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A multiplicity result for the jumping nonlinearity problem. (English) Zbl 1163.35386

Summary: We consider a class of nonlinear problems of the form \(Au+g(x,u)=f\), where \(A\) is an unbounded self-adjoint operator on a Hilbert space \(H\) of \(L^2(\Omega)\)-functions, \(\Omega\subset\mathbb R^N\) an arbitrary domain, and \(g:\Omega\times\mathbb R\to\mathbb R\) is a “jumping nonlinearity” in the sense that the limits \(\lim_{s\to-\infty} \frac{g(x,s)}{s}=a\), \(\lim_{s\to\infty} \frac{g(x,s)}{s}=b\) exist and “jump” over the principal eigenvalue of the operator \(-A\). Under rather general conditions on the operator \(L\) and for suitable \(a<b\), we prove some multiplicity results. Applications are given to the wave equation, and elliptic equations in the whole space \(\mathbb R^N\).

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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