An elliptic equation with an indefinite sublinear boundary condition

Advances in Nonlinear Analysis

article

Humberto Ramos Quoirin¹  Kenichiro Umezu² 

[1] Universidad de Santiago de Chile;Department of Mathematics, Faculty of Education, Ibaraki University

关键词: Semilinear elliptic equation;    indefinite type problem;    nonlinear boundary condition;    asymptotic profiles;   

DOI  :  10.1515/anona-2016-0023

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We investigate the problem { - Δ ⁢ u = | u | p - 2 ⁢ u in  Ω , ∂ ⁡ u ∂ ⁡ ? = λ ⁢ b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u on  ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\lvert u\rvert^{p-% 2}u&&\displaystyle\phantom{}\text{in ${\Omega}$},\\ \displaystyle\frac{\partial u}{\partial\mathbf{n}}&\displaystyle=\lambda b(x)% \lvert u\rvert^{q-2}u&&\displaystyle\phantom{}\text{on ${\partial\Omega}$},% \end{aligned}\right. where Ω is a bounded and smooth domain of ℝ N {\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ), 1 0} , and b ∈ C 1 + α ⁢ ( ∂ ⁡ Ω ) {b\in C^{1+\alpha}(\partial\Omega)} for some α ∈ ( 0 , 1 ) {\alpha\in(0,1)} . We show that ∫ ∂ ⁡ Ω b 0 {\lambda>0} sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in C ⁢ ( Ω ¯ ) {C(\overline{\Omega})} as λ → 0 {\lambda\to 0} . When p < 2 * {p<2^{*}} we also provide the asymptotic profiles of these solutions.

【 授权许可】

CC BY