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Mountain pass solutions and an indefinite superlinear elliptic problem of \(\mathbb R^N\). (English) Zbl 1254.35066

Summary: We consider the elliptic problem \[ -\Delta u-\lambda u= a(x)g(u), \]
with \(a(x)\) sign-changing and \(g(u)\) behaving like \(u^p\), \(p > 1\). Under suitable conditions on \(g(u)\) and \(a(x)\), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space \(\mathbb R^N\). More precisely, we show that there exists \(\Lambda>0\) such that this equation on \(\mathbb R^N\) has no positive solution for \(\lambda>\Lambda\), at least two positive solutions for \(\lambda\in (0,\Lambda)\), and at least one positive solution for \(\lambda\in (-\infty, 0]\cup \{\Lambda\}\).
Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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