Infinitely many positive solutions for p-Laplacian equations with singular and critical growth terms

Abstract

Abstract In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p-Laplacian type involving a singularity and a critical Sobolev exponent { − Δ p u = u p ∗ − 1 + λ | u | γ − 1 u , in Ω , u = 0 , on ∂ Ω , $$ \textstyle\begin{cases} -\Delta _{p}u=u^{p^{*}-1}+\frac{\lambda}{|u|^{\gamma -1}u}, & \text{in} ~\Omega , \\ u=0, &\text{on}~\partial \Omega , \end{cases} $$ where Ω is a bounded domain, p ∗ = N p N − p $p^{\ast}=\frac{Np}{N-p}$ ( N ≥ 3 $N \geq 3$ ) is the critical Sobolev exponent and λ > 0 $\lambda > 0$ . Based on the cutoff technique, we prove that the above problem possesses infinitely many positive solutions with 0 0 $\lambda > 0$ small enough.

Keywords

• p-Laplacian equation
• Singular nonlinearity
• Critical point theory