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Ground state solutions for some indefinite variational problems. (English) Zbl 1178.35352

Summary: We consider the nonlinear stationary Schrödinger equation \(-\Delta u+V(x)u=f(x,u)\) in \(\mathbb R^N\). Here, \(f\) is a superlinear, subcritical nonlinearity, and we mainly study the case where both \(V\) and \(f\) are periodic in \(x\) and 0 belongs to a spectral gap of \(-\Delta +V\). Inspired by previous work of Y. Li et al. [Ann. Inst. Henri Poincarè, Anal. Non Linéaire 23, No. 6, 829–837 (2006; Zbl 1111.35079)] and A. Pankov [Milan J. Math. 73, 259–287 (2005; Zbl 1225.35222)], we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities \(f\) which are allowed to have weaker asymptotic growth than usually assumed. For odd \(f\), we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35A15 Variational methods applied to PDEs
49J35 Existence of solutions for minimax problems
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