An elliptic equation with an indefinite sublinear boundary condition

We investigate the problem \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 {\mathbb{R}^{N}} ( {N\geq 2} ), {1<q<2<p} , {\lambda>0} , and {b\in C^{1+\alpha}(\partial\Omega)} for some {\alpha\in(0,1)} . We show that {\int_{\partial\Omega}b<0} is a necessary and sufficient condition for the existence of nontrivial non-negative solutions of this problem. Under the additional condition {b^{+}\not\equiv 0} we show that for {\lambda>0} sufficiently small this problem has two nontrivial non-negative solutions which converge to zero in {C(\overline{\Omega})} as {\lambda\to 0} . When {p<2^{*}} we also provide the asymptotic profiles of these solutions.