# zbMATH — the first resource for mathematics

Existence results for degenerate $$p(x)$$-Laplace equations with Leray-Lions type operators. (English) Zbl 1373.35123
The paper deals with the Dirichlet problem $\begin{cases} -\text{div\,}a(x,\nabla u)=\lambda f(x,u) & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases}$ where $$\Omega\subset \mathbb{R}^N$$ is a bounded domain with Lipschitz smooth boundary, $$a$$ and $$f$$ are Carathéodory functions and $$\lambda$$ is a positive parameter. The differential operator is modeled on the $$p(x)$$-Laplacian $$\text{div\,}\big(\omega(x)|\nabla u|^{p(x)-2}\nabla u\big)$$ with $$p\in C(\overline{\Omega},(1,\infty))$$ and where $$\omega$$ is a suitable measurable and positive function in $$\Omega.$$
By means of standard techniques based on direct methods and critical point theories in the Calculus of Variations, the authors prove existence and multiplicity of solutions to the above problem when the differential operator is monotone and the nonlinearity is nonincreasing.

##### MSC:
 35J60 Nonlinear elliptic equations
##### Keywords:
$$p(x)$$-Laplacian; Dirichlet problem
Full Text:
##### References:
 [1] Ambrosetti, A; Rabinowitz, P H, Dual variational methods in critical point theory and applications, J Funct Anal, 14, 349-381, (1973) · Zbl 0273.49063 [2] Bonanno, G; Molica Bisci, G, Three weak solutions for elliptic Dirichlet problems, J Math Anal Appl, 382, 1-8, (2011) · Zbl 1225.35067 [3] Boureanu, M M; Udrea, D N, Existence and multiplicity results for elliptic problems with p(.)-growth conditions, Nonlinear Anal Real World Appl, 14, 1829-1844, (2013) · Zbl 1271.35045 [4] Chen, Y; Levine, S; Rao, M, Variable exponent, linear growth functionals in image restoration, SIAM J Appl Math, 66, 1383-1406, (2006) · Zbl 1102.49010 [5] Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Heidelberg: Springer-Verlag, 2011 · Zbl 1222.46002 [6] Fan, X, Global C1,a regularity for variable exponent elliptic equations in divergence form, J Differential Equations, 235, 397-417, (2007) · Zbl 1143.35040 [7] Fan, X, Existence and uniqueness for the p(x)-Laplacian-Dirichlet problems, Math Nachr, 284, 1435-1445, (2011) · Zbl 1234.35111 [8] Fan, X; Han, X, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN, Nonlinear Anal, 59, 173-188, (2004) · Zbl 1134.35333 [9] Ho, K; Sim, I, Existence and some properties of solutions for degenerate elliptic equations with exponent variable, Nonlinear Anal, 98, 146-164, (2014) · Zbl 1286.35117 [10] Ho, K; Sim, I, Existence and multiplicity of solutions for degenerate $$p$$($$x$$)-Laplace equations involving concave-convex type nonlinearities with two parameters, Taiwanese J Math, 19, 1469-1493, (2015) · Zbl 1360.35076 [11] Kim, I H; Kim, Y H, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math, 147, 169-191, (2015) · Zbl 1322.35009 [12] Kim, Y H; Wang, L; Zhang, C, Global bifurcation of a class of degenerate elliptic equations with variable exponents, J Math Anal Appl, 371, 624-637, (2010) · Zbl 1198.35089 [13] Kovăčik, O; Răkosnik, J, On spaces lp(x) and wk,p(x), Czechoslovak Math J, 41, 592-618, (1991) · Zbl 0784.46029 [14] Le, V K, On a sub-supersolutionmethod for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal, 71, 3305-3321, (2009) · Zbl 1179.35148 [15] Liu D, Wang X, Yao J. On ($$p$$_{1}($$x$$), $$p$$_{2}($$x$$))-Laplace equations. ArXiv:1205.1854v1, 2012 [16] Mihăilescu, M; Rădulescu, 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) · Zbl 1149.76692 [17] Ružička M. Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer-Verlag, 2000 · Zbl 0962.76001 [18] Sim, I; Kim, Y H, Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents, 695-707, (2013) · Zbl 1305.35035 [19] Tan, Z; Fang, F, On superlinear $$p$$($$x$$)-Laplacian problems without ambrosetti and Rabinowitz condition, Nonlinear Anal, 75, 3902-3915, (2012) · Zbl 1241.35047 [20] Willem M. Minimax Theorems. Boston: Birkhäuser, 1996 · Zbl 0856.49001 [21] Zeidler E. Nonlinear Functional Analysis and Its Applications II/B. New York: Springer, 1990 · Zbl 0684.47029 [22] Zhao J F. Structure Theory of Banach Spaces (in Chinese). Wuhan: Wuhan University Press, 1991 [23] Zhikov, V V, On some variational problems, Russ J Math Phys, 5, 105-116, (1997) · Zbl 0917.49006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.