Properties of positive solutions to nonlocal problems with negative exponents in unbounded domains
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 -Laplacian and where , , , is a positive constant and is an unbounded epigraph domain in given by is the complement of in .
Recall that the fractional -Laplacian is defined for as where is a normalizing positive constant, stays for principal value and
When , becomes the fractional Laplacian 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 if as with , is a constant. In this paper we will see that such kind of results still hold for the nonlinear nonlocal fractional -Laplacian problem (1) with .
For and , Eq. (1) formally becomes the related elliptic equation Du–Guo [18] proved that there is a critical power depending on such that the equation has no stable positive solution for but it admits a family of stable positive solutions when . 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 -Laplace equation with . In [23], Guo–Wei proved that for , a positive solution to (3) is radially symmetric about some point in if and only if for some . This result is sharp for standard Laplacian operator though their method may not be applicable for nonlocal operator. For the special case , (3) is related to the study of singular minimal hypersurfaces with symmetry (see [31]) and one can easily check that (with ) 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 -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 -Laplacian. One typical application in [9] is below. Consider the Lane–Emden type problem in with , assume that is a positive solution to (4) and Then must be radially symmetric and radially decreasing with respect to some point in . For more interesting works involving the fractional -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 . The domain formed by the moving plane and , that is 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 in [9, Theorem 3.1] for Dirichlet type problem (4) in a ball domain is not necessary, actually only 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 in the first step of moving planes in . 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 , we need some proper asymptotic assumptions on solutions.
Our first result reads as follows.
Theorem 1.1 Assume that a positive function satisfies Eq. (1) with the asymptotic property where , is a constant. Then must be radially symmetric and monotone increasing with respect to some point in .
Remark 1.1 If presents the asymptotic property (5), since , then must be smaller than . Hence when our assumption on automatically becomes . As a generalization, we study the following nonlocal problem where , , , . There is one more difficulty since is a singular point. Our result for (6) is contained in the following.
Theorem 1.2 Assume that is a positive solution to (6) with the asymptotic property as , where is a positive constant, and . Then 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 in the previous paper [7]. In the statement of Theorem 1.2 in [7], is assumed to be monotone increasing about the origin, but if it is replaced by assuming , 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 which is more general than problem (2) where .
We obtain a monotonicity result for problem (7) as follows.
Theorem 1.3 Assume that is a solution of problem (7), then is monotonically decreasing with respect to .
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 which may depend on , and to denote a general positive constant which may differ from line to line.
Section snippets
Symmetry and monotonicity results in
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 , , choosing any direction to be the direction, we set be the moving plane, be the region to the left of the plane and be the reflection of with respect to . To compare the values of and , we define Obviously, presents an antisymmetric property,
Monotonicity of solutions in an unbounded parabolic domain
In this section, we prove Theorem 1.3. For , denote be the moving plane, and be the region below the plane and the reflection of with respect to respectively. Set
Proof of Theorem 1.3 From the first equation in (7), we deduce that for , Notice that in , hence, if at some point in , then the
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 11571057).
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