Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R}_+^n\) with a nonlinear boundary condition
The authors are interested in the existence and nonexistence of positive classical solutions of the problem \[ -\Delta u= au^p\quad\text{in }\mathbb{R}^n_+,\quad -{\partial u\over\partial x_n}= u^q\quad\text{on }\partial\mathbb{R}^n_+, \] where \(a\geq 0\), \(p>1\), \(q>1\) and \(n>2\). For \(p= (n+2)/(n- 2)\) and \(q= n/(n-2)\), it has been known that positive solutions exist and they can be expressed explicitly. The main aim of the paper is to prove the existence of positive solutions for \(p\geq (n+2)/(n- 2)\) and \(q\geq n/(n- 2)\), and also the nonexistence for (i) \(p\leq (n+2)/(n- 2)\) and \(q\leq n/(n- 2)\) with at least one strict inequality, (ii) \(p< n/(n- 2)\) or (iii) \(q< n/(n- 1)\) in the case \(a>0\). In the existence proof, a positive solution is obtained as a limit of eigenfunctions \(u^R\) of the auxiliary eigenvalue problem with an appropriate boundary condition: \[ -\Delta u= \lambda u+a|u|^{p- 1}u\quad\text{in }B^+_R= \{x\in \mathbb{R}^n:|x|< R, x_n>0\}, \] where the Rabinowitz bifurcation theorem and a Pohozaev identity play a key role. The nonexistence is proved by a variant of the moving plane method in (i), and from some blow-up results for parabolic problems in (ii) and (iii).
scientific article; zbMATH DE number 1215632
scientific article