Abstract
In this paper, we give some properties about the $(2,p)$ -Laplacian operator ( $p>1$ , $p\ne2$ ), and consider the existence of solutions to two kinds of partial differential equations related to the $(2,p)$ -Laplacian operator by those properties. Specifically, we establish an existence result of positive solutions using fixed point index theory and an existence result of nodal solutions via the quantitative deformation lemma.
Highlights
1 Introduction and main results Recently, much attention has been paid to the existence of solutions to the following quasilinear elliptic problems of (q, p)-Laplacian type:
By a solution u of ( . ), we mean that u, belonging to some Sobolev space, solves ( . ) in the weak sense, i.e., u satisfies
By a non-negative nontrivial solution u of ( . ), we mean that u is a solution of ( . ), u = and u(x) ≥ for x ∈ ; if u is a solution of ( . ) with u± =, where u+ = max{u, } and u– = max{u, }, we say that u is a sign-changing solution of ( . )
Summary
As an application of the property (b), in Section , we will investigate the existence of least energy sign-changing solution of the following equation: In order to obtain sign-changing solutions, it is common to assume that the nonlinearity satisfies g(t)t ≥ , t ∈ R. If the completely continuous operator B : U → V has no fixed point on ∂U, there exists an integer i(B, U, V ), which is regarded as the fixed point index, and the following statements hold: (i) If B : U → U is a constant mapping, i(B, U, V ) = .
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