A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities

Abstract

We study the existence and non-existence of nontrivial weak solution of $${\Delta }^{2}u-\mu \frac{u}{|x{|}^4}=\frac{|u{|}^{q_{\beta }-2}u}{|x{|}^{\beta }}+|u{|}^{q-2}u\text{in RN,}$$ where $N\ge 5$, $q_{\beta }=\frac{2(N-\beta )}{N-4}$, $0<\beta <4$, $1<q\le 2^{\ast \ast }$ and $\mu <{\mu }_1:=(\frac{N(N-4)}{4}{)}^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q<2^{\ast \ast }$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{\ast \ast }$ and $-(N-2{)}^2\le \mu <{\mu }_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz and the variational methods.

12 pages

References in corpus (2)

• Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials
• Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential

Cited by in corpus (2)

• Borderline variational problems involving fractional Laplacians and critical singularities
• Entire solutions for a class of fourth order semilinear elliptic equations with weights