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An elliptic problem with jumping nonlinearities. (English) Zbl 1161.35395

Summary: It is proved that the elliptic problem \[ -\Delta u=f(x,u)\;\text{in}\;\Omega,\quad u=0\;\text{on}\;\partial\Omega, \] on a bounded domain \(\Omega\subset\mathbb R^ N\) with smooth boundary has two sign-changing solutions and one positive solution if \(f\in\mathbb{C}^ 1(\overline\Omega\times\mathbb R,\mathbb R)\) satisfies \(\mu_ i<f_ u'(x,0)<\mu_ {i+1}\) for some \(i\geq 2,\) \[ \limsup_ {u\to+\infty}f(x,u)/u<\mu_ 1<\liminf_ {u\to-\infty}f(x,u)/u, \] and \[ \limsup_ {u\to-\infty}f(x,u)/u<\infty, \] where \(\mu_ 1<\mu_ 2\leq\mu_ 3\leq\cdots\) are the eigenvalues of \(-\Delta\) with Dirichlet zero boundary conditions on \(\Omega\) counting their multiplicity.

MSC:

35J60 Nonlinear 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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