A note on nonautonomous Schrödinger equations with inhomogeneous nonlinearities
DOI:
https://doi.org/10.12775/TMNA.2024.021Słowa kluczowe
Schrödinger, symmetric, positive solutionsAbstrakt
We study the existence of positive ground or bound state solutions for a class of nonlinear Schrödinger equations: \begin{equation*}\label{P_a} -\Delta u+\lambda u=a(x)f(u), \quad u \in H^1\big(\mathbb{R}^N\big), \ N \geq 3, \end{equation*} where the function a is positive, symmetric under some group action $G$, and $\lambda$ is a positive constant. The inhomogeneous nonlinearity $f$, under very mild assumptions, is asymptotically linear or superlinear and subcritical at infinity, with $f(s)/s$, $s> 0$, not satisfying any monotonicity condition.Bibliografia
N. Ackermann, M. Clapp and F. Pacella, Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains, Comm. Partial Differential Equations 38 (2013), no. 5, 751–779.
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and aplications, J. Funct. Anal. 14 (1973), 349–381.
A. Azzollini and A. Pomponio, On the Schrödinger equation in RN under the effect of a general nonlinear term, Indiana Univ. Math. J. 58 (2009), no. 3, 1361–1378.
A. Bahri and P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 3, 365–413.
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20 (1995), no. 9–10, 1725–1741.
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on RN , Arch. Rational Mech. Anal. 124 (1993), no. 3, 261–276.
V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), no. 4, 283–300.
H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I existence of a ground state solution, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345.
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
M. Clapp and L.A. Maia, A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Differential Equations 260 (2016), no. 4, 3173–3192.
M. Clapp, L.A. Maia and B. Pellacci, An upper bound for the least energy of a signchanging solution to a zero mass problem, ArXiv: 2209.10706.
D.G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in RN , J. Differential Equations 173 (2001), no. 2, 470–494.
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer–Verlag, Berlin, New York, 1990.
D G. de Figueiredo, P.L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63.
G. Évéquoz and T. Weth, Entire solutions to nonlinear scalar field equations with indefinite linear part, Adv. Nonlinear Stud. 12 (2012), no. 2, 281–314.
B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN , Adv. in Math. Suppl. Stud. 7a (1981), 369–402.
J. Hirata, A positive solution of a nonlinear elliptic equation in RN with G-symmetry, Adv. Differential Equations 12 (2007), no. 2, 173–199.
J. Hirata, A positive solution of a nonlinear Schrödinger equation with G-symmetry, Nonlinear Analysis 69 (2008), no. 9, 3174–3189.
L. Jeanjean and K. Tanaka, A remark on least energy solutions in RN , Proc. Amer. Math. Soc. 131 (2003), no. 8, 2399–2408.
L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. 21 (2004), no. 3, 287–318.
R. Lehrer and L.A. Maia, Positive solutions of asymptotically linear equations via Pohozaev manifold, J. Funct. Anal. 266 (2013), no. 1, 213–246.
R. Lehrer, L.A. Maia and M. Squassina, Asymptotically linear fractional Schrödinger equations, Complex Var. Elliptic Equ. 60 (2015), no. 4, 529–558.
P.L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145; 223–283.
P. L. Lions, Lagrange multipliers, Morse indices and compactness. Variational methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 161–184.
L.A. Maia and B. Pellacci, Positive solutions for asymptotically linear problems in exterior domains, Ann. Mat. Pura Appl. (4) 196 (2017), no. 4, 1399–1430.
J. Mederski, Nonradial solutions of nonlinear scalar field equations, Nonlinearity 33 (2020), 6349–6380.
M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517.
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Licencja
Prawa autorskie (c) 2024 Raquel Lehrer, Liliane Maia, Ricardo Ruviaro
Utwór dostępny jest na licencji Creative Commons Uznanie autorstwa – Bez utworów zależnych 4.0 Międzynarodowe.
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
