On critical double phase Kirchhoff problems with singular nonlinearity

Abstract

AbstractThe paper deals with the following double phase problem $$\begin{aligned} \begin{aligned}&-m \left[ \int _\Omega \left( \frac{|\nabla u|^p}{p} + a(x) \frac{|\nabla u|^q}{q}\right) \,\mathrm {d}x\right] {\text{div}} \left( |\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \right) \\&\quad = \lambda u^{-\gamma } +u^{p^*-1}&\quad \text {in } \Omega ,\\&u > 0&\quad \text {in } \Omega ,\\&u = 0&\quad \text {on } \partial \Omega , \end{aligned} \end{aligned}$$ - m ∫ Ω | ∇ u | p p + a ( x ) | ∇ u | q q d x div | ∇ u | p - 2 ∇ u + a ( x ) | ∇ u | q - 2 ∇ u = λ u - γ + u p ∗ - 1 in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain with Lipschitz boundary $$\partial \Omega $$ ∂ Ω , $$N\ge 2$$ N ≥ 2 , m represents a Kirchhoff coefficient, $$1<p<q<p^*$$ 1 < p < q < p ∗ with $$p^*=Np/(N-p)$$ p ∗ = N p / ( N - p ) being the critical Sobolev exponent to p, a bounded weight $$a(\cdot )\ge 0$$ a ( · ) ≥ 0 , $$\lambda >0$$ λ > 0 and $$\gamma \in (0,1)$$ γ ∈ ( 0 , 1 ) . By the Nehari manifold approach, we establish the existence of at least one weak solution.