Existence of positive solutions to some nonlinear Neumann problems
The paper deals with nontrivial nonnegative solutions of the nonlinear Neumann problem \[ -\Delta u= b(x) u^{s-1}\quad\text{in }\Omega,\quad{\partial u\over\partial\nu}= a(x) u^{q-1}\quad\text{on }\partial\Omega. \] The solutions appear as critical points of the corresponding energy functional. A critical point is then obtained by using the first author's method [Sov. Math., Dokl. 20, 912-916 (1979; Zbl 0456.47053)] in the form \(u= r(v)v\), where \(v\) is a solution of a maximization problem with constraint and \(r(v)\) is a root of the associated bifurcation equation.
scientific article; zbMATH DE number 1557547