Abstract
In this paper, we study a class of nonlocal semilinear elliptic problems with inhomogeneous strong Allee effect. By means of variational approach, we prove that
Highlights
In this paper, we study the following problem −M Ω |∇u|2 dx∆u = λf (x, u) in Ω, u=0 on ∂Ω, (1.1)where Ω is a bounded smooth domain of RN with N ≥ 1, the nonlocal coefficient M(t) is a continuous function of t = |∇u|2 dx.We shall give a positive answer to a conjecture byLiu, Wang and Shi of [1].The problem (1.1) is related to a model introduced by Kirchhoff [2]
Under the special case of problem (1.3) with a = 1, b = 0 and f (x, u) satisfies inhomogeneous strong Allee effect growth pattern, Liu, Wang and Shi [1] proved that the problem
We prove some nonexistence results for the nonlocal problem (1.1)
Summary
Under the special case of problem (1.3) with a = 1, b = 0 and f (x, u) satisfies inhomogeneous strong Allee effect growth pattern, Liu, Wang and Shi [1] proved that the problem. 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 shall give a positive answer to the conjecture by Liu, Wang and Shi. We note that, in [19], the authors studied the existence of positive solutions for a nonlocal elliptic problem (which different from (1.1)).
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