Advances in Nonlinear Analysis | |
An elliptic equation with an indefinite sublinear boundary condition | |
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
【 预 览 】
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