A Neumann problem with asymmetric nonlinearity and a related minimizing problem. (English) Zbl 0910.34035

Villegas, Salvador

J. Differ. Equations 145, No. 1, 145-155 (1998).

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.

Reviewer: Walter Šeda (Bratislava)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
35J60	Nonlinear elliptic equations

Keywords:

nonlinear Neumann boundary value problems; critical point theory; Palais-Smale condition

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