup-2u=fx,u-Γxuq-2u,x∈RN,where N≥2, pα∗>q>p≥2, α∈(0,1), and -Δpα is the fractional p-Laplacian. V(x), Γ(x), and f(x,u):RN×R→R satisfy the following assumptions:(V)
V∈C(RN,R),V0≔infx∈RN⁡V(x)>0, there exists a constant d0>0 such that (2)limy→∞⁡measx∈RN:x-y≤d0,Vx≤M=0,∀M>0,where meas (·) denotes the Lebesgue measure in RN;(Γ)
Γ∈L∞(RN),Γ(x)≥0 for a.e. x∈RN;(f1)
f(x,t):RN×R→R is measurable, continuous in t∈R for a.e. x∈RN and there are C>0 and 2≤p<q<r<pα∗ such that (3)fx,t≤C1+tr-1for  allt∈R,x∈RN,where pα∗=Np/N-pα;(f2)
f(x,t)=o(tp-1) as t→0 uniformly in x∈RN;(f3)
F(x,t)/tq→∞ uniformly in x as t→∞, where F(x,t)=∫0tf(x,τ)dτ;(f4)
t↦f(x,t)/tq-1 is nondecreasing on (-∞,0)∪(0,∞).Whenp=2, (1) arises in the study of the nonlinear Fractional Schrödinger equation(4)-Δαu+Vxu=fx,u-Γxuq-2u,x∈RN. Problems with this type have occurred in many different fields, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [1–4].Whenα=1 and Γ=0, (4) reduces to be the nonlinear Schrödinger equation(5)-Δu+Vxu=fx,u,x∈RN. Using the Nehari-type monotonicity condition, Szulkin and Weth [5] obtained the existence of ground state solutions for (5). But in this paper, the Nehari manifold is usually not smooth and the Nehari-type monotonicity condition for the nonlinearity is not satisfied; then the Nehari manifold method is invalid. In this paper, we are aimed to obtain ground state solutions for (1) by the so-called non-Nehari manifold method, which is established by Tang [6, 7]. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing sequence for the energy functional outside the Nehari manifold by using the diagonal method.Now, we are ready to state the main result of this paper.Theorem 1.
Suppose that(V),(Γ) and (f1)-(f4) hold. Then (1) has a nontrivial ground state solution.


## 2. Preliminaries

In the paper, we will denoteon(1) by the infinitesimal as n→+∞. For the sake of simplicity, the norm of the space Lp(RN) will be denoted by ·p, and integrals over the whole RN will be written ∫.We define the Gagliardo seminorm by(6)uα,p=∫ux-uypx-yN+αpdxdy1/p,where u:RN→R is a measurable function. Then fractional Sobolev space Wα,p(RN) is given by (7)Wα,pRN=u∈LpRN:uis  measurable  anduα,p<∞ endowed with the norm (8)uα,p=uα,pp+upp1/p. For the basic properties of fractional Sobolev spaces, we refer the interested reader to [8]. By condition (V), we define the fractional Sobolev space with potential V(x) by (9)E≔u∈Wα,p:∫Vxupdx<∞ endowed with the norm (10)u≔uα,pp+∫Vxupdx1/p. The energy functional J:E→R defined by (11)Ju=1p∫ux-uypx-yN+αpdxdy+1p∫Vxuxpdx-∫Fx,u-1qΓxuqdx. Under our hypotheses, J is well defined on E. It is well known that J∈C1(E,R), and its derivative is given by(12)J′u,v=∫R2Nux-uyp-2ux-uyvx-vyx-yN+αpdxdy+∫RNVxup-2uvdx-∫fx,uv-Γxuq-2uvdx, for u,v∈E. It is standard to verify that the weak solutions of (1) correspond to the critical points of J. Now, we review the main embedding result for the space E.Lemma 2 ([9, Lemma 1]).
Under assumption(V), the embedding E↪Lr(RN) is compact for any r∈[p,pα∗).In the following lemma, we will show thatJ has Mountain Pass geometric structure.Lemma 3.
Suppose that(V),(Γ), and (f1)-(f4) hold.
(i) There isδ0>0 such that ρ0≔infu=δ0⁡J(u)>J(0)=0.
(ii) For anyu≠0, there exists t>0 such that J(tu)<0.Proof.
(i) By(f1) and (f2), we have(13)fx,u≤εup-1+Cεur-1,Fx,u≤εpup+Cεrur.By E↪Ls(RN) for s∈[p,pα∗) and (13), we have(14)Ju=1pup-∫Fx,udx+1q∫Γxuqdx≥1pup-εC1up-CεC2ur. By the arbitrariness of ε and p<r, we get the conclusion.
(ii) Fixu≠0; by (f3), we have (15)Jtutq=1ptq-pup-∫Fx,tutuquq-1qΓxuqdx⟶-∞, as t→+∞. Thus, there exists t>0 such that J(tu)<0.Now, we define the Nehari manifold by(16)N≔u∈E∖0:J′u,u=0. It is easy to prove that N is not empty. And we have the following lemma.Lemma 4.
Suppose that(V),(Γ) and (f1)-(f4) hold. Let θ∈[0,1)∪(1,∞) and u∈N; then J(θu)<J(u).Proof.
By(f4), we have(17)fx,s≤fx,ttq-1sq-1,s<t;fs≥fx,ttq-1sq-1,s>t. Then(18)Fx,t-Fx,θt=∫θttfx,sds≤1-θqqfx,tt,∀θ≥0,t∈R.Let h(θ)=pθq-qθp; then h′(θ)=pq(θq-1-θp-1). By a simple calculation, we have h′(1)=0 and h(θ)>h(1)=p-q for all θ∈[0,1)∪(1,∞). Thus,(19)θp-1p-θq-1q=p-q-hθpq<0.Let u∈N, it follows from (18) and (19) that(20)Jθu=Ju+Jθu-Ju-θq-1qJ′u,u=Ju+θp-1p-θq-1qup+∫θq-1qfx,uu-Fx,θu+Fx,udx<Ju.Lemma 5.
Suppose that(V),(Γ) and (f1)-(f4) hold; let m≔infN⁡J; then there exist {un}∈E,c∗∈(ρ0,m] satisfying (21)Jun⟶c∗,1+unJ′un⟶0.Proof.
By (i) of Lemma3, there exist δ0>0 and ρ0>0 such that (22)u∈E,u=δ0⟹Ju≥ρ0. Choose vk∈N such that(23)m≤Jvk<m+1k,k∈N.Since J(tvk)<0 for large t>0, Mountain Pass Lemma implies that there exists uk,nn∈N⊂E satisfying(24)Juk,n⟶ck,J′uk,n1+uk,n⟶0,k∈N,where ck∈[ρ0,supt≥0⁡J(tvk)]. By Lemma 4, we have J(vk)=supt≥0⁡J(tvk). Hence, by (23) and (24), we have(25)Juk,n⟶ck<m+1k,J′uk,n1+uk,n⟶0,k∈N. Now, we can choose a sequence {nk}⊂N such that(26)Juk,nk<m+1k,J′uk,nk1+uk,nk<1k,k∈N. Let uk=uk,nk,k∈N. Then, going if necessary to a subsequence, we have (27)Jun⟶c∗∈ρ0,m,J′un1+un⟶0.


## 3. Proof of Theorem1

Proof of Theorem1.
In view of Lemma5, we find a Cerami sequence {un} satisfying (21). By (18), we have(28)1qfx,unun-Fx,un≥0,for  allx∈RN,un∈E.Combining (21) and (28), for n big enough, we have(29)c∗+1≥Jun-1qJ′un,un=1p-1qunp+∫1qfx,unun-Fx,undx≥1p-1qunp. It follows that un is bounded. Passing to a subsequence, we have un⇀u0 in E. By Lemma 2, we have un→u0 in Lr(RN) for r∈[p,pα∗). Then, by (13) and the Hölder inequality, we have(30)∫fx,un-fx,u0un-u0dx≤ε∫unp-1+u0p-1un-u0dx+Cε∫unr-1+u0r-1un-u0dx≤εunpp-1+u0pp-1un-u0p+Cεunrr-1+u0rr-1un-u0r⟶0and(31)∫Γxunq-2un-u0q-2u0un-u0dx≤Γ∞∫unq-1+u0q-1un-u0dx≤unqq-1+u0qq-1un-u0q⟶0.It follows from (30), (31) and Simon inequality ((ap-2a-bp-2b)(a-b)≥1/2p-2a-bp) that(32)J′un-J′u0,un-u0=∫1x-yN+αpunx-unyp-2unx-uny-u0x-u0yp-2u0x-u0yunx-uny-u0x+u0ydxdy+∫Vxunp-2un-u0p-2uun-u0dx-∫fx,un-fx,u0un-u0dx+∫Γxunq-2un-u0q-2u0un-u0dx≥12p-2∫unx-uny-u0x-u0ypx-yN+αpdxdy+12p-2∫Vxun-u0pdx+on1=12p-2un-u0p+on1.On the other hand, by J′un,un-u0→0 and un⇀u0, we have(33)J′un-J′u0,un-u0⟶0.Combining (32) and (33), we have un→u0 in E. Then, by J∈C1(E,R), we have J′(u0)=0. By (28), Lemma 5, and Fatou’s lemma, we have(34)m≥c∗=limn→∞⁡Jun-1qJ′un,un=1p-1qlimn→∞⁡unp+limn→∞⁡∫1qfx,unun-Fx,undx≥1p-1qu0p+∫1qfx,u0u0-Fx,u0dx=Ju0-1qJ′u0,u0=Ju0. This shows that J(u0)≤m and so J(u0)=m=infN⁡J>0.



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*Source: 7935706-2018-07-18.xml*