Multiple solutions for perturbed quasilinear elliptic problems
DOI:
https://doi.org/10.12775/TMNA.2022.069Keywords
$(p, q)$-quasilinear elliptic equation, asymptotically $(q-1)$-linear problem, q)$-Laplacian operator, variational methods, essential value, perturbed problem, pseudo-genus, quasi-eigenvalue, regularity of solutionsAbstract
We investigate the existence of multiple solutions for the $(p,q)$-quasilinear elliptic problem \[ \begin{cases} -\Delta_p u -\Delta_q u\ =\ g(x, u) + \varepsilon\ h(x,u) & \mbox{in } \Omega,\\ u=0 & \mbox{on } \partial\Omega,\\ \end{cases} \] where $1< p< q< +\infty$, $\Omega$ is an open bounded domain of ${\mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$, $\varepsilon\in{\mathbb R}$ and $h\in C(\overline\Omega\times{\mathbb R},{\mathbb R})$. In spite of the possible lack of a variational structure of this problem, from suitable assumptions on $g(x,u)$ and appropriate procedures and estimates, the existence of multiple solutions can be proved for small perturbations.References
A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), 725–728.
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), 981–1012.
R. Bartolo, Multiplicity results for a class of quasilinear elliptic problems, Mediterr. J. Math. 11 (2014), 1099–1113.
R. Bartolo, A.M. Candela and A. Salvatore, p-Laplacian problems with nonlinearities interacting with the spectrum, NoDEA 20 (2013), 1701–1721.
R. Bartolo, A.M. Candela and A. Salvatore, Perturbed asymptotically linear problems, Ann. Mat. Pura Appl. 193 (2014), 89–101.
R. Bartolo, A.M. Candela and A. Salvatore, An existence result for perturbations of (p, q)-quasilinear elliptic problems, Recent Advances in Mathematical Analysis (A.M. Candela et al., eds.), Trends Math. (to appear), DOI: 10.1007/978-3-031-20021-2 8.
V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533–572.
A.M. Candela and G. Palmieri, Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations 34 (2009), 495–530.
L. Cherfils and Y. Il’yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22.
S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations 30 (2005), 1191–1203.
C.V. Coffman, A minimum-maximum principle for a class of nonlinear integral equations, J. Analyse Math. 22 (1969), 391–419.
F. Colasuonno, Multiple solutions for asymptotically q-linear (p, q)-Laplacian problems, Math. Meth. Appl. Sci. (2021), 1–19.
M. Degiovanni and S. Lancelotti, Perturbations of even nonsmooth functionals, Differential Integral Equations 8 (1995), 981–992.
M. Degiovanni and S. Lancelotti, Perturbations of critical values in nonsmooth critical point theory, Well-Posed Problems and Stability in Optimization (Y. Sonntag, ed.), Serdica Math. J. 22 (1996), 427–450.
E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850.
G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. 58 (2001), 339–378.
E.R. Fadell and S. Husseini, Relative cohomological index theories, Adv. Math. 64 (1987), 1–31.
E.R. Fadell and P.H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139–174.
J. Garcı́a Azorero and I. Peral Alonso, Existence and nonuniqueness for the pLaplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), 1389–1430.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983.
M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879–902.
C.J. He and G.B. Li, The regularity of weak solutions to nonlinear scalar field containing p&q-Laplacians, Ann. Acad. Sci. Fenn. Math. 33 (2008), 337–371.
G. Li and H.S. Zhou, Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as up−1 at infinity, J. London Math. Soc. 65 (2002), 123–138.
S. Li and Z. Liu, Perturbations from symmetric elliptic boundary value problems, J. Differential Equations 185 (2002), 271–280.
S. Li and Z. Liu, Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents, Topol. Methods Nonlinear Anal. 28 (2006), 235–261.
G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0, Proc. Amer. Math. Soc. 109 (1990), 157–164.
P. Lindqvist, Addendum: “On the equation div(|∇u|p−2 ∇u) + λ|u|p−2 u = 0”, Proc. Amer. Math. Soc. 116 (1992), 583–584.
E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index, Nonlinear Anal. 71 (2009), 3654–3660.
D. Mugnai and N.S. Papageorgiu, Wang’s multiplicity results for superlinear (p, q)equations without the Ambrosetti–Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919–4937.
K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003), 301–309.
K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst. 13 (2005), 743–753.
P.H. Rabinowitz, Variational methods for nonlinear eigenvalues problems, Eigenvalues of Non-Linear Problems (C.I.M.E., III Ciclo, Varenna, 1974), (G. Prodi, ed.), Edizioni Cremonese, Roma, 1974, pp. 139–195.
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, 1986.
J.T. Schwartz, Generalizing the Lusternik–Schnirelmann theory of critical points, Commun. Pure Appl. Math. 17 (1964), 307–315.
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4rd edition, Ergeb. Math. Grenzgeb. (4), vol. 34, Springer–Verlag, Berlin, 2008.
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.
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