## 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

### Citations:

Zbl 1111.35079; Zbl 1225.35222
Full Text:

### References:

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