| Popis: |
We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger [math] -Laplace equation ¶ ( − Δ ) Φ u + V ( x ) φ ( u ) = ξ ( x ) f ( u ) in ℝ d , ¶ where [math] is an [math] -function, [math] is the [math] -Laplacian operator, [math] is a continuous function, [math] is a function with sign-changing on [math] and the nonlinearity [math] is sublinear as [math] . During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces. ¶ We also study the existence and multiplicity of solutions to the general fractional [math] -Laplacian equations of Kirchhoff type ¶ M ( ∫ ℝ 2 d Φ ( u ( x ) − u ( y ) K ( | x − y | ) ) d x d y N ( | x − y | ) ) ( − Δ ) Φ K , N u = f ( x , u ) in Ω , u = 0 in ℝ d ∖ Ω , ¶ where [math] is an open bounded subset of [math] with smooth boundary [math] , [math] , and [math] is a continuous function and [math] is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory. |
