Existence and multiplicity results for double phase problem

By Wulong Liu, Guowei Dai. Journal of Differential Equations, 2018

283 citations recorded by the source corpus.

Abstract By variational method, we obtain various existence and multiplicity results for the following double phase problem { − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω . In particular, we find a sign-changing ground state solution. Some properties of double phase operator are also obtained.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/a3b28c5afe5b1ec9ff76ffb07faca87a95887547

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In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Existence of solutions for singular double phase problems via the Nehari manifold method (2021, background): “Finally, existence results for double phase problems with homogeneous Dirichlet or nonlinear Neumann boundary conditions without singular term can be found in the papers of Colasuonno–Squassina [13], El Manouni–Marino–Winkert [22], Gasiński–Papageorgiou [26], Gasiński–Winkert [28–30], Liu–Dai [34–”
An existence result for singular Finsler double phase problems (2020, background): “Recent existence results for double phase problems in the case if F is the Euclidean norm and without singular term can be found, for example, in the papers of Colasuonno-Squassina [11], Liu-Dai [30], Gasiński-Papageorgiou [24], Gasiński-Winkert [25–27], Perera-Squassina [41] and the references th”
Existence Results for Double Phase Problem in Sobolev–Orlicz Spaces with Variable Exponents in Complete Manifold (2021, background): “In the case of single valued equations, Liu-Dai in [17] discussed double phase problems and proved the existence and the multiplicity of the results, with the sign-changing solutions by variational method.”
New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems (2022, methodology): “We refer to the papers of BahrouniRădulescu-Winkert [3] (Baouendi-Grushin operator), Colasuonno-Squassina [15] (double phase eigenvalue problems), Farkas-Winkert [31] (Finsler double phase problems), Gasiński-Papageorgiou [32] (locally Lipschitz right-hand sides), Gasiński-Winkert [34, 35] (convecti”
Existence of solutions for double phase obstacle problems with multivalued convection term (2020, background): “The properties of the operator A : W 1,H0 (Ω) → W 1,H 0 (Ω)∗ are summarized in the following proposition, see Liu-Dai [20].”
Existence and multiplicity of solutions for double‐phase Robin problems (2020, background): “Existence and multiplicity results for two phase problems were proved recently by Cencelj, Rădulescu and Repovš [4] (problems with variable growth), Colasuonno and Squassina [5] (eigenvalue problems), Liu and Dai [13] (existence of solutions for problems with superlinear reaction), Papageorgiou, Răd”
Constant sign solutions for double phase problems with superlinear nonlinearity (2019, methodology): “Moreover, the application of the Nehari manifold method does not seem to work for our problem because of the sign of the nonlinearity f , compare with the assumption in GasińskiPapageorgiou [11] and Liu-Dai [15].”
Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold (2020, methodology): “The proof of this proposition follows the idea of the proof of Theorem 1.4 in Liu-Dai [26] and exploits the quantitative deformation lemma of Willem, see Jabri [23, Theorem 4.2].”
Existence and uniqueness results for double phase problems with convection term (2019, methodology): “By applying variational methods, Liu-Dai [19] treated double phase problems and proved existence and multiplicity results.”
On double phase Kirchhoff problems with singular nonlinearity (2021, background): “[13] (singular Finsler double phase problems), Fiscella [15] (Hardy potentials), Gasiński and Papageorgiou [19] (locally Lipschitz right-hand side), Gasiński and Winkert [20–22] (convection and superlinear problems), Liu and Dai [28] (Nehari manifold approach), Liu et al.”