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Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions. (English) Zbl 1222.35090
Summary: We consider the following quasilinear elliptic exterior problem
$\begin{cases} -\text{div}\big(a(x)|\nabla u|^{p-2}\nabla u\big)+ g(x)|u|^{q-2}u= h(x)|u|^{s-2}u+ \lambda H(x)|u|^{r-2}u, &x\in\Omega,\\ a(x)|\nabla u|^{p-2} \frac{\partial u}{\partial\nu}+ b(x)|u|^{p-2}u=0, &x\in\Gamma=\partial\Omega, \end{cases}$
where $$\Omega$$ is a smooth exterior domain in $$\mathbb R^N$$, and $$\nu$$ is the unit vector of the outward normal on the boundary $$\Gamma$$, $$1<p<N$$, $$1<s<p<r<p^*= Np/(N-p)$$. By the variational principle and the mountain pass theorem, we establish the existence of infinitely many solutions if $$q>r$$ and at least one solution if 1$$<q<s$$.

##### MSC:
 35J62 Quasilinear elliptic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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