Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities

Abstract

This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, $\begin{array}{cc}{\Delta }_p^{2}u={\left|u\right|}^{\gamma -1}u+\mu \left({\left|u\right|}^{-1-\alpha }/{\left|x\right|}^{\beta }\right)u& \text{in}\Omega \end{array}$ and $\begin{array}{c}u=\partial u/\partial n=0\text{on}\partial \Omega ,\end{array}$ where $4<2p<N,0\in$ $\Omega ,-\infty <\mu <{\mu }_{\ast },=\left(N-2p\right)\left(1-\alpha \right)/pN,p<\gamma <p^{\ast }=pN/N-2p$ is the critical Sobolev exponent, $0\le \beta <N\left(\gamma +\alpha \right)/\left(\gamma +1\right),0<\alpha <1$ . Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in $E$ by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when $\Omega \subset {ℝ}^N$ be a bounded, smoothandstrictlystar-shapeddomain, $\beta =0\text{and}\gamma \ge {\gamma }_{\ast },=\text{pN}\left(1-\alpha \right)/\left(\text{N}-2\text{p}\right)\left(1-\alpha \right)-\mu \text{Np}>{\text{p}}^{\ast }=\text{pN}/\text{N}-2\text{p}$ .