具有不定线性部分的非线性标量场方程的整体解
Entire solutions to nonlinear scalar field equations with indefinite linear part
摘要：
本文考虑定常半线性薛定谔方程$-Δu + a(x) u = f(x,u)$，$u\in H^1(\R^N)$，其中$a$和$f$是收敛到某个极限的连续函数$a_\infty>0$和$f_\infty=f_\infty(u)$是$|x|\to\infty$ .在薛定谔算子$-Δ+a$具有负特征值的不定情形下，我们结合约化方法和拓扑论证证明了在弱单边渐近估计下问题解的存在性.在这种情况下，不需要达到最小能量水平。在本文的第二部分中，我们证明了基态解的存在性下更严格的假设$a$和$f$。我们强调，对于我们的一些结果，我们也允许零躺在频谱的$-Δ+ a$。
英文摘要：
We consider the stationary semilinear Schrödinger equation $-Δu + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, 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 Schrödinger operator $-Δ+a$ 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 $-Δ+ a$.
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