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Solutions in spectral gaps for a nonlinear equation of Schrödinger type. (English) Zbl 0804.35033

Summary: We study the existence of a nontrivial \(H^ 2 (\mathbb{R}^ N)\) solution for an equation of the form \[ -\Delta u(x) + p(x) u(x) - f \bigl( x,u(x) \bigr) = \lambda u(x), \quad x \in \mathbb{R}^ N,\;\lambda \in \mathbb{R}, \] where \(p \in L^ \infty (\mathbb{R}^ N)\) is a periodic function. We assume that the operator \(-\Delta + p - \lambda : H^ 2 (\mathbb{R}^ N) \subset L^ 2 (\mathbb{R}^ N) \to L^ 2 (\mathbb{R}^ N)\) is strongly indefinite and invertible and that \(f(x,\cdot):\mathbb{R} \to \mathbb{R}\) is odd and satisfies some superlinear but subcritical growth conditions. We extend the class of nonlinearities which has been studied up to now. In particular, under standard technical restrictions, the existence of a solution is derived, when \(\lim_{| x | \to \infty} f(x,s) \equiv \tilde f(s)>0\) exists for all \(s \in \mathbb{R}\), if we assume that \(f(x,s) \geq \overline f(s)\) for all \(s \in \mathbb{R}\) and a.e. on \(\mathbb{R}^ N\).

MSC:

35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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