Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane
Abstract
A nonlinear scalar field equation is studied, which takes the form − △u=g(u) in RN, u ∈ E, where E is a suitable Banach space.. The nonlinear term g(s) has subcritical growth at infinity. Near s=0, the nonlinear term g(s) grows like sq for any q>0 if N=2 and for any q ∈ (0, (N+2)/(N−2)) if N≥ 3. Through the technique of adding one dimension of space, by mountain pass theorem and symmetric mountain theorem, we prove the multiplicity of radial solutions if N≥ 2 and the multiplicity of nonradial solutions if N≥ 4. Unlike [1] and [2], the proof presented here does not rely on any decomposition results. During the treatment for N=2, we provide a new proof for certain Gagliardo-Nirenberg type inequality on R2, which is originally due to [3]. Based on this inequality we get a Trudinger-Moser type inequality involving gradient norm through Taylor expansion. Eventually we construct a special path to obtain the least energy level for solutions. [1] J. Mederski, Nonradial solutions of nonlinear scalar field equations. Nonlinearity 33 (2020), 6349--6380. [2] L. Jeanjean, S.S. Lu, Nonlinear scalar field equations with general nonlinearity. Nonlinear Anal. 190 (2020), 111604. [3] D.E. Edmunds, A.A.Ilyin, Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995) 71--74.
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BibTeX
@misc{lu2026radial,
title = {{Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane}},
author = {Qiuping Lu and Shuolei Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02524},
eprint = {2611.02524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02524},
}
RIS
TY - UNPB
TI - Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane
AU - Qiuping Lu
AU - Shuolei Liu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02524
UR - https://arxiv.org/abs/2611.02524
AB - A nonlinear scalar field equation is studied, which takes the form − △u=g(u) in R^N, u ∈ E, where E is a suitable Banach space.. The nonlinear term g(s) has subcritical growth at infinity. Near s=0, the nonlinear term g(s) grows like s^q for any q>0 if N=2 and for any q ∈ (0, (N+2)/(N−2)) if N≥ 3. Through the technique of adding one dimension of space, by mountain pass theorem and symmetric mountain theorem, we prove the multiplicity of radial solutions if N≥ 2 and the multiplicity of nonradial solutions if N≥ 4. Unlike [1] and [2], the proof presented here does not rely on any decomposition results. During the treatment for N=2, we provide a new proof for certain Gagliardo-Nirenberg type inequality on R^2, which is originally due to [3]. Based on this inequality we get a Trudinger-Moser type inequality involving gradient norm through Taylor expansion. Eventually we construct a special path to obtain the least energy level for solutions. [1] J. Mederski, Nonradial solutions of nonlinear scalar field equations. Nonlinearity 33 (2020), 6349--6380. [2] L. Jeanjean, S.S. Lu, Nonlinear scalar field equations with general nonlinearity. Nonlinear Anal. 190 (2020), 111604. [3] D.E. Edmunds, A.A.Ilyin, Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995) 71--74.
ER -