In:
manuscripta mathematica, Springer Science and Business Media LLC, Vol. 167, No. 1-2 ( 2022-01), p. 345-363
Abstract:
We study the nonlocal nonlinear problem $$\begin{aligned} \left\{ \begin{array}[c]{lll} (-\Delta )^s u = \lambda f(u) & {} \text{ in } \Omega , \\ u=0 & {}\text{ on } \mathbb {R}^N{\setminus }\Omega , \quad (P_{\lambda }) \end{array} \right. \end{aligned}$$ ( - Δ ) s u = λ f ( u ) in Ω , u = 0 on R N \ Ω , ( P λ ) where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^N$$ R N , $$N 〉 2s$$ N 〉 2 s , $$0 〈 s 〈 1$$ 0 〈 s 〈 1 ; $$f:\mathbb {R}\rightarrow [0,\infty )$$ f : R → [ 0 , ∞ ) is a nonlinear continuous function such that $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 and $$f(t)\sim |t|^{p-1}t$$ f ( t ) ∼ | t | p - 1 t as $$t\rightarrow 0^+$$ t → 0 + , with $$2 〈 p+1 〈 2^*_s$$ 2 〈 p + 1 〈 2 s ∗ ; and $$\lambda $$ λ is a positive parameter. We prove the existence of two nontrivial solutions $$u_{\lambda }$$ u λ and $$v_{\lambda }$$ v λ to ( $$P_{\lambda }$$ P λ ) such that $$0\le u_{\lambda } 〈 v_{\lambda }\le 1$$ 0 ≤ u λ 〈 v λ ≤ 1 for all sufficiently large $$\lambda $$ λ . The first solution $$u_{\lambda }$$ u λ is obtained by applying the Mountain Pass Theorem, whereas the second, $$v_{\lambda }$$ v λ , via the sub- and super-solution method. We point out that our results hold regardless of the behavior of the nonlinearity f at infinity. In addition, we obtain that these solutions belong to $$L^{\infty }(\Omega )$$ L ∞ ( Ω ) .
Type of Medium:
Online Resource
ISSN:
0025-2611
,
1432-1785
DOI:
10.1007/s00229-021-01275-w
Language:
English
Publisher:
Springer Science and Business Media LLC
Publication Date:
2022
detail.hit.zdb_id:
3448-4
detail.hit.zdb_id:
1459409-2
detail.hit.zdb_id:
2113265-3
SSG:
17,1
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