Abstract
In this paper, we establish the existence of at least three weak solutions for a parametric double eigenvalue quasi-linear ellipticpx,qx-Kirchhoff-type potential system. Our approach is based on a variational method, and a three critical point theorem is obtained by Bonano and Marano.
Highlights
System (1) is a generalization of the elliptic equation associated with the following Kirchhoff equation, introduced by Kirchhoff in [1]: z2u ρ zt2 −
|zu/zx|2dx |zu/zx|2dx, which depends on and the equation the is no (ρ0/h)+ average longer a pointwise equation. e parameters in equation (3) have the following meanings: E is Young’s modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross section, and ρ0 is the initial tension. e p(x)-Laplacian operator possesses more complicated nonlinearities than p-Laplacian operator mainly due to the fact that it is not homogeneous. e study of various mathematical problems involving variable exponents has received a strong rise of interest in recent years
E existence and multiplicity of solutions for the elliptic systems involving the p(x)-Kirchhoff model have been studied by many authors, where the nonlinear source F has different mixed growth conditions
Summary
System (1) is a generalization of the elliptic equation associated with the following Kirchhoff equation, introduced by Kirchhoff in [1]: z2u ρ zt2 −. By using the mountain pass theorem, the authors in [24] showed the existence of nontrivial solutions for system (1) when (p, q) ∈ [C(RN)]2(N ≥ 2), M1(t) and M2(t) are continuous functions such that M1(t) M2(t), a(x) b(x) 0, λ 1, and F ∈ C1(RN × R2, R) verifies some mixed growth conditions. E goal of this work is to establish the existence of a definite interval in which λ lies such that system (1) admits at least three weak solutions by applying the following very recent abstract critical point result of Bonanno and Marano For any p ∈ C+(RN), we define the variable exponent Lebesgue space as
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