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A Neumann problem with asymmetric nonlinearity and a related minimizing problem. (English) Zbl 0910.34035
Let be $$f\in C([0,1] \times\mathbb{R}, \mathbb{R})$$, $$h\in L^2(0,1)$$, $$\lim_{s\to-\infty} (f(x,s)/s) < {\pi^2 \over 4}$$ uniformly in $$x\in [0,1]$$, and $$\lim_{s\to- \infty} f(x,s)<-\int^1_0 h(x)dx <\lim_{s\to \infty} f(x,s)$$ uniformly in $$x\in [0,1]$$, then the problem $$-u''= f(x,u) +h(x)$$, $$x\in(0,1)$$, $$u'(0)= u'(1)=0$$, has a weak solution. The proof is based on variational methods. A partial differential equation version of these problems is also considered.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 35J60 Nonlinear elliptic equations
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##### References:
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