No abstract

In this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N with $N\ge 3$ N ≥ 3 , $a,b>0$ a , b > 0 , $1< q<2$ 1 < q < 2 and $\lambda >0$ λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ b ↘ 0 .

We are concerned with the following nonlocal problem involving critical Sobolev exponent − a − b ∫ Ω ∇ u 2 d x Δ u = λ u q − 2 u + δ u 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in ℝ 4 , a , b > 0 , 1 < q < 2 , δ , and λ are positive parameters. We prove the existence of two positive solutions and obtain uniform estimates of extremal values for the problem. Moreover, the blow-up and the asymptotic behavior of these solutions are also discussed when b ↘ 0 and δ ↘ 0 . In the proofs, we apply variational methods.

In this paper, we study the following nonlocal problem − a − b ∫ Ω   ∇ u 2 d x Δ u = λ u + f x u p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b > 0 are constants, 1 < p < 2 , λ > 0 , f ∈ L ∞ Ω is a positive function, and Ω is a smooth bounded domain in ℝ N with N ≥ 3 . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f ∞ is small enough.