Abstract
The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form \(0\le f(r,u)\le a_1|u|^{p(r)-1}\), if \(u\ge 0\), where \(r = |x|\), \(p(r ) = 2^* + r^{\alpha }\), with \(\alpha > 0\), and \(2^* = 2N/(N - 2)\) is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do Ó et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math 12, 623–727 (1959)
Alves, C.O., de Figueiredo, D.G.: Nonvariational elliptic systems via Galerkin methods. In: Haroske, D., Runst, T., Schmeisser, H.J. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. The Hans Triebel Anniversary Volume (2003)
Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal., 122, 519–543 (1994)
Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123(11), 3555–3561 (1995)
Bianchi, G., Chabrowski, J., Szulkin, A.: On symmetric solutions of an elliptic equation with nonlinearity involving critical sobolevexponent. Nonlinear Anal. 25, 41–59 (1995)
Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983)
Brézis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)
Cabré, X., Majer, P.: Truncation of nonlinearities in some supercritical elliptic problems. C. R. Acad. Sci. Paris Ser. I Math. 322, 1157–1162 (1996)
Cao, D., Li, S., Liu, Z.: Nodal solutions for a supercritical semilinear problem with variable exponent. Calc. Var. Partial Differ. Equ. 57, 38 (2018)
de Araujo, A.L.A., Faria, L.F.O.: Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term. J. Differ. Equ. 267, 4589–4608 (2019)
de Araujo, A.L.A., Montenegro, M.: Existence of solution for a nonvariational elliptic system with exponential growth in dimension two. J. Differ. Equ. 264, 2270–2286 (2018)
de Araujo, A.L.A., Montenegro, M.: Existence of solution for a general class of elliptic equations with exponential growth. Ann. Mat. 195, 1737–1748 (2016)
Diening, L., Harjulehto, P., Hasto, P., Ružička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin (2011)
do Ó, J.M., Ruf, B., Ubilla, P.: On supercritical Sobolev type inequalities and related elliptic equations. Calc. Var. Partial Differ. Equ. 55, 83 (2016)
Fan, X., Zhao, Y., Zhao, D.: Compact imbedding theorems with symmetry of strauss-lions type for the space \(W^{1,p(x)}(\Omega )\). J. Math. Anal. Appl. 333–348 (2001)
Gilbarg, D., Trundiger, N.S.: Elliptic Partial Differential Equations of Second Order, 3rd edn. Springer, New York (2001)
Kajikiya, R.: Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations. Arch. Math. (Basel) 91(5), 427–435 (2008)
Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, London (1989)
Kuhestani, N., Moameni, A.: Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties. Calc. Var. 57, 54 (2018). https://doi.org/10.1007/s00526-018-1333-y
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, London (1968)
Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian System. Springer, Berlin (1989)
Melrose, R.: 18.102 Introduction to Functional Analysis. Spring. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA (2009)
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser Verlag (2007)
Strauss, W.A.: On weak solutions of semilinear hyperbolic equations. An. Acad. Brasil. Ciênc 42, 645–651 (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P.Rabinowitz.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A.L.A de Araujo and J. L. F. Melo Gurjão were partially supported by FAPEMIG/FORTIS, L. F.O Faria was partially supported by FAPEMIG CEX APQ 02374/17.
Rights and permissions
About this article
Cite this article
de Araujo, A.L.A., Faria, L.F.O. & Melo Gurjão, J.L.F. Positive solutions of nonlinear elliptic equations involving supercritical Sobolev exponents without Ambrosetti and Rabinowitz condition. Calc. Var. 59, 147 (2020). https://doi.org/10.1007/s00526-020-01800-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-020-01800-x