Existence and nonexistence of positive solutions for nonlocal problems with inhomogeneous strong Allee effect

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Abstract

In this paper, we deal with the nonlocal elliptic problem with inhomogeneous strong Allee effect-MΩ1pupdxΔpu=λf(x,u),inΩ,u=0,onΩ,where the nonlocal coefficient MΩ1pupdx is a continuous function of Ω12u2dx. By means of variational approach, we prove that the problem has at least two positive solutions for large λ under suitable hypotheses about nonlinearity. We also prove some nonexistence results.

Introduction

In this paper, we study the following problem-MΩ1pupdxΔu=λf(x,u),inΩ,u=0,onΩ,where Ω is a bounded smooth domain of RN with N1, the nonlocal coefficient M(t) is a continuous function of t=Ω1pupdx.

The problem (1.1) is a related to a model introduced by Kirchhoff [2]. More precisely, Kirchhoff proposed a model given by the equationρ2ut2-ρ0h+E2L0Lux2dx2ux2=0,where ρ,ρ0,h,E,L 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 ρ0h+E2L0Lux2dx, and hence the equation is no longer a pointwise identity. The equation-a+bΩu2dxΔu=f(x,u),inΩ,u=0,onΩis 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 f(x,u)ug(x,u) represents a density dependent growth if g(x,u) is a function depending on the population density u. While traditionally g(x,u) 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 a=1,b=0 and f(x,u) satisfies inhomogeneous strong Allee effect growth pattern, Liu et al. [1] proved that the equation-Δu=λf(x,u),inΩ,u=0,onΩhas at least two positive solutions for large λ if 0c(x)f(x,s)ds>0 for x in an open subset of Ω, where c(x)C1(Ω¯) such that f(x,c(x))=0 (see the assumption of (f2)). They also prove some nonexistence results. In particular, they conjecture that the nonexistence holds if 0c(x)f(x,s)ds0 for any xΩ¯ (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 f(x,u) satisfies inhomogeneous strong Allee effect growth pattern and the nonlocal coefficient M(t) 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 M(t) 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 C2 function under our assumptions. In order to overcome this difficulty, we divided Ω into B1 and B2 by comparing the value of c(x) with b̲, 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 X=W01,p(Ω) with the norm u=Ωupdx1/p.

Hereafter, f(x,t) and M(t) are always supposed to verify the following assumptions:

  • (f1)

    f(x,u)C(Ω¯×R+) and f(x,·)C1(R+) for any xΩ¯;

  • (f2)

    there exist b(x)C(Ω¯),c(x)C1(Ω¯) such that 0<b(x)<c(x) and f(x,0)=f(x,b(x))=f(x,c(x))=0 for any xΩ¯;

  • (f3)

    for a.e. xΩ¯,f(x,s)<0 for any s(0,b(x))(c(x),+) and f(x,s)>0 for any s(b

Proof of Theorems 2.1 and 2.2

In this section we shall prove Theorem 2.1, Theorem 2.2.

Lemma 3.1

If M(t) satisfies (M), and f(x,u) satisfies (f1)–(f3) and (2.1), then for λ large enough, Iλ(·) has a global minimum point u1 such that Iλ(u1)<0.

Proof

Since 0c(x)f(x,s)ds>0 in Ω1, then there exists a measurable set Ω0Ω with positive measure, such that 0c(x)f(x,s)ds>0 in Ω0, and 0c(x)f(x,s)ds0 in ΩΩ0. From (M) and the definition of M^(t), we have M^(t)m0t. In view of Lemma 2.1, we haveIλ(u)=M^Ω1pupdx-λΩF(x,u)dxm0Ω1pupdx-λΩ0u(x)f

Proof of a conjecture and some examples

In [1], Liu, Wang and Shi conjecture that the nonexistence holds with a weaker condition:0c(x)f(x,s)ds0for anyxΩ¯.In fact, as we shall see in the following proposition, the condition (4.1) is more strong than 0c¯f¯(s)ds0. Therefore, by Theorem 2.2, the conjecture is right.

Proposition 4.1

If f(x,u) satisfies (f1)–(f3) and 0c(x)f(x,s)ds0 for any xΩ¯, we have 0c¯f¯(s)ds0.

Proof

From (f1)–(f3), we can easily see that f(x,s)0 when s[c(x),c¯]. Thus, we have c(x)c¯f(x,s)ds0. Then, for any xΩ¯, we have00c(x)f

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