Solutions for a \(p(x)\)-Kirchhoff type equation with Neumann boundary data. (English) Zbl 1225.35079
Summary: This paper is concerned with the existence and multiplicity of solutions to a class of \(p(x)\)-Kirchhoff type problems with Neumann boundary data of the following form
\[ \begin{cases} -M\left(\int_\Omega \frac{1}{p(x)} \big(|\nabla u|^{p(x)}+|u|^{p(x)}\big)\,dx\right) \big(\text{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)- |u|^{p(x)-2}\big)= f(x,u) &\text{in }\Omega,\\ \frac{\partial u}{\partial v}=0 &\text{on }\partial\Omega. \end{cases} \]
By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on \(f\) and \(M\), we obtain a number of results on the existence and multiplicity of solutions for the problem. In particular, we also obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solutions and the regularity of weak solutions of the problem are considered.
\[ \begin{cases} -M\left(\int_\Omega \frac{1}{p(x)} \big(|\nabla u|^{p(x)}+|u|^{p(x)}\big)\,dx\right) \big(\text{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)- |u|^{p(x)-2}\big)= f(x,u) &\text{in }\Omega,\\ \frac{\partial u}{\partial v}=0 &\text{on }\partial\Omega. \end{cases} \]
By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on \(f\) and \(M\), we obtain a number of results on the existence and multiplicity of solutions for the problem. In particular, we also obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solutions and the regularity of weak solutions of the problem are considered.
MSC:
| 35J35 | Variational methods for higher-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35D30 | Weak solutions to PDEs |
| 35B09 | Positive solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
variational method; \(p(x)\)-Kirchhoff type equation; nonlocal problems; Neumann boundary value problemReferences:
| [1] | Ru̇z˘ic˘ka, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0968.76531 |
| [2] | Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 9, 33-66 (1987) · Zbl 0599.49031 |
| [3] | Antontsev, S. N.; Shmarev, S. I., A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60, 515-545 (2005) · Zbl 1066.35045 |
| [4] | Antontsev, S. N.; Rodrigues, J. F., On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52, 19-36 (2006) · Zbl 1117.76004 |
| [5] | Chen, Y.; Levine, S.; Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 4, 1383-1406 (2006) · Zbl 1102.49010 |
| [6] | Kováčik, O.; Rákosník, J., On spaces \(L^{p(x)}(\Omega)\) and \(W^{k, p(x)}(\Omega)\), Czechoslovak Math. J., 41, 116, 592-618 (1991) · Zbl 0784.46029 |
| [7] | P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, In: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Work-shop, Jyväskylä, 2003, pp. 85-93.; P. Harjulehto, P. Hästö, An overview of variable exponent Lebesgue and Sobolev spaces, In: D. Herron (Ed.), Future Trends in Geometric Function Theory, RNC Work-shop, Jyväskylä, 2003, pp. 85-93. · Zbl 1046.46028 |
| [8] | Samko, S., On a progress in the theory of Lebesgue spaces with variable exponent maximal and singular operators, Integral Transforms Spec. Funct., 16, 461-482 (2005) · Zbl 1069.47056 |
| [9] | Zhikov, V. V.; Jikov, V. V.; Kozlov, S. M.; Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag: Springer-Verlag Berlin, Translated from the Russian by G.A. Yosifian · Zbl 0838.35001 |
| [10] | Zhikov, V. V., On some variational problems, Russ. J. Math. Phys., 5, 105-116 (1997) · Zbl 0917.49006 |
| [11] | Fan, X. L., On the sub-supersolution methods for \(p(x)\)-Laplacian equations, J. Math. Anal. Appl., 330, 665-682 (2007) · Zbl 1206.35103 |
| [12] | Fan, X. L.; Han, X. Y., Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(R^N\), Nonlinear Anal., 59, 173-188 (2004) · Zbl 1134.35333 |
| [13] | Fan, X. L.; Shen, J. S.; Zhao, D., Sobolev embedding theorems for spaces \(W^{k, p(x)}(\Omega)\), J. Math. Anal. Appl., 262, 749-760 (2001) · Zbl 0995.46023 |
| [14] | Fan, X. L.; Zhang, Q. H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems, Nonlinear Anal., 52, 1843-1852 (2003) · Zbl 1146.35353 |
| [15] | Fan, X. L.; Zhang, Q. H.; Zhao, D., Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302, 306-317 (2005) · Zbl 1072.35138 |
| [16] | Fan, X. L.; Zhao, D., On the Spaces \(L^{p(x)}\) and \(W^{m, p(x)}\), J. Math. Anal. Appl., 263, 424-446 (2001) · Zbl 1028.46041 |
| [17] | Fan, X. L.; Zhao, Y. Z.; Zhang, Q. H., A strong maximum principle for \(p(x)\)-Laplace equations, Chinese J. Contemp. Math., 24, 3, 277-282 (2003) · Zbl 1112.35079 |
| [18] | Fan, X. L., Solutions for \(p(x)\)-Laplacian Drichlet problems with singular coefficients, J. Math. Anal. Appl., 312, 464-477 (2005) · Zbl 1154.35336 |
| [19] | Yao, J., Solutions for Neumann boundary value problems involving \(p(x)\)-Laplace operators, Nonlinear Anal., 68, 1271-1283 (2008) · Zbl 1158.35046 |
| [20] | Kirchhoff, G., Mechanik (1883), Teubner: Teubner Leipzig · JFM 08.0542.01 |
| [21] | Lions, J. L., On some equations in boundary value problems of mathematical physics, (Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977). Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud., vol. 30 (1978), North-Holland: North-Holland Amsterdam), 284-346 · Zbl 0404.35002 |
| [22] | Arosio, A.; Pannizi, S., On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348, 305-330 (1996) · Zbl 0858.35083 |
| [23] | Cavalcante, M. M.; Cavalcante, V. N.; Soriano, J. A., Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6, 701-730 (2001) · Zbl 1007.35049 |
| [24] | D’Ancona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108, 247-262 (1992) · Zbl 0785.35067 |
| [25] | He, X.; Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70, 1407-1414 (2009) · Zbl 1157.35382 |
| [26] | Chipot, M.; Rodrigues, J. F., On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26, 447-467 (1992) · Zbl 0765.35021 |
| [27] | Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30, 4619-4627 (1997) · Zbl 0894.35119 |
| [28] | Alves, C. O.; Corrêa, F. J.S. A., On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8, 43-56 (2001) · Zbl 1011.35058 |
| [29] | Corrêa, F. J.S. A.; Menezes, S. D.B.; Ferreira, J., On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147, 475-489 (2004) · Zbl 1086.35038 |
| [30] | Corrêa, F. J.S. A.; Figueiredo, G. M., On a elliptic equation of \(p\)-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 74, 263-277 (2006) · Zbl 1108.45005 |
| [31] | Dreher, M., The Kirchhoff equation for the \(p\)-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64, 217-238 (2006) · Zbl 1178.35006 |
| [32] | Dreher, M., The ware equation for the \(p\)-Laplacian, Hokkaido Math. J., 36, 21-52 (2007) · Zbl 1146.35060 |
| [33] | Dai, G.; Liu, D., Infinitely many positive solutions for a \(p(x)\)-Kirchhoff-type equation, J. Math. Anal. Appl., 359, 704-710 (2009) · Zbl 1173.35463 |
| [34] | Dai, G.; Wei, J., Infinitely many non-negative solutions for a \(p(x)\)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Anal., 73, 3420-3430 (2010) · Zbl 1201.35181 |
| [35] | Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math., 113, 401-410 (2000) · Zbl 0946.49001 |
| [36] | Dai, G.; Hao, R., Existence of solutions for a \(p(x)\)-Kirchhoff-type equation, J. Math. Anal. Appl., 359, 275-284 (2009) · Zbl 1172.35401 |
| [37] | Fan, X. L., On nonlocal \(p(x)\)-Laplacian Dirichlet problems, Nonlinear Anal., 72, 3314-3323 (2010) · Zbl 1189.35127 |
| [38] | Autuori, G.; Pucci, P.; Salvatori, M. C., Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl., 352, 149-165 (2009) · Zbl 1175.35013 |
| [39] | Corrêa, F. J.S. A.; Menezes, S. D.B., Existence of solutions to nonlocal and singular elliptic problems via Galerkin method, Electron. J. Differential Equations, 2004, 1-10 (2004) · Zbl 1217.35063 |
| [40] | Ma, T. F., Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63, e1967-e1977 (2005) · Zbl 1224.35140 |
| [41] | Corrêa, F. J.S. A.; de Morais Filho, D. C., On a class of nonlocal elliptic problems via Galerkin method, J. Math. Anal. Appl., 310, 177-187 (2005) · Zbl 1136.35378 |
| [42] | Carrier, G. F., On the non-linear vibration problem of the elastic string, Quart. Appl. Math., 3, 157-165 (1945) · Zbl 0063.00715 |
| [43] | Cousin, A. T.; Frota, C. L.; Lar’kin, N. A.; Medeiros, L. A., On the abstract model of the Kirchhoff-Carrier equation, Commun. Appl. Anal., 1, 389-404 (1997) · Zbl 0894.35069 |
| [44] | E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Berlin, Heidelberg, New York, 1985.; E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Berlin, Heidelberg, New York, 1985. · Zbl 0583.47051 |
| [45] | Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001 |
| [46] | Fan, X. L.; Zhao, D., A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36, 295-318 (1996) · Zbl 0927.46022 |
| [47] | Fan, X. L., Global \(C^{1, \alpha}\) regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235, 397-417 (2007) · Zbl 1143.35040 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
