Abstract
The aim of this paper is to prove the existence of at least one nontrivial weak solution for equations involving the \((p(\cdot ),q(\cdot ))\)-Laplace operator. The approach is variational and based on the critical point theory.
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References
Bonanno, G.: Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal. 1, 205–220 (2012)
Bonanno, G., Chinnì, A.: Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent. J. Math. Anal. Appl. 418, 812–827 (2014)
Bonanno, G., Marano, S.A.: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1–10 (2010)
Diening, L., Harjulehto, P., Hästö, P., Rŭzĭcka, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture notes in math, vol. 2017. Springer, Heidelberg (2011)
Gasiński, L., Papageorgiou, N.S.: Anisotropic nonlinear Neumann problems. Calc. Var. 42, 323–354 (2011)
Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)
Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)
Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and variational methods with applications to nonlinear boundary value problems. Springer, New York (2014)
Papageorgiou, N.S., Vetro, C.: Superlinear \((p(z), q(z))\)-equations. Complex Var. Elliptic Equ. 64, 8–25 (2019)
Tan, Z., Fang, F.: On superlinear \(p(x)\)-Laplacian problems without the Ambrosetti and Rabinowitz condition. Nonlinear Anal. 75, 3902–3915 (2012)
Vetro, C.: Weak solutions to Dirichlet boundary value problem driven by \(p(x)\)-Laplacian-like operator. Electron. J. Qual. Theory Differ. Equ. 2017(98), 1–10 (2017)
Zhou, Q.-M.: On the superlinear problems involving \(p(x)\)-Laplacian-like operators without AR-condition. Nonlinear Anal. Real World Appl. 21, 161–169 (2015)
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The authors wish to thank the knowledgeable referee for his/her important remarks.
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Vetro, C., Vetro, F. On Problems Driven by the \((p(\cdot ),q(\cdot ))\)-Laplace Operator. Mediterr. J. Math. 17, 24 (2020). https://doi.org/10.1007/s00009-019-1448-1
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DOI: https://doi.org/10.1007/s00009-019-1448-1