On a p(x)-biharmonic singular problem with changing sign weight and with no-flux boundary condition
Resumo
In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at least two nontrivial weak solutions.
Downloads
Referências
A. Ambrosetti, PH. Rabinowitz. Dual variational methodes in critical points theory and applications. J. Funct. Anal. 04:349-381, (1973).
MM. Boureanu, V. Rădulescu, D. Repovš. On a p(.)−biharmonic problem with no-flux boundary condition. Compu. Math. appl. 72:2505-2515, (2016).
A. Callegari, A. Nachman, A nonlinear singular boundary value problem in the theory of pseudoplastic fuids, SIAM J. Appl. Math. 38 275–281, (1980).
D. Cruz-Uribe, A. Fiorenza. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer, Basel; (2013).
L. Diening, P. Harjulehto , P. Hästö, et al. Lebesgue and Sobolev Spaces with Variable Exponents . in: lecture Notes in Mathematics, Springer-verlag. Berlin. Heidelberg; (2011).
L. Diening. Maximal function on generalized Lebesgue spaces Lp(x). Math. Inequal. Appl. 7:245-253, (2004).
DE. Edmunds, J. Lang, A. Nekvinda. On Lp(x) norms. Proc. R. Soc. Lond. Ser. A . 455:219-225, (1999).
DE. Edmunds, J. Rákosník. Sobolev embeddings with variable exponent. stud, Math. 143:267-293, (2000).
AR. El Amrouss, A. Ourraoui. Existence of solutions for a boundary problem involving p(x)−biharmonic operator. Bol. Soc. Parana. Mat. 31:179-192, (2013).
A. El Khalil, M. Laghzal , M.D. Morchid Alaoui, A. Touzani: Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential. Adv. Nonlinear Anal. 9(1), 1130–1144, (2020).
XL. Fan, D. Zhao. On the spaces Lp(x) and Wm,p(x). J. Math. Anal appl. 263:424-446, (2001).
XL. Fan, X Han. Existence and multiplicity of solutions for p(x)-laplacian equations in RN. Nonlinear Anal, 59:173-188, (2004).
XL. Fan. Solution for P(x)−Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 321:464-477, (2005).
XL. Fan. Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312, 464–477, (2005).
D. de Figueiredo. Lectures on the Ekeland Variational Principle with Applications and Detours, TATA Institute, Springer-Verlag, Heidelberg, (1989).
G. Fragnelli. Positive periodic solutions for a system of anisotropic parabolic equation. J. Math. Anal. Appl. 367:204-228, (2010).
M. Ghergu, V. Rădulescu. Ground state solutions for the singular Lane–Emden–Fowler equation with sublinear convection term, J. Math. Anal. Appl. 333, 265–273, (2007).
K. Kefi, V. Rădulescu. Small perturbations of nonlocal biharmonic problems with variable exponent and competing nonlinearities. Rend.Lincei Mat. Appl. 29:439-463, (2018).
O. Kováčik, J. Rákosník. On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J. 41:592-618, (1991).
M. Makvand Chaharlang, A. Razani. A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese Journal of Mathematics 23, 589–599, (2019).
E. Mitidieri. A simple approach to Hardy’s inequalities, Math. Notes 67, 479–486, (2000).
V. Rădulescu. Combined effects in nonlinear singular elliptic problems with convenction, Rev. Roum. Math. Pures Appl. 53, (5–6), 543–553, (2008).
A. Zang, Y. Fu. Interpolation inequalities for derivatives in variable exponent Lebesgue Sobolev spaces. Nonl. Anal. T. M. A. 69:3629-3636, (2008).
VV. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory,Izv. Akad. Nauk SSSR Ser. Mat. 50:675-710, (1986).
Copyright (c) 2024 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
