## 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

### Keywords:

weak solution; Sobolev space; quasilinear problem
Full Text:

### References:

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