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Two non-trivial solutions for a non-homogeneous Neumann problem: An Orlicz-Sobolev space setting. (English) Zbl 1172.35026
Let $\Omega\subset \mathbb{R}^N (N\geq 3)$ be a smooth bounded domain, $\nu$ be the outer normal of $\partial \Omega$ and $\lambda>0$ is a parameter. Suppose that the function $a: (0,\infty)\rightarrow \mathbb{R}$ is such that $$\phi(t)=a(|t|)t \text{ if } t\neq 0 \text{ and } \phi(t)=0 \text{ if } t=0$$ is an odd, strictly increasing homeomorphism from $\mathbb{R}$ to $\mathbb{R}$. Then the authors of the paper proved that the Neumann problem $$\cases -\text{div}(a(|\nabla u(x)|)\nabla u(x)) + a(|u(x)|)u(x) =\lambda f(x,u(x)), &\text{ for } x\in \Omega,\\ \partial u(x) / \partial \nu =0, &\text{ for } x \in \partial\Omega, \endcases$$ has at least two non-trivial weak solutions for any $\lambda$ in some open interval $\Lambda$ and the norms of the solutions are bouned above by a constant, if the following further conditions are satisfied: {\parindent5mm \item{$\Phi$} $1<\liminf_{t\rightarrow \infty} \frac{t\phi(t)}{\Phi(t)} \leq \sup_{t>0}\frac{t\phi(t)}{\Phi(t)} <\infty \text{ and } N<p_0<\liminf_{t\rightarrow \infty} \frac{\log(\Phi(t))}{\log(t)},$ $\text{ where } \Phi(t)=\int_0^t \phi(s)ds \text{ and } p_0=\inf_{t>0} \frac{t\phi(t)}{\Phi(t)}.$ \item{f0} $\exists c_0>0 \text{ and } s\in (0,p_0-1) \text{ such that } |f(x,t)| \leq c_0 (1+|t|^s) \text{ for every } (x,t) \in \Omega\times\mathbb{R}.$ \item{f1} $\exists b \in \mathbb{R} \text{ such that } B_F=\int_\Omega F(x,b)dx >0 \text{ with } F(x,t)=\int_0^tf(x,s)ds \text{ for } t\in \mathbb{R}.$ \item{f2} $\exists \delta>0 \text{ such that } f(x,t)t\leq 0 \text{ for every } x\in \Omega \text{ and } t \in [-\delta, \delta].$\par}

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods 35J70 Degenerate elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35B45 A priori estimates for solutions of PDE
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