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Weak solutions of quasilinear problems with nonlinear boundary condition. (English) Zbl 0972.35038

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 \[ -\text{div}(a(x)|\nabla u|^{p- 2}\nabla u)= \lambda(1+|x|)^{\alpha_1}|u|^{p- 2}u+(1+|x|)^{\alpha_2}|u|^{q- 2}u\quad\text{in }\Omega, \]
\[ a(x)|\nabla u|^{p- 2}\nabla u\cdot n+ b(x)|u|^{p-2} u=g(x, u)\quad\text{on }\Gamma. \] Under more simple assumptions, the existence of eigensolutions for the eigenvalue problem \[ -\text{div}(a(x)|\nabla u|^{p- 2}\nabla u)= \lambda[(1+|x|^{\alpha_1})|p|^{p- 2} u+(1+|x|)^{\alpha_2}|u|^{q- 2}u]\quad\text{in }\Omega, \]
\[ a(x)|\nabla u|^{p- 2}\nabla u\cdot n+ b(x)|u|^{p- 2}u= \lambda g(x,u)\quad\text{on }\Gamma \] is proved.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
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