Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth

Abstract

Abstract Using variational methods we prove the existence of nonnegative solutions for the following class of quasilinear problems given by: − div ( | x | − ϒ p | ∇ u | p − 2 ∇ u ) + | x | − b p ∗ | u | p − 2 u = λ | x | − b p ∗ a ( x ) g ( u ) + γ | x | − b p ∗ | u | p ∗ − 2 u in R N , for the subcritical case ( γ = 0 $\gamma =0$ ) and also for the critical case ( γ = 1 $\gamma =1$ ). The functions a : R N → R and g : R → R are continuous functions that satisfy some additional conditions, 1 < p < N $1 < p < N$ , 0 ≤ ϒ < N − p p $0 \leq \Upsilon < \frac{N-p}{p}$ , ϒ < b ≤ ϒ + 1 $\Upsilon < b \leq \Upsilon +1$ , p ∗ = p ∗ ( ϒ , b ) = p N N − d p $p^{*}=p^{*}(\Upsilon ,b)=\frac{pN}{N -d p}$ with d = 1 + ϒ − b $d = 1 + \Upsilon - b$ .

Keywords

• Variational methods
• Caffarelli
• Kohn
• Nirenberg problem
• Nonnegative solutions