Ground states for irregular and indefinite superlinear Schrödinger equations

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• 作者：Nils Ackermann ; ^{nils@ackermath.info" class="auth_mail" title="E-mail the corresponding author} ; Juliá ; n Chagoya ^{julian.chagoya@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J61 ; 35J20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：261
• 期：9
• 页码：5180-5201
• 全文大小：364 K

文摘

We consider the existence of a ground state for the subcritical stationary semilinear Schrödinger equation −Δu+u=a(x)|u|^p−2u$-\mathrm{\Delta }u+u=a(x)|u{|}^{p-2}u$ in H¹$H^1$, where a∈L^∞(R^N)$a\in L^{\infty }({\mathbb{R}}^N)$ may change sign. Our focus is on the case where loss of compactness occurs at the ground state energy. By providing a new variant of the Splitting Lemma we do not need to assume the existence of a limit problem at infinity, be it in the form of a pointwise limit for a   as |x|→∞$|x|\to \infty$ or of asymptotic periodicity. That is, our problem may be irregular at infinity. In addition, we allow a to change sign near infinity, a case that has never been treated before.