# zbMATH — the first resource for mathematics

An existence result for $$( p , q )$$-Laplace equations involving sandwich-type and critical growth. (English) Zbl 1445.35183
Summary: We investigate the existence of a nontrivial nonnegative solution to $$( p , q )$$-Laplace equations involving two nonlinear terms, one grows as $$s$$ with $$q < s < p$$ and the other possibly has critical growth. This interesting case cannot be appeared in single $$m$$-Laplace equations and has not been studied under even the subcritical growth. Our argument is based on the concentration-compactness principle by P. L. Lions and the Ekeland variational principle.
##### MSC:
 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
Full Text:
##### References:
 [1] Bonheure, D.; d’Avenia, P.; Pomponio, A., On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346, 877-906 (2016) · Zbl 1365.35170 [2] Pomponio, A.; Watanabe, T., Some quasilinear elliptic equations involving multiple $$p$$-Laplacians, Indiana Univ. Math. J., 2199-2224 (2018) · Zbl 1417.35048 [3] Benci, V.; Fortunato, D.; Pisani, L., Solitons like solutions of a lorentz invariant equation in dimension 3, Rev. Math. Phys., 10, 315-344 (1998) · Zbl 0921.35177 [4] Derrick, G. H., Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5, 1252-1254 (1964) [5] Cherfils, L.; Il’yasov, Y., On the stationary solutions of generalized reaction diffusion equations with $$p \& q$$-Laplacian, Commun. Pure Appl. Anal., 4, 9-22 (2005) · Zbl 1210.35090 [6] Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50, 675-710 (1986), English translation in Math. USSR-Izv. 29 (1987) 33-66 [7] Li, G.; Zhang, G., Multiple solutions for the $$p \& q$$-Laplacian problem with critical exponent, Acta Math. Sci., 29B, 4, 903-918 (2009) · Zbl 1212.35125 [8] Yin, H.; Yang, Z., Multiplicity of positive solutions to a $$p - q$$-Laplacian equation involving critical nonlinearity, Nonlinear Anal., 75, 3021-3035 (2012) · Zbl 1235.35123 [9] Candito, P.; Marano, S. A.; Perera, K., On a class of critical $$( p , q )$$-Laplacian problems, NoDEA Nonlinear Differential Equations Appl., 22, 1959-1972 (2015) · Zbl 1328.35053 [10] Marano, S. A.; Mosconi, S., Some recent results on the Dirichlet problem for $$( p , q )$$-Laplace equations, Discrete Contin. Dyn. Syst. Ser. S, 11, 279-291 (2018) · Zbl 1374.35137 [11] Kawohl, B.; Lucia, M.; Prashanth, S., Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differential Equations, 12, 4, 407-434 (2007) · Zbl 1158.35069 [12] Ho, K.; Kim, Y.-H.; Sim, I., Existence results for Schrödinger $$p ( \cdot )$$-Laplace equations involving critical growth in $$\mathbb{R}^N$$, Nonlinear Anal., 182, 20-44 (2019) · Zbl 1421.35132 [13] Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case, Part 1, Rev. Mat. Iberoam., 1, 1, 145-201 (1985) · Zbl 0704.49005 [14] Ho, K.; Sim, I., On degenerate $$p ( x )$$-Laplace equations involving critical growth with two parameters, Nonlinear Anal., 132, 95-114 (2016) · Zbl 1331.35114 [15] Komiya, Y.; Kajikiya, R., Existence of infinitely many solutions for the $$( p , q )$$-Laplace equation, NoDEA Nonlinear Differential Equations Appl., 23, 49 (2016), 23pp · Zbl 1358.35022 [16] Ho, K.; Sim, I., Existence and multiplicity of solutions for degenerate $$p ( x )$$-Laplace equations involving concave-convex type nonlinearities with two parameters, Taiwanese J. Math., 19, 5, 1469-1493 (2015) · Zbl 1360.35076 [17] Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 2, 519-543 (1994) · Zbl 0805.35028 [18] de Figueiredo, D. G.; Gossez, J.-P.; Ubilla, P., Local superlinearity and sublinearity for the $$p$$-Laplacian, J. Funct. Anal., 257, 721-752 (2009) · Zbl 1178.35176
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.