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Ackermann, Nils; Chagoya, Julián

Ground states for irregular and indefinite superlinear Schrödinger equations. (English) Zbl 1347.35114

J. Differ. Equations 261, No. 9, 5180-5201 (2016).
Summary: We consider the existence of a ground state for the subcritical stationary semilinear Schrödinger equation \(-\Delta u+u=a(x)| u|^{p-2}u\) in \(H^1\), where \(a \in L^\infty(\mathbb R^N)\) 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 \(| x|\to \infty\) 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.

Cited in 4 Documents

MSC:

35J61 Semilinear elliptic equations
35J20 Variational methods for second-order elliptic equations

Keywords:

stationary Schrödinger equation; ground state; indefinite superlinear; subcritical; no limit problem
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References:

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