Nonexistence and existence of solutions for a supercritical p-Laplacian elliptic problem

Abstract

In this paper, we obtain a general supercritical Sobolev inequality in W0,rad1,p(B) ${W}_{0,rad}^{1, p}\left(B\right)$ , where B is the unit ball in RN ${\mathbb{R}}^{N}$ . In order to discuss the sharp growth, we consider p-Laplacian elliptic problem −Δpu=uq(r) in B,u>0 in B,u=0 on ∂B, $$\begin{cases}-{{\Delta}}_{p}u={u}^{q\left(r\right)}\quad \hfill & \quad \text{in}\quad B,\hfill \\ u{ >}0\quad \hfill & \quad \text{in}\quad B,\hfill \\ u=0\quad \hfill & \quad \text{on}\quad \partial B,\hfill \end{cases}$$ where Δp u = div(|∇u|p−2∇u), 1 < p < N, p*=NpN−p ${p}^{{\ast}}=\frac{Np}{N-p}$ , r = |x| and the variable exponent q(r) ≥ p* − 1 is a differentiable function in [0,1]. We give a Pohozaev identity and prove nonexistence of positive solutions to supercritical p-Laplacian equation in the unit ball B, which is significant for studying the nonlinear problems with variable exponential growth. Moreover, we also give the result of existence for a general nonlinearity case. Finally, we also give some results for a class of concave-convex nonlinear problem.

Keywords

• supercritical sobolev inequality
• sharp growth
• pohozaev identity
• nonexistence and existence result
• 35j60
• 35b33
• 46e35