Topological Methods in Nonlinear Analysis

Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in $\mathbb{R}^{n}$

Manassés de Souza, João Marcos do Ó, and Tarciana Silva

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In this paper, using variational methods, we establish the existence and multiplicity of weak solutions for nonhomogeneous quasilinear elliptic equations of the form $$ -\Delta_n u + a(x)|u|^{n-2}u= b(x)|u|^{n-2}u+g(x)f(u)+\varepsilon h \quad \mbox{in }\mathbb{R}^n , $$% where $n \geq 2$, $ \Delta_n u \equiv {\rm div}(|\nabla u|^{n-2}\nabla u)$ is the $n$-Laplacian and $\varepsilon$ is a positive parameter. Here the function $g(x)$ may be unbounded in $x$ and the nonlinearity $f(s)$ has critical growth in the sense of Trudinger-Moser inequality, more precisely $f(s)$ behaves like $e^{\alpha_0 |s|^{n/(n-1)}}$ when $s\to+\infty$ for some $\alpha_0 > 0$. Under some suitable assumptions and based on a Trudinger-Moser type inequality, our results are proved by using Ekeland variational principle, minimization and mountain-pass theorem.

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Topol. Methods Nonlinear Anal., Volume 45, Number 2 (2015), 615-639.

First available in Project Euclid: 30 March 2016

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de Souza, Manassés; Marcos do Ó, João; Silva, Tarciana. Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in $\mathbb{R}^{n}$. Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 615--639. doi:10.12775/TMNA.2015.029.

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  • Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $N$-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 393–413.
  • Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger–Moser inequality in $\mathbb{R}^N$ and its applications, Internat. Mathematics Research Notices 13 (2010), 2394–2426.
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I, Arch. Ration. Mech. Anal. 82 (1983), 313–345.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
  • D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Comm. Partial Differential Equations 17 (1992), 407–435.
  • J. Chabrowski, Variational Methods for Potential Operator Equations. With Applications to Nonlinear Elliptic Equations, de Gruyter Studies in Mathematics, 24. Walter de Gruyter & Co., Berlin, 1997.
  • D.G. de Figueiredo, J.M. do Ó and B. Ruf, Elliptic equations and systems with critical Trudinger–Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), 455–476.
  • D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.
  • M. de Souza, On a singular elliptic problem involving critical growth in $\mathbb{R}^N$, NoDEA Nonlinear Differential Equations Appl. 18 (2011), 199–215.
  • ––––, Existence and multiplicity of solutions for a singular semilinear elliptic problem in $\mathbb{R}^2$, Electron. J. Differential Equations 98 (2011), 1–13.
  • M. de Souza and J.M. do Ó, On a class of singular Trudinger–Moser type inequalities and its applications, Math. Nachr. 284 (2011), 1754–1776.
  • J.M. do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in $\mathbb{R}^N$ with nonlinearities in critical growth range, Differ. Integral Equ. 9 (1996), 967–979.
  • ––––, $N$-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.
  • J.M. do Ó, E.S. Medeiros and U.B. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), 286–304.
  • J.M. do Ó, M. de Souza, E. de Medeiros and U.B. Severo, An improvement for the Trudinger–Moser inequality and applications, J. Differential Equations 256 (2014), 1317–1349.
  • J.M. do Ó, E. Medeiros and U.B. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^N$, J. Differential Equations 246 (2009), 1363–1386.
  • J. Giacomoni and K. Sreenadh, A multiplicity result to a nonhomogeneous elliptic equation in whole space $\mathbb{R}^2$, Adv. Math. Sci. Appl. 15 (2005), 467–488.
  • O. Kavian, Introduction á la Théorie des Points Critiques et Applications aux Problèmes Eelliptiques, Springer–Verlag, Paris, 1993.
  • N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal. 262, (2012), 1132–1165.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer–Verlag, Berlin (1989).
  • J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.
  • S.I. Pohozaev, The Sobolev embedding in the case $pl = n$, Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Section, Moscov. Energet. Inst. (1965), 158–170.
  • P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
  • B. Sirakov, Existence and multiplicity of solutions of semi-linear elliptic equations in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations 11 (2000), 119–142.
  • N.S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.
  • M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
  • Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 262, (2012), 1679–1704.