Abstract
In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.
Highlights
In this paper, we will discuss the nonlocal elliptic problem involving (p(x), q(x))-biharmonic operator: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ Δ2p(x)u(x) − M1 |∇u(x)|p(x) Ω p(x) dxΔp(x)u(x) + ρ1(x)|u|p(x)−2u(x) λFu(x, u, v), in Ω, ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩Δ2q(x)v(x) u Δu
It well known that the variable exponent case possess more complicated properties than the constant exponent case, and some methods used in the (p, q)-biharmonic case cannot be applied to the (p(x), q(x))-biharmonic case. erefore, Allaoui et al [9] have made a great contribution to such problems, and they continued to extend (p, q)-biharmonic operator in [8] to (p(x), q(x))-biharmonic case, on the basis of Ricceri’s variational principle [10] and the basic theory of Sobolev space, and the following system is solved:
There are few results on the existence and multiplicity of solutions for (p(x), q(x))-Kirchhoff type problems under Navier boundary condition. erefore, inspired by the above research, in the present paper, our target is to show the existence and multiplicity of solutions of problem (1), according to the variational principle proposed by Ricceri and the basic results of Lebesgue or Sobolev spaces with a variable exponent
Summary
In [3], when the nonlinear functions F and G satisfy certain conditions, Li and Tang studied the (p, q)-biharmonic problem, ⎧⎪⎪⎪⎨ Δ|Δu|p−2Δu λFu(x, u, v) + μGu(x, u, v), ⎪⎪⎪⎩ Δ|Δv|q−2Δv λFv(x, u, v) + μGv(x, u, v), u Δu v Δv 0, x ∈ Ω, x ∈ Ω, x ∈ zΩ, (2) Erefore, Allaoui et al [9] have made a great contribution to such problems, and they continued to extend (p, q)-biharmonic operator in [8] to (p(x), q(x))-biharmonic case, on the basis of Ricceri’s variational principle [10] and the basic theory of Sobolev space, and the following system is solved:
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