Weak solutions of quasilinear problems with nonlinear boundary condition (Q5925829)

Let \(\Omega\subset \mathbb{R}^N\) be an unbounded domain with (possible noncompact) smooth boundary \(\Gamma\) and \(n\) is the unit outward normal on \(\Gamma\). At certain (extensive) assumptions there is proved the existence of solutions for the boundary value problem NEWLINE\[NEWLINE-\text{div}(a(x)|\nabla u|^{p- 2}\nabla u)= \lambda(1+|x|)^{\alpha_1}|u|^{p- 2}u+(1+|x|)^{\alpha_2}|u|^{q- 2}u\quad\text{in }\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINEa(x)|\nabla u|^{p- 2}\nabla u\cdot n+ b(x)|u|^{p-2} u=g(x, u)\quad\text{on }\Gamma.NEWLINE\]NEWLINE Under more simple assumptions, the existence of eigensolutions for the eigenvalue problem NEWLINE\[NEWLINE-\text{div}(a(x)|\nabla u|^{p- 2}\nabla u)= \lambda[(1+|x|^{\alpha_1})|p|^{p- 2} u+(1+|x|)^{\alpha_2}|u|^{q- 2}u]\quad\text{in }\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINEa(x)|\nabla u|^{p- 2}\nabla u\cdot n+ b(x)|u|^{p- 2}u= \lambda g(x,u)\quad\text{on }\GammaNEWLINE\]NEWLINE is proved.