Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$left{begin{array}{ll}Delta_p u = |u|^{q-2}u, & quad x in Omega , |nabla u|^{p-2} frac{partial u}{partial nu}= lambda |u|^{p-2}u, &quad x in partial Omega, end{array}right.$$ where ({Omega subset mathbb{R}^N}) is a bounded smooth domain, ({nu}) is the outward unit normal at ({partial Omega}) and ({lambda > 0}) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all ({lambda > lambda_n}) , ({lambda_n}) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all ({lambda_n}) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those ({lambda_n}) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case ({p neq 2}) and related problems under spatially heterogeneous reactions are also addressed.