Existence and nonexistence of positive solutions for nonlocal problems with inhomogeneous strong Allee effect☆
Introduction
In this paper, we study the following problemwhere is a bounded smooth domain of with , the nonlocal coefficient is a continuous function of .
The problem (1.1) is a related to a model introduced by Kirchhoff [2]. More precisely, Kirchhoff proposed a model given by the equationwhere are constants, which extends the classical D’Alembert’s wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinguishing feature of Eq. (1.2) is that the equation contains a nonlocal coefficient , and hence the equation is no longer a pointwise identity. The equationis related to the stationary analogue of Eq. (1.2). Eq. (1.3) received much attention only after Lions [3] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].
In the context of population biology, the nonlinear function represents a density dependent growth if is a function depending on the population density u. While traditionally is assumed to be declining to reflect the crowding effect of the increasing population, Allee suggested that physiological and demographic precesses often possess an optimal density, with the response decreasing as either higher or lower densities. Such growth pattern is called an Allee effect. If the growth rate per capita is negative when u is small, we call it a strong Allee effect; if the growth rate per capita is small than the maximum but still positive for small u, we call it a weak Allee effect (for detail, see [16] or [17]).
Under the special case of Eq. (1.3) with and satisfies inhomogeneous strong Allee effect growth pattern, Liu et al. [1] proved that the equationhas at least two positive solutions for large λ if for x in an open subset of , where such that (see the assumption of ()). They also prove some nonexistence results. In particular, they conjecture that the nonexistence holds if for any (see Remark 1.7 of [1]).
Motivated by above, we generalize existence and nonexistence results for the elliptic Eq. (1.4) to the case of nonlocal elliptic Eq. (1.1). More precisely, if satisfies inhomogeneous strong Allee effect growth pattern and the nonlocal coefficient satisfies some suitable conditions, we establish the existence of at least two positive solutions for the nonlocal problem (1.1) with λ large enough. We also prove some nonexistence results for the nonlocal problem (1.1).
We point out the nonlocal coefficient raises some of the essential difficulties. For example, the way of proving the geometry condition of Mountain Pass Theorem in [1] cannot be used here because the functional of (1.1) is not function under our assumptions. In order to overcome this difficulty, we divided into and by comparing the value of with , then use poincaré inequality to prove it (see Lemma 3.3).
This paper is organized as follows. In Section 2, we present our main results and some necessary preliminary lemmata. In Sections 3, we use variational method and sub-supersolution method to prove the main results. In Section 4, we give some examples which satisfy our hypotheses.
Section snippets
Main results and preliminaries
In this section, we give our main results and some necessary preliminary lemmata which will be used in the following proof. For simplicity we write with the norm .
Hereafter, and are always supposed to verify the following assumptions:
- ()
and for any ;
- ()
there exist such that and for any ;
- ()
for a.e. for any and for any
Proof of Theorems 2.1 and 2.2
In this section we shall prove Theorem 2.1, Theorem 2.2. Lemma 3.1 If satisfies (M), and satisfies ()–() and (2.1), then for λ large enough, has a global minimum point such that . Proof Since in , then there exists a measurable set with positive measure, such that in , and in . From (M) and the definition of , we have . In view of Lemma 2.1, we have
Proof of a conjecture and some examples
In [1], Liu, Wang and Shi conjecture that the nonexistence holds with a weaker condition:In fact, as we shall see in the following proposition, the condition (4.1) is more strong than . Therefore, by Theorem 2.2, the conjecture is right. Proposition 4.1 If satisfies ()–() and for any , we have . Proof From ()–(), we can easily see that when . Thus, we have . Then, for any , we have
References (22)
- et al.
On a class of problems involving a nonlocal operator
Appl. Math. Comput.
(2004) - et al.
Some remarks on nonlocal elliptic and parabolic problems
Nonlinear Anal.
(1997) - et al.
Existence of solutions for a -Kirchhoff-type equation
J. Math. Anal. Appl.
(2009) - et al.
Infinitely many positive solutions for a -Kirchhoff-type equation
J. Math. Anal. Appl.
(2009) On nonlocal -Laplacian Dirichlet problems
Nonlinear Anal.
(2010)- et al.
Infinitely many positive solutions for Kirchhoff-type problems
Nonlinear Anal.
(2009) Boundary regularity for solutions of degenerate elliptic equations
Nonlinear Anal.
(1988)- et al.
Existence and nonexistence of positive solutions of semilinear elliptic equation with inhomogeneous strong Allee effect
Appl. Math. Mech. Engl. Ed.
(2009) Mechanik
(1883)- J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Proceedings of International...
On the well-posedness of the Kirchhoff string
Trans. Am. Math. Soc.
Cited by (1)
Multiple positive solutions for nonlocal problems involving a sign-changing potential
2017, Electronic Journal of Differential Equations
- ☆
Research supported by the NSFC (Nos. 11261052, 11201378).