Infinitely many positive solutions for a class of semilinear elliptic equations

Hong-Ge Chen; Jiaojiao Zhang; Jie Zhao

doi:10.3934/dcds.2022130

Article Contents

2022, Volume 42, Issue 12: 5909-5935. Doi: 10.3934/dcds.2022130

This issue Previous Article Upper and lower $ L^2 $-decay bounds for a class of derivative nonlinear Schrödinger equations Next Article Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

Infinitely many positive solutions for a class of semilinear elliptic equations

1.
Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
2.
University of Chinese Academy of Sciences, Beijing 100049, China
3.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

^*Corresponding author: Hong-Ge Chen
^*Corresponding author: Hong-Ge Chen

Received: June 2022

Revised: August 2022

Early access: September 8, 2022

Published online: September 8, 2022

The first author is supported by NSFC (Grant No. 12071364) and China Postdoctoral Science Foundation (Grant No. 2021M703282).

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Abstract

For $ N\geq 3 $ and $ 1<p<\frac{N+2}{N-2} $, we consider the following semilinear elliptic equation

$ \begin{equation*}\label{0-1} \left\{ \begin{aligned} -\Delta u& = V(|x|)(|u-1|^p-1)~~\mathrm{in}~\mathbb{R}^N, \\ u&>0~~\mathrm{in}~\mathbb{R}^N, \quad u\in H^1(\mathbb{R}^N), \end{aligned} \right. ~~~{(1)} \end{equation*}$

where $ V(r) $ is a positive bounded function satisfying

$ V(r) = 1+\frac{a_1}{r^\alpha}+\frac{a_2}{r^{\alpha+1}}+O\left(\frac{1}{r^{\alpha+1+\theta}}\right)\quad\mathrm{as}\ r\to+\infty. $

Here $ a_{1}, \theta>0 $, $ a_{2}\in \mathbb{R} $ and $ \alpha>2(\min\left\{1, (p-1)\right\})^{-1} $ are some constants. By the finite dimensional Lyapunov-Schmidt reduction method, we show that $ (1) $ has infinitely many non-radial positive solutions.

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Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 35B08, 35B09.

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[1]	A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1972), 231-246. doi: 10.1007/BF02412022.
[2]	D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. II, J. Math. Phys., 12 (1971), 945-952. doi: 10.1063/1.1665686.
[3]	D. L. T. Anderson and G. H. Derrick, Stability of time-dependent particlelike solutions in nonlinear field theories. I, J. Math. Phys., 11 (1970), 1336-1346. doi: 10.1063/1.1665265.
[4]	W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5.
[5]	W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601. doi: 10.3934/dcds.2017242.
[6]	A. Bahri, Topological results on a certain class of functionals and application, J. Functional Analysis, 41 (1981), 397-427. doi: 10.1016/0022-1236(81)90083-5.
[7]	A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32. doi: 10.1090/S0002-9947-1981-0621969-9.
[8]	A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. doi: 10.1002/cpa.3160410803.
[9]	A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205-1215. doi: 10.1002/cpa.3160450908.
[10]	I. Bendahou, Z. Khemiri and F. Mahmoudi, On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem, Discrete Contin. Dyn. Syst., 40 (2020), 2367-2391. doi: 10.3934/dcds.2020118.
[11]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[12]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.
[13]	M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24 (1975), 837-846. doi: 10.1512/iumj.1975.24.24066.
[14]	D. Cao, S. Peng and S. Yan, Singularly Perturbed Methods for Nonlinear Elliptic Problems, Cambridge University Press, Cambridge, 2021. doi: 10.1017/9781108872638.
[15]	G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168. doi: 10.1007/s00526-004-0293-6.
[16]	G. Cerami, R. Molle and D. Passaseo, Multiplicity of positive and nodal solutions for scalar field equations, J. Differential Equations, 257 (2014), 3554-3606. doi: 10.1016/j.jde.2014.07.002.
[17]	G. Cerami, D. Passaseo and S. Solimini, Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 32 (2015), 23-40. doi: 10.1016/j.anihpc.2013.08.008.
[18]	W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 252 (2012), 2425-2447. doi: 10.1016/j.jde.2011.09.032.
[19]	M. del Pino and J. Wei, An introduction to the finite and infinite dimensional reduction methods, in Geometric Analysis Around Scalar Curvatures, World Scientific, 2016, 35-118. doi: 10.1142/9789813100558_0002.
[20]	M. del Pino, J. Wei and W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.
[21]	G. C. Dong and S. Li, On the existence of infinitely many solutions of the Dirichlet problem for some nonlinear elliptic equations, Sci. Sinica Ser. A, 25 (1982), 468-475.
[22]	X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE, forthcoming book, 2020, available at the webpage of the authors, https://www.ub.edu/pde/xros/Llibre-ellipticPDE.pdf
[23]	P. H. Frampton, Consequences of vacuum instability in quantum field theory, Phys. Rev. D, 15 (1977), 2922-2928. doi: 10.1103/PhysRevD.15.2922.
[24]	B. Gidas, Euclidean Yang-Mills and related equations, in Bifurcation Phenomena in Mathematical Physics and Related Topics, NATO Advanced Study Institutes Series (Series C - Mathematical and Physical Sciences), vol 54. Springer, Dordrecht, 1980,243-267. doi: 10.1007/978-94-009-9004-3_15.
[25]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.
[26]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.
[27]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[28]	K. McLeod, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb{R}^{N}$. II, Trans. Amer. Math. Soc., 339 (1993), 495-505. doi: 10.1090/S0002-9947-1993-1201323-X.
[29]	M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.
[30]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.
[31]	P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769. doi: 10.1090/S0002-9947-1982-0662065-5.
[32]	M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364. doi: 10.1007/BF01299609.
[33]	M. Struwe, Variational Methods, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9.
[34]	L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc., 362 (2010), 4581-4615. doi: 10.1090/S0002-9947-10-04955-X.
[35]	L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc., 102 (2011), 1099-1126. doi: 10.1112/plms/pdq051.
[36]	J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1.
[37]	J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^{N}$, J. Funct. Anal., 258 (2010), 3048-3081. doi: 10.1016/j.jfa.2009.12.008.