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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}

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
35J70Degenerate elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35B45A priori estimates for solutions of PDE
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