Some one-dimensional elliptic problems with constraints
DOI:
https://doi.org/10.12775/TMNA.2025.060Keywords
Poly-harmonic Schrödinger equations, homoclinic solutions, normalised solutions, least-energy solutions, bifurcation theory, variational methodsAbstract
Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} \biggl(-\dfrac{d^2}{d x^2}\biggr)^m u + \lambda G'(u) = F'(u),\\ \noalign{\vskip5pt} \displaystyle \int_{\mathbb{R}} K(u) dx = \rho, \end{cases} \end{equation*} in the following cases: $m=1$ or $2G(s) = K(s) = s^2$. In the former, we follow a bifurcation argument; in the latter, we use variational methods.References
P. Antonopoulos and P. Smyrnelis, On minimizers of the Hamiltonian system u00 = ∇W (u), and on the existence of heteroclinic, homoclinic and periodic orbits, Indiana Univ. Math. J. 65 (2016), no. 5, 1503–1524.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345.
B. Bieganowski, P. d’Avenia and J. Schino, Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities, J. Differential Equations 441 (2025), 113489.
B. Bieganowski and J. Mederski, Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth, J. Funct. Anal. 280 (2021), no. 11, paper no. 108989, 26 pp.
B. Bieganowski, J. Mederski and J. Schino, Normalized solutions to at least mass critical problems: singular polyharmonic equations and related curl-curl problems, J. Geom. Anal. 34 (2024), no. 10, paper no. 322, 32 pp.
B. Bieganowski, O. H. Miyagaki and J. Schino, Normalized solutions to polyharmonic equations with Hardy-type potentials and exponential critical nonlinearities, Commun. Contemp. Math. 28 (2026), no. 10, article no. 2550088.
X. Chang, M. Liu and D. Yan, Normalized ground state solutions of nonlinear Schrödinger equations involving exponential critical growth, J. Geom. Anal. 33 (2023), no. 3, paper no. 83, 20 pp.
E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137.
E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 24–51.
P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.
L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (1997), no. 10, 1633–1659.
L. Jeanjean and S.-S. Lu, A mass supercritical problem revisited, Calc. Var. Partial Differential Equations 59 (2020), no. 5, paper no. 174, 43 pp.
L. Jeanjean and S.-S. Lu, Normalized solutions with positive energies for a coercive problem and application to the cubicquintic nonlinear Schrödinger equation, Math. Model Meth. Appl. Sci. 32 (2022), no. 8, 1557–1588.
J. Jeanjean and S.-S. Lu, On global minimizers for a mass constrained problem, Calc. Var. Partial Differential Equations 61 (2022), no. 6, paper no. 214, 18 pp.
L. Jeanjean, J. Zhang and X. Zhong, A global branch approach to normalized solutions for the Schrödinger equation, J. Math. Pures Appl. (9) 183 (2024), 44–75.
Q. Li, V.D. Rădulescu and W. Zhang, Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth, Nonlinearity 37 (2024), no. 2, paper no. 025018, 28 pp.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéare 1 (1984), no. 4, 223–283.
Y. Liu and L. Zhao, Normalized solutions for Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differential Equations 63 (2024), no. 4, paper no. 99, 37 pp.
Y. Lu and X. Zhang, Existence of normalized positive solution of nonhomogeneous biharmonic Schrödinger equations: mass-supercritical case, J. Fixed Point Theory Appl. 26 (2024), no. 2, paper no. 25, 13 pp.
J. Mederski: Nonradial solutions for nonlinear scalar field equations, Nonlinearity 33 (2020), no. 12, 6349–6380. J. Mederski, General class of optimal Sobolev inequalities and nonlinear scalar field equations, J. Differential Equations 281 (2021), 411–441.
J. Mederski and J. Schino, Least energy solutions to a cooperative system of Schrödinger equations with prescribed L2 -bounds: at least L2 -critical growth, Calc. Var. Partial Differential Equations 61 (2022), no. 1, paper no. 10, 31 pp.
J. Mederski and J. Schino, Normalized solutions to Schrödinger equations in the strongly sublinear regime, Calc. Var. Partial Differential Equations 63 (2024), no. 5, paper no. 137, 20 pp.
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162.
T.V. Phan, Blowup for biharmonic Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys. 69 (2018), no. 2, paper no. 31, 11 pp.
J. Schino, Normalized ground states to a cooperative system of Schrödinger equations with generic L2 -subcritical or L2 -critical nonlinearity, Adv. Differential Equations 27 (2022), no. 7–8, 467–496.
M. Shibata, A new rearrangement inequality and its application for L2 -constraint minimizing problems, Math. Z. 287 (2017), no. 1–2, 341–359.
J. Siemianowski, Brezis–Kato type regularity results for higher order elliptic operators, Topol. Methods Nonlinear Anal. 65 (2025), no. 1, 203–217.
N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations 269 (2020), no. 9, 6941–6987.
C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. (3) 45 (1982), no. 1, 169–192.
Z. Zhang, J. Liu and Q. Guan, Existence and multiplicity of normalized solutions to biharmonic Schrödinger equations with subcritical growth, Bull. Iranian Math. Soc. 49 (2023), no. 6, paper no. 80, 26 pp.
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