Existence of multiple solutions for a Schrödinger logarithmic equation via Lusternik-Schnirelmann category. (English) Zbl 1528.35035
Summary: This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form
\[
\begin{cases}
\begin{aligned}
&- \varepsilon^2 \Delta u + V (x) u = u \log u^2, \quad \text{in } \mathbb{R}^N,\\
&u \in H^1 (\mathbb{R}^N),
\end{aligned}
\end{cases}
\tag{\(P_\varepsilon\)}
\]
where \(V: \mathbb{R}^N\to\mathbb{R}\) is a continuous function that satisfies some technical conditions and \(\varepsilon\) is a positive parameter. We will establish the multiplicity of solution for \((P_\varepsilon)\) by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is \(C^1\).
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Adams, A. and Fournier, J. F., Sobolev Spaces, 2nd edn. (Academic Press, 2003). · Zbl 1098.46001 |
| [2] | Almeida, V., Betancor, J. and Mesa-Rodríguez, L., Variation operators associated with the semigroups generated by Schrödinger operators with inverse square potentials, Anal. Appl.21 (2023) 353-383. · Zbl 1507.42015 |
| [3] | Alves, C. O. and da Silva, A. R., Multiplicity and concentration behavior of solutions for a quasilinear problem involving N-functions via penalization methods, Electron. J. Differ. Equ.158 (2016) 1-24. · Zbl 1383.35093 |
| [4] | Alves, C. O. and de Morais Filho, D. C., Existence and concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys.69 (2018) 144-165. · Zbl 1411.35090 |
| [5] | Alves, C. O. and Ji, C., Existence an concentration of positive solutions for a logarithmic Schrödinger equation via penalization method, Calc. Var. Partial Differential Equations59 (2020) 21, https://doi.org/10.1007/s00526-019-1674-1. · Zbl 1433.35023 |
| [6] | Alves, C. O. and Ji, C., Multi-peak positive solutions for a logarithmic Schrödinger equation via variatinal methods, Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2494-8 |
| [7] | Alves, C. O. and Ji, C., Multiple positive solutions for a Schrödinger logarithmic equation, Discrete Contin. Dyn. Syst.40(5) (2020) 2671-2685. · Zbl 1435.35028 |
| [8] | Alves, C. O. and Figueiredo, G. M., Existence and multiplicity of positive solutions to a \(p\)-Laplacian equation in \(\mathbb{R}^N\), Differ. Integral Equ.19 (2006) 143-162. · Zbl 1212.35107 |
| [9] | Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973) 349-381. · Zbl 0273.49063 |
| [10] | Ambrosetti, A., Badiale, A. M. and Cingolani, S., Semiclassical stats of nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal.140 (1997) 285-300. · Zbl 0896.35042 |
| [11] | Bergfeldt, A. and Staubach, W., On the regularity of multilinear Schrödinger integral operators, Anal. Appl.21 (2023) 385-427. · Zbl 1511.35110 |
| [12] | Cazenave, T., Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal.7 (1983) 1127-1140. · Zbl 0529.35068 |
| [13] | Cingolani, S. and Lazzo, M., Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal.10 (1997) 1-13. · Zbl 0903.35018 |
| [14] | dAvenia, P., Montefusco, E. and Squassina, M., On the logarithmic Schrödinger equation, Commun. Contemp. Math.16 (2014) 1350032. · Zbl 1292.35259 |
| [15] | Degiovanni, M. and Zani, S., Multiple solutions of semilinear elliptic equations with one-sided growth conditions, Math. Comput. Model.32 (2000) 1377-1393. · Zbl 0970.35038 |
| [16] | del Pino, M. and Dolbeault, J., The optimal Euclidean Lp-Sobolev logarithmic inequality, J. Funct. Anal.197 (2003) 151-161. · Zbl 1091.35029 |
| [17] | del Pino, M. and Felmer, P. L., Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ.4 (1996) 121-137. · Zbl 0844.35032 |
| [18] | Fefferman, C. L., Lee Thorp, J. P. and Weinstein, M. I., Honeycomb Schrödinger operators in the strong binding regime, Commun. Pure Appl. Anal.71(6) (2018) 1178-1270. · Zbl 1414.35061 |
| [19] | Fukagai, N., Ito, M. and Narukawa, K., Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on \(\mathbb{R}^N\), Funkcial. Ekvac.49 (2006) 235-267. · Zbl 1387.35405 |
| [20] | Ji, C. and Szulkin, A., A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl.437 (2016) 241-254. · Zbl 1333.35010 |
| [21] | Ji, C. and Xue, Y., Existence and concentration of positive solutions for a fractional logarithmic Schrödinger equation, Differential Integral Equations35 (2022) 677-704. · Zbl 1524.35028 |
| [22] | Oh, J. Y., Existence of semi-classical bound state of nonlinear Schrödinger equations with potential on the class \(( V )_a\), Commun. Partial Differ. Equ.13 (1988) 1499-1519. · Zbl 0702.35228 |
| [23] | Oh, J. Y., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Partial Differ. Equ.131 (1990) 223-253. · Zbl 0753.35097 |
| [24] | Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys.43 (1992) 270-291. · Zbl 0763.35087 |
| [25] | Rao, M. N. and Ren, Z. D., Theory of Orlicz Spaces (Marcel Dekker, New York, 1985). |
| [26] | Squassina, M. and Szulkin, A., Multiple solution to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differ. Equ.54 (2015) 585-597. · Zbl 1326.35358 |
| [27] | Szulkin, A., Minmax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. Henri Poincare Sec. C3(2) (1986) 77-109. · Zbl 0612.58011 |
| [28] | Vzquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12 (1984) 191-201. · Zbl 0561.35003 |
| [29] | Willem, M., Minimax Theorems (Birkhäuser, Boston, 1996). · Zbl 0856.49001 |
| [30] | X. An, Semiclassical states for fractional logarithmic Schrödinger equations, preprint arXiv:2006.10338v3 [math.AP]. |
| [31] | Zloshchastiev, K. G., Logarithmic nonlinearity in the theories of quantum gravity: Origin of time and observational consequences, Gravit. Cosmol.16 (2010) 288-297. · Zbl 1232.83044 |
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.
