Entire solutions to nonlinear scalar field equations with indefinite linear part

Details Entire solutions to nonlinear scalar field equations with indefinite linear part Entire solutions to nonlinear scalar field equations with indefinite linear part

2011-09-21

arxiv.org/abs/1109.4550v1

Mathematics

Analysis of PDEs

31 pages

Scientific paper

We consider the stationary semilinear Schr\"odinger equation $-\Delta u + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In the indefinite setting where the Schr\"odinger operator $-\Delta +a$ has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on $a$ and $f$. We stress that for some of our results we also allow zero to lie in the spectrum of $-\Delta + a$.

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