Multiple ordered solutions for a class of quasilinear problem with oscillating nonlinearity
Tóm tắt
In this paper, we use truncation argument combined with method of minimization, argument of comparison, topological degree arguments and sub-supersolutions method to show existence of multiple positive solutions (which are ordered in the
$$C(\overline{\Omega })$$
-norm) for the following class of problems:
$$\begin{aligned} \left\{ \begin{aligned} -&\Delta u - \kappa \Delta (u^{2}) u +\mu |u|^{q-2}u = \lambda f(u)+h(u) \ \ \text{ in } \ \ \Omega , \\ u&=0 \ \ \text{ on } \ \ \partial \Omega , \end{aligned} \right. \end{aligned}$$
where
$$\Omega $$
is a bounded smooth domain of
$$\mathbb {R}^N$$
$$(N\ge 1), \kappa ,\mu ,\lambda > 0,q\ge 1$$
are parameters, the nonlinearity
$$f: \mathbb {R}\rightarrow \mathbb {R}$$
is a continuous function that can change sign and satisfies an area condition and
$$h: \mathbb {R}\rightarrow \mathbb {R}$$
is a general nonlinearity.
Tài liệu tham khảo
Alarcón, S., Iturriaga, L., Ritorto, A.: Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros. Manuscr. Math. 167, 345–363 (2022). https://doi.org/10.1007/s00229-021-01275-w
Ambrosetti, A., Hess, P.: Positive solutions of asymptotically linear elliptic eigenvalue. Math. Anal. Appl. 73(73), 411–422 (1980)
Arcoya, D., Boccardo, L., Orsina, L.: Critical points for functionals with quasilinear singular Euler–Lagrange equations. Calc. Var. Partial Differ. Equ. 47, 159–180 (2013)
Bass, F., Nasanov, N.N.: Nonlinear electromagnetic spin waves. Phys. Rep. 189, 165–223 (1990)
Borovskii, A., Galkin, A.: Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP 77, 562–573 (1983)
Brandi, H., Manus, C., Mainfray, G., Lehner, T., Bonnaud, G.: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B 5, 3539–3550 (1993)
Brizhik, L., Eremko, A., Piette, B., Zakrzewski, W.J.: Static solutions of a Ddimensional modified nonlinear Schrödinger equation. Nonlinearity 16, 1481–1497 (2003)
Brown, K.J., Budin, H.: Multiple positive solutions for a class of nonlinear boundary value problems. J. Math. Anal. Appl. 60, 329–338 (1977)
Brown, K.J., Budin, H.: On the existence of positive solutions for a class of semilinear elliptic boundary value problems. SIAM J. Math. Anal. 60, 875–883 (1979)
Chen, X.L., Sudan, R.N.: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys. Rev. Lett. 70, 2082–2085 (1993)
Chen, S., Santos, C.A., Yang, M., Zhou, J.: Bifurcation analysis for a modified quasilinear equation with negative exponent. Adv. Nonlinear Anal. 11, 684–701 (2022)
Chipot, M., Roy, P.: Existence results for some functional elliptic equations. Differ. Integr. Equ. 27, 289–300 (2014)
Cintra, W., Medeiros, E., Severo, U.: On positive solutions for a class of quasilinear elliptic equations. Z. Angew. Math. Phys. 70, 79 (2019)
Colin, M., Jeanjean, L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56, 213–226 (2004)
Corrêa, F.J.S.A., Carvalho, M.L., Gonçalves, J.V.A., Silva, K.O.: Positive solutions of strongly nonlinear elliptic problems. Asymptot. Anal. 93, 1–20 (2015). https://doi.org/10.3233/ASY-141278
Corrêa, F.J.S.A., Corrêa, A.S.S., Santos Junior, J.R.: Multiple ordered positive solutions of an elliptic problem involving the p-q-Laplacian. J. Convex Anal. 21(4), 1023–1042 (2014)
Corrêa, F.J.S.A., de Lima, R.N., Nóbrega, A.B.: On positive solutions of elliptic equations with oscillating nonlinearity in \(\mathbb{R} ^N\). Mediterr. J. Math. 19, 62 (2022). https://doi.org/10.1007/s00009-022-01993-9
Corrêa, F.J.S.A., dos Santos, G.C.G., Tavares, L.S.: Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method. Z. Angew. Math. Phys. 72, 99 (2021). https://doi.org/10.1007/s00033-021-01532-8
Dancer, E.N.: Multiple fixed points of positive mappings. J. Reine Angew. Math. 371, 46–66 (1986)
Dancer, E., Schmitt, K.: On positive solutions of semilinear elliptic equations. Proc. Am. Math. Soc. 101, 445–452 (1987)
de Figueiredo, D.G.: On the existence of multiple ordered solutions for nonlinear eigenvalue problems. Nonlinear Anal. TMA 11, 481–492 (1987)
do Ó, J.M., Moameni, A.: Solutions for singular quasilinear Schrödinger equations with one parameter. Commun. Pure Appl. Anal. 9, 1011–1023 (2010)
do Ó, J.M., Severo, U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. 38, 275–315 (2010). https://doi.org/10.1007/s00526-009-0286-6
dos Santos, G., Figueiredo, G.M., Severo, U.B.: Multiple solutions for a class of singular quasilinear problems. J. Math. Anal. Appl. 480(2), 123405 (2019)
dos Santos, G., Figueiredo, G.M., Pimenta, M.T.O.: Multiple ordered solutions for a class of problems involving the 1-Laplacian operator. J. Geom. Anal. 32, 140 (2022). https://doi.org/10.1007/s12220-022-00881-8
Fang, X.-D., Szulkin, A.: Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ. 254, 2015–2032 (2013)
Figueiredo, G.M., Júnior, J.R.S., Suarez, A.: Structure of the set of positive solutions of a non-linear Schrödinger equation. Isr. J. Math. 227, 485–505 (2018)
Figueiredo, G.M., Ruviaro, R., Junior, J.O.: Quasilinear equations involving critical exponent and concave nonlinearity at the origin. Milan J. Math. (2020). https://doi.org/10.1007/s00032-020-00315-63
Figueiredo, G.M., Severo, U.B., Siciliano, G.: Multiplicity of positive solutions for a quasilinear Schrödinger equation with an almost critical nonlinearity. Adv. Nonlinear Stud. 20(4), 933–963 (2020)
García-Melián, J., Iturriaga, L.: Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros. Isr. J. Math. 210, 233–244 (2015)
Hartmann, B., Zakrzewski, W.J.: Electrons on hexagonal lattices and applications to nanotubes. Phys. Rev. B 68, 184302 (2003)
Hasse, R.W.: A general method for the solution of nonlinear soliton and kink Schroödinger equation. Z. Phys. B 37, 83–87 (1980)
Hess, P.: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Commun. Partial Differ. Equ. 6, 951–961 (1981)
Iturriaga, L., Massa, E., Sánchez, J., Ubilla, P.: Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros. J. Differ. Equ. 248, 309–327 (2010)
Jing, Y., Liu, Z., Wang, Z.-Q.: Multiple solutions of a parameter-dependent quasilinear elliptic equation. Calc. Var. Partial Differ. Equ. 55, 150 (2016)
Kavian, O.: Introduction á la théorie des points critiques et applications aux problémes elliptíques, Mathématiques and Applications, vol. 13. Springer, Paris (1993)
Kurihura, S.: Large-amplitude quasi-solitons in super fluids films. J. Phys. Soc. Jpn. 50, 3262–3267 (1981)
Laedke, E., Spatschek, K.: Evolution theorem for a class of perturbed envelope soliton solutions. J. Math. Phys. 24, 2764–2769 (1963)
Lima Alves, R.: Existence of positive solution for a singular elliptic problem with an asymptotically linear nonlinearity. Mediterr. J. Math. 18, 4 (2021)
Liu, J., Wang, Y., Wang, Z.-Q.: Soliton solutions for quasilinear Schrödinger equations II. J. Differ. Equ. 187, 473–493 (2003)
Liu, J., Liu, X., Wang, Z.-Q.: Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete Contin. Dyn. Syst. Ser. S 14, 1779–1799 (2021)
Loc, N.H., Schmitt, K.: On positive solutions of quasilinear elliptic equations. Differ. Integr. Equ. 22, 829–842 (2009)
Makhankov, V.G., Fedyanin, V.K.: Non-linear effects in quasi-one-dimensional models of condensed matter theory. Phys. Rep. 104, 1–86 (1984)
Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Quispel, G.R.W., Capel, H.W.: Equation of motion for the Heisenberg spin chain. Phys. A 110, 41–80 (1982)
Ritchie, B.: Relativistic self-focusing and channel formation in laser-plasma interactions. Phys. Rev. E 50, 687–689 (1994)
Santos, C.A., Yang, M., Zhou, J.: Global multiplicity of solutions for a modified elliptic problem with singular terms. Nonlinearity 34, 7842–7871 (2021)
Xiaohui, Y.: Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros. Commun. Pure Appl. Anal. 12, 451–459 (2013)