Existence of solutions for a nonhomogeneous Kirchhoff-Schrödinger type equation in $\mathbb{R}^{2}$ involving unbounded or decaying potentials

Francisco S. B. Albuquerque, Anouar Bahrouni, Uberlandio B. Severo

DOI: http://dx.doi.org/10.12775/TMNA.2020.013

Abstract


In this paper, we consider the following nonhomogeneous Kirchhoff-Schrödinger equation: $$ m\bigg(\int_{\mathbb{R}^{2}}|\nabla u|^2{d}x +\int_{\mathbb{R}^{2}}V(|x|)u^2{d}x \bigg) [-\Delta u + V(|x|)u] = Q(|x|)f(u) + \varepsilon h(x), $$% for $ x\in\mathbb{R}^2$, where $m$, $ V$, $ Q$ and $f$ are continuous functions, $\varepsilon$ is a small parameter and $h\neq 0$. When $f$ has exponential growth by means of a Trudinger-Moser type inequality, the Mountain Pass Theorem and Ekeland's Variational Principle in weighted Sobolev spaces are applied in order to establish the existence of at least two weak solutions for this equation.

Keywords


Kirchhoff-Schrödinger equation; Trudinger-Moser inequality; Exponential growth

Full Text:

PREVIEW FULL TEXT

References


Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (1990), 393–413.

Adimurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, Nonlinear Differ. Equ. Appl. 13 (2007), 585–603.

F.S.B. Albuquerque, C.O. Alves and E.S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R2 , J. Math. Anal. Appl. 409 (2014), 1021–1031.

F.S.B. Albuquerque and S. Aouaoui, A weighted Trudinger–Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space, Topol. Methods Nonlinear Anal. 54, (2019), 109–130.

C.O. Alves, F.J.S.A. Corrêa and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl. 2 (2010), 409–417.

C.O. Alves, F.J.S.A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl. 18 (2011), 369–405

B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl. 394 (2012), 488–495.

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619–4627.

D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.

J.M. do Ó, Semilinear Dirichlet problems for the N -Laplacian in RN with nonlinearities in critical growth range, Differential Integral Equations 9 (1996), 967–979.

J.M. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.

J.M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), 286–304.

I. Ekeland, On the variational principle, J. Math. Anal. App. 47 (1974), 324–353.

G.M. Figueiredo, Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. App. 401 (2013), 706–713.

G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), 23–39.

X.M. He and W.M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), 1407–1414.

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), 2285–2294.

A. Li and J. Su, Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in R3 , Z. Angew. Math. Phys. (6) 66 (2015), 3147–3158.

D. Liu and P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation, Nonlinear Anal. 75 (2012), 5032–5038.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.

J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger-type equations, J. Math. Anal. Appl. 417 (2014), 65–79.

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Kirchhoff–Schrödinger-type equations with radial potential, Nonlinear Anal. 75 (2012), 3470–3479.

R.S. Palais, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19–30.

V. Radulescu and D. Smets, Critical singular problems on infinite cones, Nonlinear Anal. 54 (2003), 1153–1164.

E. Silva and S. Soares, Liouville–Gelfand type problems for the N -Laplacian on bounded domains of RN , Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999), 1–30.

W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), 571–583.

J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 201–219.

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 281–304.

E. Tonkes, Solutions to a perturbed critical semilinear equation concerning the N Laplacian in RN , Comment. Math. Univ. Carolin. 40 (1999), 679–699.

N.S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.

V.I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Sov. Math., Dokl. 2 (1961), 746–749.

F. Wang and Y. An, Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition, Bound. Value Probl. 2009, Article ID 540360.

J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), 2314–2351.

X. Wu, Existence of nontrivial solutions and high energy solutions for Kirchhoff–Schrödinger-type equations in RN , Nonlinear Anal. 12 (2011), 1278–1287.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism