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On the existence of the fundamental eigenvalue of an elliptic problem in \(\mathbb R^N\). (English) Zbl 1136.47040

The nonlinear eigenvalue problem \(-\Delta u+F'(u)=\lambda u\) in \(\mathbb{R}^N\), \(N\geq 3\), where \(F\) is \(C^2\) and such that \(F(0)= F'(0)= F''(0)= 0\), is studied in this paper. The main result says that \(F\) satisfies \(|F'(s)|\leq c_1|s|^{q-1}+ c_2|s|^{p-1}\) with \(2< q\leq p< 2^*= 2N/(N- 2)\) and \(F(s)\geq c_1 s^2- c_2|x|\gamma\), \(c_1,c_2\geq 0\), \(\gamma< 2+ {4\over N}\), and if there exists \(s_0\) such that \(F(s_0)= 0\), then there exists \(\overline r\) such that for any \(r>\overline r\) there exists a solution \(\overline u\in H^1(\mathbb{R}^N)\), with \(\overline u> 0\) radially symmetric such that \(\| u\|_{L^2}= r\) with \(\lambda> 0\).
Several variants of this result are also proved. An interesting application to orbital stability of standing waves of an associated nonlinear Schrödinger equation are given. The proof of the first result involves suitable variational arguments. For the second, a proof is given by using a “splitting lemma” (an idea due to M. Struwe) as an alternative approach to concentration-compactness methods.

MSC:

47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
35Q55 NLS equations (nonlinear Schrödinger equations)
47J35 Nonlinear evolution equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
47J30 Variational methods involving nonlinear operators
35B35 Stability in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 0535.35025
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References:

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