×

A remark on least energy solutions in \(\mathbb R^N\). (English) Zbl 1094.35049

Summary: We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in \(\mathbb R^N\): \[ -\Delta u = g(u),\, u \in H^1(\mathbb R^N), \] where \(N\geq 2\). Without the assumption of the monotonicity of \(t\to \frac{g(t)}{t}\), we show that the mountain pass value gives the least energy level.

MSC:

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

References:

[1] Shinji Adachi and Kazunaga Tanaka, Trudinger type inequalities in \?^{\?} and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051 – 2057. · Zbl 0980.46020
[2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313 – 345. , https://doi.org/10.1007/BF00250555 H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347 – 375. · Zbl 0533.35029 · doi:10.1007/BF00250556
[3] Henri Berestycki, Thierry Gallouët, and Otared Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 5, 307 – 310 (French, with English summary). · Zbl 0544.35042
[4] S. Coleman, V. Glaser, and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), no. 2, 211 – 221.
[5] Manuel del Pino and Patricio L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121 – 137. · Zbl 0844.35032 · doi:10.1007/BF01189950
[6] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, preprint. · Zbl 1060.35012
[7] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109 – 145 (English, with French summary). P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223 – 283 (English, with French summary). · Zbl 0541.49009
[8] Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819 – 851. · Zbl 0754.35042 · doi:10.1002/cpa.3160440705
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.