Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent

AbstractIn this paper we investigate the following Kirchhoff type elliptic boundary value problem involving a critical nonlinearity: $$\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=\mu g(x,u)+u^5, u>0& \text{in }\Omega,\\ u=0& \text{on }\partial \Omega,\end{array}\right. {\rm {(K1)}}$$-(a+b∫Ω|∇u|2dx)Δu=μg(x,u)+u5,u>0inΩ,u=0on∂Ω,(K1)here $${\Omega \subset \mathbb{R}^3}$$Ω⊂R3 is a bounded domain with smooth boundary $${\partial \Omega, a,b \geq 0}$$∂Ω,a,b≥0 and a + b > 0. Under several conditions on $${g \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})}$$g∈C(Ω¯×R,R) and $${\mu \in \mathbb{R}}$$μ∈R, we prove the existence and nonexistence of solutions of (K1). This is some extension of a part of Brezis–Nirenberg’s result in 1983.

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