Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with nonlinear boundary conditio
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Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with nonlinear boundary conditions Somayeh Yazdani1 · Ghasem Alizadeh Afrouzi1,2 · Jamal Rezaee Roshan1 Sayyed Hashem Rasouli3
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Received: 9 April 2020 / Accepted: 14 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this work, we peruse a class of Kirchhoff type problems with nonlinear boundary conditions on a bounded domain, and earn the uniqueness and existence of positive solutions for a class of those problems by the variational methods. Also, we earn the multiplicity of positive solutions of those problems by using the Nehari manifold. Keywords Kirchhoff type equation · Weak solution · Nehari manifold Mathematics Subject Classification 35D05 · 35J60 · 58J32
1 Introduction and statement of main result In this article, we discuss the following Kirchhoff type problem |∇u|2 d x u = αu 3 + βu −λ in , − c+d |∇u|2 d x = g(x)|u|q−2 u on ∂ c+d
(1.1)
B
Jamal Rezaee Roshan [email protected] Somayeh Yazdani [email protected] Ghasem Alizadeh Afrouzi [email protected] Sayyed Hashem Rasouli [email protected]
1
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
2
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
3
Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani University of Technology, Babol, Iran
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S. Yazdani et al.
where ⊂ R3 is a bounded domain with a smooth boundary ∂, c, d ≥ 0 and c + d > 0, α, β > 0 are four parameters, the exponents 0 < λ < 1, q ≥ 1 and g(x) is a continuous function. When c ≥ 0 and d > 0 , problem (1.1) is named the singular Kirchhoff type problem. Kirchhofftype problems are often referred to as being nonlocal because of the presence of the term ( |∇u|2 d x)u and the integral over the entire domain which pointes out that the equation in problem (1.1) is no longer a pointwise identity. When c = 0, d > 0 problem (1.1) is named degenerate, the otherwise is named non-degenerate. The existence and multiplicity of solutions for the following problem − c+d |∇u|2 d x u = αu 3 + βu −λ in ,
(1.2)
c > 0 in , u = 0 on ∂
on a smooth bounded domain ⊂ R3 , has been studied in [16]. Actually, this kind of problem can be steped backward to the search of Kirchhoff. In [13], Kirchhoff considered the model of the form E ∂ 2 u P0 + ρ 2 − ∂t h 2L
L
0
∂u 2 ∂ 2 u dx =0 ∂x ∂x2
(1.3)
where ρ, P0 , h, E, and L are all positive constants. This equation expands the classical d’Alembert’s wave equation by investigating the results of changes in the length of the strings during the vibrations. Problem (1.1) is related to the changeless analogue of problem (1.3). After Kirchhoff’s search, various models of Kirchhoff type have been studied by many authors. We refer the readers to [1,3,5,7,8,15,20,26,27]. In [5], by the variational methods, Bensedik and Bouchekif worked the p
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