Abstract
We study the multiplicity of concentrating solutions for the following class of (p, q)-Laplacian problems
where ε > 0 is a small parameter, \(\gamma \in \{0,1\},\,1 < p < q < N,\,\,{q^*} = {{Nq} \over {N - q}}\) is the critical Sobolev exponent, \({\Delta _s}u = {\rm{div}}(|\nabla u{|^{s - 2}}\nabla u)\), with s ∈ {p, q}, is the s-Laplacian operator, V: ℝN → ℝ is a positive continuous potential such that inf∂Λ V > infΛ V for some bounded open set Λ ⊂ ℝN, and f: ℝ → ℝ is a continuous nonlinearity with subcritical growth. The main results are obtained by combining minimax theorems, penalization technique and Ljusternik–Schnirelmann category theory. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. As far as we know, all these results are new.
Similar content being viewed by others
References
C. O. Alves and A. R. da Silva, Multiplicity and concentration behavior of solutions for a quasilinear problem involving N-functions via penalization method, Electronic Journal of Differential Equations 2016 2016, Article no. 158.
C. O. Alves and A. R. da Silva, Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz–Sobolev space, Journal of Mathematical Physics 57 (2016), Article no. 111502.
C. O. Alves, J. M. do Ó and M. A. S. Souto, Local mountain-pass for a class of elliptic problems in ℝNinvolving critical growth, Nonlinear Analysis 46 (2001), 495–510.
C. O. Alves and G. M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in ℝNvia penalization method, Advanced Nonlinear Studies 5 (2005), 551–572.
C. O. Alves and G. M. Figueiredo, Existence and multiplicity of positive solutions to a p-Laplacian equation in ℝN, Differential and Integral Equations 19 (2006), 143–162.
C. O. Alves and G. M. Figueiredo, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Advanced Nonlinear Studies 11 (2011), 265–294.
A. Ambrosetti and P. H Rabinowitz, Dual variational methods in critical point theory and applications, Journal of Functional Analysis 14 (1973), 349–381.
V. Ambrosio and V. D. Rădulescu, Fractional double-phase patterns: concentration and multiplicity of solutions, Journal de Mathématiques Pures et Appliquées 142 (2020), 101–145.
V. Ambrosio and D. Repovš, Multiplicity and concentration results for a (p, q)-Laplacian problem in ℝN, Zeitschrift für Angewandte Mathematik und Physik 72 (2021), Article no. 33.
L. Beck and G. Mingione, Lipschitz bounds and non-uniform ellipticity, Communications on Pure and Applied Mathematics 73 (2020), 944–1034.
V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calculus of Variations and Partial Differential Equations 2 (1994), 29–48.
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Mathematics 36 (1983), 437–477.
J. Chabrowski and J. Yang, Existence theorems for elliptic equations involving supercritical Sobolev exponent, Advances in Differential Equations 2 (1997), 231–256.
L. Cherfils and V. Il’yasov, On the stationary solutions of generalized reaction difusion equations with p&q-Laplacian, Communications on Pure and Applied Analysis 4 (2005), 9–22.
S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topological Methods in Nonlinear Analysis 10 (1997), 1–13.
M. Colombo and G. Mingione, Regularity for double phase variational problems, Archive for Rational Mechanics and Analysis 215 (2015), 443–496.
M. Del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calculus of Variations and Partial Differential Equations 4 (1996), 121–137.
G. M. Figueiredo, Existence of positive solutions for a class of p&q elliptic problems with critical growth on ℝN, Journal of Mathematical Analysis and Applications 378 (2011), 507–518.
G. M. Figueiredo and M. Furtado, Positive solutions for some quasilinear equations with critical and supercritical growth, Nonlinear Analysis 66 (2007), 1600–1616.
G. M. Figueiredo and M. Furtado, Positive solutions for a quasilinear Schrödinger equation with critical growth, Journal of Dynamics and Differential Equations 24 (2012), 13–28.
G. M. Figueiredo and J. R. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM. Control, Optimisation and Calculus and Variations 20 (2014), 389–415.
N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Communications in Partial Differential Equations 18 (1993), 153–167.
C. He and G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing p&q-Laplacians, Annales Academiæ Scientiarum Fennicæ. Mathematica 33 (2008), 337–371.
C. He and G. Li, The existence of a nontrivial solution to the p&q-Laplacian problem with nonlinearity asymptotic to up−1at infinity in ℝN, Nonlinear Analysis 68 (2008), 1100–1119.
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York–London, 1968.
G. Li and G. Zhang, Multiple solutions for the p&q-Laplacian problem with critical exponent, Acta Mathematica Scientia. Series B 29 (2009), 903–918.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 (1984), 223–283.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Revista Matemática Iberoamericana 1 (1985), 145–201.
P. Marcellini, Regularity under general and p, q-growth conditions, Discrete and Continuous Dynamical Systems. Series S 13 (2020), 2009–2031.
P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, Journal of Mathematical Analysis and Applications 501 (2021), Article no. 124408.
G. Mingione and V. D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, Journal of Mathematical Analysis and Applications 501 (2021), Article no. 125197.
J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Communications in Pure and Applied Mathematics 13 (1960), 457–468.
D. Mugnai and N. S. Papageorgiou, Wang’s multiplicity result for superlinear (p, q)-equations without the Ambrosetti–Rabinowitz condition, Transactions of the American Mathematical Society 366 (2014), 4919–4937.
J. M. do Ó, On existence and concentration of positive bound states of p-Laplacian equations in ℝNinvolving critical growth, Nonlinear Analysis 62 (2005), 777–801.
N. Papageorgiou and V. D. Rădulescu, Resonant (p, 2)-equations with asymmetric reaction, Analysis and Applications 13 (2015), 481–506.
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, On a class of parametric (p, 2)-equations, Applied Mathematics and Optimization 75 (2017), 193–228.
N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Double-phase problems with reaction of arbitrary growth, Zeitschrift für Angewandte Mathematik und Physik 69 (2018), Article no. 108.
P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana University Mathematics Journal 23 (1973), 729–754.
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Zeitschrift für Angewandte Mathematik und Physik 43 (1992), 270–291.
J. Simon, Régularité de la solution d’un problème aux limites non linéaires, Toulouse. Faculté des Sciences. Annales. Mathématiques 3 (1981), 247–274.
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, Somerville, MA, 2010, pp. 597–632.
N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Communications on Pure and Applied Mathematics 20 (1967), 721–747.
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Communications in Mathematical Physics 53 (1993), 229–244.
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser, Boston, MA, 1996.
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 50 (1986), 675–710; English translation in Mathematics of the USSR-Izvestiya 29 (1987), 33–66.
Acknowledgements
The first author was partly supported by the GNAMPA Project 2020 entitled: Studi di Problemi Frazionari Nonlocali tramite Tecniche Variazionali. The second author was supported by the grant “Nonlinear Differential Systems in Applied Sciences” of the Romanian Ministry of Research, Innovation and Digitization, within PNRR-III-C9-2022-I8/22.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ambrosio, V., Rădulescu, V.D. Multiplicity of concentrating solutions for (p, q)-Schrödinger equations with lack of compactness. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2619-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-024-2619-8