Elsevier

Nonlinear Analysis

Volume 200, November 2020, 112086
Nonlinear Analysis

Properties of positive solutions to nonlocal problems with negative exponents in unbounded domains

https://doi.org/10.1016/j.na.2020.112086Get rights and content

Abstract

We study nonlinear nonlocal problems involving the fractional p-Laplacian operator and nonlinearities with negative exponents in the whole space or in an unbounded epigraph domain. The symmetry and monotonicity results are proved by using a direct method of moving planes.

Introduction

The study of symmetry and monotonicity of solutions to nonlinear equations with fractional order in bounded or unbounded domains has attracted considerable attention in recent years, see e.g. [1], [4], [10], [16], [17], [19]. But there are few results for nonlinear nonlocal problems involving nonlinearities with negative exponents. Analytical problems with negative exponents arise naturally in the thin film equations and electrostatic micro-electromechanical system (MEMS) device [2], [13], [14], [21], singular minimal surface equations [31], Lichnerowicz equations in general relativity [29], [30], and prescribed curvature equations in conformal geometry [36]. Both research directions lead us to study nonlinear nonlocal problems with negative exponents. The main purpose of this paper is to study the qualitative properties of positive solutions to the following nonlinear nonlocal problems involving the fractional p-Laplacian (Δ)psu(x)+uq(x)=0,xRn,and (Δ)psu(x)+uq(x)=0,xΩ,0<u<c,xΩ,u=c,xΩc,where 0<s<1, p2, q>0, c is a positive constant and Ω is an unbounded epigraph domain in Rn given by Ω={x=(x,xn)Rn|xn>|x|2,x=(x1,x2,,xn1)}, Ωc is the complement of Ω in Rn.

Recall that the fractional p-Laplacian (Δ)ps is defined for uCloc1,1Lsp as (Δ)psu(x)=Cn,spPVRn|u(x)u(y)|p2[u(x)u(y)]|xy|n+spdy,where Cn,sp is a normalizing positive constant, PV stays for principal value and Lsp={uLlocp1|Rn|u(x)|p11+|x|n+spdx<+}.

When p=2, (Δ)ps becomes the fractional Laplacian (Δ)s which has a wide application in different subjects (see e.g. [15], [32]). In this case, problem (1) was studied by Cai–Ma [7], the authors obtained the symmetry and monotonicity of positive solutions for q>0 if u(x)=a|x|m+o(1) as |x| with 2sq+1<m<1, a>0 is a constant. In this paper we will see that such kind of results still hold for the nonlinear nonlocal fractional p-Laplacian problem (1) with p>2.

For s=1 and p=2, Eq. (1) formally becomes the related elliptic equation Δu+uq=0,xRn.Du–Guo [18] proved that there is a critical power pc depending on n such that the equation has no stable positive solution for q=pc but it admits a family of stable positive solutions when 0<qpc. This is also one of the results in the earlier paper of Ma–Wei [28]. Later, Guo–Mei [22] obtained that such kind of results still hold for p-Laplace equation Δpu+uq=0 with 1<pn. In [23], Guo–Wei proved that for qn4n, a positive solution u to (3) is radially symmetric about some point in Rn if and only if lim|x|u(x)|x|2q+1=λ for some λ>0. This result is sharp for standard Laplacian operator though their method may not be applicable for nonlocal operator. For the special case q=1, (3) is related to the study of singular minimal hypersurfaces with symmetry (see [31]) and one can easily check that u(x)=C|x| (with C=1n1) is a radially symmetric solution. This will also guide us to propose suitable conditions for the symmetry properties of solutions to our nonlocal problems.

To deal with the nonlocality of the fractional Laplacian, Caffarelli and Silvestre [5] introduced an extension method to localize fractional equations by constructing a Dirichlet to Neumann operator of a degenerate elliptic equation in one higher dimensional space. Another method is to consider the corresponding equivalent integral problems and apply the moving planes of integral forms. A series of results were obtained by these two methods, see [3], [11], [12], [27] and the references therein. Thanks to the works [10] (also [20], [24]), we can use a much simpler method to work directly on the fractional Laplacian problems, that is the so-called direct method of moving planes. One can refer to [7], [26], [33], [37] and the references therein for its application to fractional Laplacian problems. However, it cannot be generalized directly to the fractional p-Laplacian problems due to the nonlinearity of the operator.

Recently, Chen–Li [9] introduced some new ideas, among which is a key boundary estimate, to overcome the difficulties arising from the fractional p-Laplacian. One typical application in [9] is below. Consider the Lane–Emden type problem (Δ)psu(x)=uq(x)in Rn with qp1, assume that uCloc1,1(Rn)Lsp is a positive solution to (4) and u(x)1|x|βfor|x|sufficiently large and for β>spqp+1.Then u must be radially symmetric and radially decreasing with respect to some point in Rn. For more interesting works involving the fractional p-Laplacian in recent years one can see [8], [25], [34], [35] and the references therein.

Inspired by these new ideas, our main concern in this paper is using this direct method of moving planes to prove the symmetry and monotonicity of positive solutions to nonlinear nonlocal problems (1), (2). For nonlocal problems in unbounded domains with the epigraph property, in this paper, we consider bounded solutions in a parabolic type domain Ω. The domain formed by the moving plane and Ω, that is Ωλ in Section 3, is bounded, hence we can apply the key boundary estimate introduced in [9] at the first step of moving planes, which is simpler than singular integral estimate at a negative minimum. By this method, we found that the assumption qp1 in [9, Theorem 3.1] for Dirichlet type problem (4) in a ball domain is not necessary, actually only q1 is needed (see in our previous paper [6, Corollary 1.2]). For the problem in the whole space, we first establish a lemma (see Lemma 2.1 in Section 2) to state the boundedness of negative minima of our target function for unbounded domain, which acts as a maximum principle of decay at infinity as p=2 in the first step of moving planes in Rn. Then, combining the key boundary estimate, we can deduce a contradiction to obtain symmetry of solutions at the limiting position of the moving plane. Different to the case of bounded domains or Lipschitz nonlinearities, to deal with our problems with negative powers in Rn, we need some proper asymptotic assumptions on solutions.

Our first result reads as follows.

Theorem 1.1

Assume that a positive function uCloc1,1(Rn)Lsp satisfies Eq. (1) with the asymptotic property u(x)=a|x|m+o(1),as|x|,where spp+q1<m<1, a>0 is a constant. Then u must be radially symmetric and monotone increasing with respect to some point in Rn.

Remark 1.1

If u presents the asymptotic property (5), since uLsp, then m must be smaller than sp. Hence when 0<sp1 our assumption on m automatically becomes spp+q1<m<sp.

As a generalization, we study the following nonlocal problem (Δ)psu(x)+|x|σuq(x)=0,xRn{0},where 0<s<1, p2, q>0, σ<0. There is one more difficulty since x=0 is a singular point. Our result for (6) is contained in the following.

Theorem 1.2

Assume that uCloc1,1(Rn{0})Lsp is a positive solution to (6) with the asymptotic property u(x)=a|x|m+o(1) as |x|, where a is a positive constant, max{0,sp+σp+q1}<m<1 and limx0u(x)=0. Then u must be radially symmetric and monotone increasing with respect to the origin.

Similar results to Theorem 1.1, Theorem 1.2 were obtained for the classical fractional Laplacian (Δ)s in the previous paper [7]. In the statement of Theorem 1.2 in [7], u is assumed to be monotone increasing about the origin, but if it is replaced by assuming limx0u(x)=0, the result still holds true. Furthermore, we can obtain not only the symmetry but also the monotonicity.

Finally, we study the following problem in the parabolic domain Ω which is more general than problem (2) (Δ)psu(x)+|x|αuq(x)=0,xΩ,0<u<c,xΩ,u=c,xΩc,where α0.

We obtain a monotonicity result for problem (7) as follows.

Theorem 1.3

Assume that uCloc1,1(Ω)C(Rn)Lsp is a solution of problem (7), then u is monotonically decreasing with respect to xn.

The paper is organized as follows. Section 2 is devoted to the proofs of Theorem 1.1, Theorem 1.2 by first establishing a lemma to show that the negative minima of our target function are contained in a ball. The proof of Theorem 1.3 is given in Section 3. The Appendix provides some key lemmas we will use.

In the following, we will use capital C which may depend on n, s and p to denote a general positive constant which may differ from line to line.

Section snippets

Symmetry and monotonicity results in Rn

In this section, we prove Theorem 1.1, Theorem 1.2. We first introduce some basic notations which will be used in the process of moving planes.

For any xRn, λR, choosing any direction to be the x1 direction, we set Tλ={xRn|x1=λ}be the moving plane, Σλ={xRn|x1<λ}be the region to the left of the plane and xλ=(2λx1,x2,,xn)be the reflection of x with respect to Tλ. To compare the values of u(x) and u(xλ), we define uλ(x)=u(xλ),vλ(x)=u(x)uλ(x).Obviously, vλ presents an antisymmetric property,

Monotonicity of solutions in an unbounded parabolic domain

In this section, we prove Theorem 1.3. For λ>0, denote Tλ={xRn|xn=λ}be the moving plane, Hλ={xRn|xn<λ} and xλ=(x1,x2,,2λxn) be the region below the plane and the reflection of x with respect to Tλ respectively. Set uλ(x)=u(xλ),vλ(x)=u(x)uλ(x), Ωλ=HλΩ,Ωλ={xΩλ|vλ(x)<0}.

Proof of Theorem 1.3

From the first equation in (7), we deduce that for xΩλ, (Δ)psu(x)(Δ)psuλ(x)=|xλ|αuλq(x)|x|αuq(x)|x|α(uλq(x)uq(x))q|x|αuq1(x)vλ(x).

Notice that vλ0 in HλΩ, hence, if vλ<0 at some point in Hλ, then the

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11571057).

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