On an indefinite nonhomogeneous equation with critical exponential growth. (English) Zbl 1520.35077
Summary: In this manuscript it is obtained existence of solution for the equation
\[
-\mathrm{div}(a(|\nabla u|^p)|\nabla u|^{p-2}\nabla u) +b(|u|^p)|u|^{p-2}u = c(x)f(u), \text{ in } {\mathbb{R}}^N,
\]
where \(1<p<N\), \(N\ge 2\), the functions \(a\), \(b:{\mathbb{R}}^+\rightarrow{\mathbb{R}}^+\) satisfy suitable conditions, \(c\) is a continuous sign-changing potential and the nonlinearity \(f\) has an exponential critical growth at infinity. In the proof we apply variational methods.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Alves, CO; de Freitas, L., Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth, Topol. Method. Nonlinear Anal., 39, 2, 243-262 (2012) · Zbl 1275.35111 |
| [2] | Alves, CO; de Freitas, LR; Soares, SHM, Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth, Differ. Integral Eq., 24, 11-12, 1047-1062 (2011) · Zbl 1249.35085 |
| [3] | Alves, CO; Figueiredo, GM, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 11, 2, 265-294 (2011) · Zbl 1231.35006 · doi:10.1515/ans-2011-0203 |
| [4] | Ambrosetti, A.; Rabinowitz, PH, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [5] | Bezerra do Ó, JM; Medeiros, ES; Severo, UB, On a quasilinear nonhomogeneous elliptic equation with critical growth in \({{\mathbb{R}}^N} \), J. Differ. Eq., 246, 4, 1363-1386 (2009) · Zbl 1159.35027 · doi:10.1016/j.jde.2008.11.020 |
| [6] | Cao, DM, Nontrivial solution of semilinear elliptic equation with critical exponent in \({ R}^2\), Commun. Partial Differ. Eq., 17, 3-4, 407-435 (1992) · Zbl 0763.35034 · doi:10.1080/03605309208820848 |
| [7] | Cherfils, L.; Ilyasov, Y., On the stationary solutions of generalized reaction diffusion equations with p & q- laplacian, Commun. Pure Appl. Anal., 4, 1, 9-22 (2005) · Zbl 1210.35090 · doi:10.3934/cpaa.2005.4.9 |
| [8] | de Figueiredo, DG; Miyagaki, OH; Ruf, B., Elliptic equations in \({ R}^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Eq., 3, 2, 139-153 (1995) · Zbl 0820.35060 · doi:10.1007/BF01205003 |
| [9] | Figueiredo, GM; Rǎdulescu, VD, Nonhomogeneous equations with critical exponential growth and lack of compactness, Opuscula Math., 40, 1, 71-92 (2020) · Zbl 1436.35176 · doi:10.7494/OpMath.2020.40.1.71 |
| [10] | Fiscella, A.; Pucci, P., \((p, N)\) equations with critical exponential nonlinearities in \({\mathbb{R} }^N\), J. Math. Anal. Appl. (2019) · Zbl 1471.35175 · doi:10.1016/j.jmaa.2019.123379 |
| [11] | He, C.; Li, G., The regularity of weak solutions to nonlinear scalar field elliptic equations containing p & q- laplacians, Ann. Acad. Sci. Fenn. Math., 33, 2, 337-371 (2008) · Zbl 1159.35023 |
| [12] | Kavian, O., Introduction à la théorie des points critiques et applications aux problèmes elliptiques (1983), Heidelberg: Springer, Heidelberg |
| [13] | Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 1077-1092 (1971) · Zbl 0213.13001 · doi:10.1007/BF01205003 |
| [14] | Willem, M., Minimax theorems (1996), Boston: Birkhäuser, Boston · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1 |
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.
