Abstract
We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When alpha+ beta<4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.
Highlights
Introduction and main resultsIn this paper, we consider the existence and multiplicity of positive solutions for the following truncated Kirchhoff-type system involving concave–convex nonlinearities: ⎧⎪⎨ –M1k( Ω |∇u|2) u = λf (x)|u|q–2u + α α+β|u|α–2u|v|β in Ω, ⎪⎩ –M2k
Problem (Eλ,μ,Mk ) is called nonlocal because of the presence of b1 Ω |∇u|2 and b2 Ω |∇v|2, and b( Ω |∇u|2) u appears in the Kirchhoff equation
In [19] the authors studied the nonlocal boundary value problem of Kirchhoff-type system, where Ω is a bounded domain in RN, N = 1, 2, 3, β ∈ R, ai, bi, λi > 0 for i = 1, 2, and p and q are two positive numbers satisfying certain conditions
Summary
Problem (Eλ,μ,Mk ) is called nonlocal because of the presence of b1 Ω |∇u|2 and b2 Ω |∇v|2, and b( Ω |∇u|2) u appears in the Kirchhoff equation. In [19] the authors studied the nonlocal boundary value problem of Kirchhoff-type system, where Ω is a bounded domain in RN , N = 1, 2, 3, β ∈ R, ai, bi, λi > 0 for i = 1, 2, and p and q are two positive numbers satisfying certain conditions. They obtained the existence of positive solutions by the Nehari manifold and mountain pass lemma and the multiplicity by using cohomological index of Fadell and Rabinowitz.
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