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Existence of multiple solutions of Schrödinger‐Kirchhoff‐type equations involving the p (.) ‐Laplacian in ℝ N
Author(s) -
Ayazoglu Mashiyev Rabil,
Akbulut Sezgin,
Akkoyunlu Ebubekir
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6626
Subject(s) - mathematics , lipschitz continuity , bounded function , type (biology) , p laplacian , mountain pass theorem , operator (biology) , laplace operator , nonlinear system , pure mathematics , mathematical analysis , ecology , biochemistry , chemistry , physics , repressor , quantum mechanics , gene , transcription factor , biology , boundary value problem
In this paper, we prove the existence of multiple solutions for the nonhomogeneous Schrödinger‐Kirchhoff‐type problem involving the p (.)‐Laplacian− 1 + b ∫ℝ N1 p ( x )∇ up ( x ) d xΔ p ( x ) u + V ( x )up ( x ) − 2 u = f ( x , u ) + g ( x ) inℝ N ,u ∈ W 1 , p ( . )ℝ N,where b ≥0 is a constant, N ≥2, Δ p (.) u := d i v (|∇ u | p (.)−2 ∇ u ) is the p (.)‐Laplacian operator, p : ℝ N → ℝ is Lipschitz continuous, V : ℝ N → ℝ is a coercive type potential, f : ℝ N × ℝ → ℝ and g : ℝ N → ℝ functions verifying suitable conditions. We propose different assumptions on the nonlinear term f : ℝ N × ℝ → ℝ to yield bounded Palais‐Smale sequences and then prove that the special sequences we found converge to critical points, respectively. The solutions are obtained by the Mountain Pass Theorem, Ekeland variational principle, and Krasnoselskii genus theory.