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Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions. (English) Zbl 1437.35224

Summary: Consider the semilinear Schrödinger equation \[\begin{cases} - \triangle u + V ( x ) u = f ( x , u ) , x \in \mathbb{R}^N , \\ u \in H^1 ( \mathbb{R}^N ) , \end{cases}\]
where both \(V(x)\) and \(f(x, u)\) are periodic in \(x\), 0 belongs to a spectral gap of \(- \triangle + V\), and \(f(x, u)\) is subcritical and allowed to be super-linear at some \(x \in \mathbb{R}^N\) and asymptotically linear at the other \(x \in \mathbb{R}^N\). In the existing works in the literature, it is commonly assumed that \(\lim_{| u | \to \infty} \frac{ \int_0^u f ( x , s ) \operatorname{d} s}{ u^2} = \infty\) uniformly in \(x \in \mathbb{R}^N\), to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition \(\lim_{| u | \to \infty} \frac{ \int_0^u f ( x , s ) \operatorname{d} s}{ u^2} = \infty \), a.e. \(x \in G\) just for some domain \(G \subset \mathbb{R}^N\).

MSC:

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