Double phase anisotropic variational problems and combined effects of reaction and absorption terms

By Qihu Zhang, Vicentiu D. Rădulescu. Journal des Mathématiques Pures et Appliquées, 2018

152 citations recorded by the source corpus.

Abstract This paper deals with the existence of multiple solutions for the quasilinear equation − div A ( x , ∇ u ) + V ( x ) | u | α ( x ) − 2 u = f ( x , u ) in R N , which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like | ξ | q ( x ) − 2 ξ for small | ξ | and like | ξ | p ( x ) − 2 ξ for large | ξ | , where 1 α ( ⋅ ) ≤ p ( ⋅ ) q ( ⋅ ) N . Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Rădulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p ( ⋅ ) and q ( ⋅ ) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/678a824186a57398eb40e6e1bc7e226b73cbedf9

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Citations in the corpus

Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves (2019, methodology): “The present paper complements our previous contributions related to double phase anisotropic variational integrals, see [33, 35].”
A new class of double phase variable exponent problems: Existence and uniqueness (2021, background): “In 2018, Zhang-Rădulescu [69] studied the following variable exponent elliptic equation − div A(x,∇u) + V (x)|u|u = f(x, u), (1.”
Fractional double-phase patterns: concentration and multiplicity of solutions (2020, methodology): “For various types of double-phase problems, we refer to the recent papers [32,55,56,14,62,63,72].”
Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold (2020, unknown): “We refer to Bahrouni-Rădulescu-Winkert [3], Cencelj-RădulescuRepovš [12], Marino-Winkert [29], Papageorgiou-Rădulescu-Repovš [35], [36], [37], Rădulescu [42], Zhang-Rădulescu [48], Zeng-Gasiński-Winkert-Bai [46], [47] and the references therein.”
A variational approach to parabolic equations under general and p,q-growth conditions (2020, unknown): “See also Rǎdulescu–Zhang [50], Cencelja–Rădulescu– Repovš [21] and De Filippis [29].”
Maximal regularity for local minimizers of non-autonomous functionals (2019, background): “Rădulescu and colleagues [22, 76, 84] have considered a functional with model case φ(x, t) = t + t, which they call “double phase” (it is different from the double phase functional of Zhikov, considered below).”
Existence of solutions for double phase obstacle problems with multivalued convection term (2020, background): “Works which are closely related to ours dealing with certain types of double phase problems can be found in Bahrouni-Rădulescu-Repovš [1], [2], Cencelj-Rădulescu-Repovš [8], Papageorgiou-Rădulescu-Repovš [25], [24], Rădulescu [28] Zhang-Rădulescu [29] and the references therein.”
Double phase problems with variable growth (2018, result): “In such a way, the present paper complements our previous related contributions to this field, see [37,36].”