Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents
Tóm tắt
In this paper, we first obtain the existence of solutions for a class of elliptic equations involving critical variable exponents and nonlinear boundary values by the mountain pass theorem and concentration compactness principle. Then, under suitable assumptions, we obtain a sequence of solutions with positive energies going towards infinity by Fountain Theorem.
Tài liệu tham khảo
Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)
Acerbi, E., Mingione, G.: Regularity results for stationary electrorheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)
Kováčik, O., Rákosnik, J.: On spaces \(L^{p(x)}\) and \(W^{k,p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)
Chabrowski, J., Fu, Y.Q.: Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 306, 604–618 (2005)
Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal., Theory Methods Appl. 52, 1843–1852 (2003)
Bonder, J.F.: Multiple solutions for the p-Laplacian equation with nonlinear boundary conditions. Electron. J. Differ. Equ. 2006, 37 (2006)
Pflüger, K.: Existence and multiplicity of solutions to a equation with nonlinear boundary condition. Electron. J. Differ. Equ. 1998, 10 (1998)
Bonder, J.F., Rossi, J.D.: Existence results for the p-Laplacian with nonlinear boundary conditions. J. Math. Anal. Appl. 263(1), 195–223 (2001)
Xiang, M.Q., Rǎdulescu, V.D., Zhang, B.L.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58, 57 (2019). https://doi.org/10.1007/s00526-019-1499-y
Xiang, M.Q., Rǎdulescu, V.D., Zhang, B.L.: Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM Control Optim. Calc. Var. 24, 1249–1273 (2018)
Xiang, M.Q., Rǎdulescu, V.D., Zhang, B.L.: A critical fractional Choquard–Kirchhoff problem with magnetic field. Commun. Contemp. Math. (2019). https://doi.org/10.1142/S0219199718500049
Xiang, M.Q., Rǎdulescu, V.D., Zhang, B.L.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31, 3228–3250 (2018)
Xiang, M.Q., Zhang, B.L., Qiu, H.: Existence of solutions for a critical fractional Kirchhoff type problem in \(R^{N}\). Sci. China Math. 60, 1647–1660 (2017)
Boureanu, M.M., Preda, F.: Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 7, 1–17 (2011)
Mihǎilescu, M., Rǎdulescu, V.: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. Lond. Ser. A 462, 2625–2641 (2006)
Fu, Y.Q.: The principle of concentration compactness in \(L^{p(x)}\) spaces and its application. Nonlinear Anal., Theory Methods Appl. 71, 1876–1892 (2009)
Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}\) and \(W^{m,p(x)}\). J. Math. Anal. Appl. 263, 424–446 (2001)
Fan, X.L.: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339, 1395–1412 (2008)
Le, V.K.: On a sub-supersolution method for variational inequalities with Leray–Lions operators in variable exponent spaces. Nonlinear Anal., Theory Methods Appl. 71, 3305–3321 (2009)
Fan, X.L., Han, X.Y.: Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Theory Methods Appl. 59, 173–188 (2004)