Existence of triple solutions for elliptic equations driven by p-Laplacian-like operators with Hardy potential under Dirichlet–Neumann boundary conditions

In this article, we focus on triple weak solutions for some p-Laplacian-type elliptic equations with Hardy potential, two parameters, and mixed boundary conditions. We show the existence of at least three distinct weak solutions by using variational methods, the Hardy inequality, and the Bonanno–Marano-type three critical points theorem under suitable assumptions, and the existence of solutions to some particular cases of this type of elliptic equations are also obtained.


Introduction
Elliptic differential equations in bounded domains with singular Hardy potential and Dirichlet-Neumann-type mixed boundary conditions are used to describe many engineering or physical phenomena and play a role in modeling in applied sciences such as the heat conduction in electrically conducting materials, singular minimal surfaces, and the non-Newtonian fluids, and the state of stress and strain on the elastic surface in mechanics and the solidification and melting of materials in industrial processes are only some examples involving mixed conditions. In particular, an intuitive example is that an iceberg is partially immersed in water, and mixed boundary conditions must be imposed on its boundary. In the underwater part, a Dirichlet boundary condition is required, while the Neumann condition is used in the remaining part of the boundary in contact with air.
Recently, researches on the numbers of the existence of weak solutions to nonlinear differential equations via variational methods have received wide attention (see, for example [1,2,6,7,[9][10][11][12]). In particular, in this very interesting paper [6], the author studied the existence of two nontrivial solutions for a class of mixed elliptic problems with Dirichlet-Neumann mixed boundary conditions and concave-convex nonlinearity has been obtained. In the detailed literature [7], the existence of at least one positive solu-tion of a class of perturbed equations with mixed boundary conditions was discussed. It is worth noting that in the papers cited, the boundary conditions are homogeneous. In this paper, we deal with the existence of at least three weak solutions to the following elliptic equations with homogeneous Neumann boundary conditions, and the results of some particular cases of this type elliptic problems are also obtained, is an open bounded subset in R N (N ≥ 3) with smooth boundary ∂ , ν is the outward normal vector field on ∂ , 1 and 2 are two smooth (N -1)-dimensional submanifolds of ∂ such that 1 where q ∈ (1, p), 0 < h(x) ∈ L β ( 2 ), β > N-1 p-1 and γ : W 1,p ( ) → L p (∂ ) is called a trace map satisfying γ (u) = u| ∂ , ∀u ∈ W 1,p ( ) ∩ C 1 ( ), that is, γ (u) is the trace of u or the generalized boundary values of u.

Preliminaries and variational structure
Throughout the paper we denote the L z -norm by u z . Let be an open, bounded subset in Obviously, W 1,p 0 ( ) is a reflexive Banach space, and the embedding W ,p 0 ( ), for some positive constantc, and A satisfy the following assumptions, (A3) There exist a 1 , a 2 > 0 such that for all x ∈ and ξ ∈ R N . From (A1) and (A3), one has a 1 |ξ | p ≤ pA(x, ξ ) ≤ a 2 |ξ | p , see [5,Remark 2.3] for details.
Define the functional I λ : W The following Bonanno-Marano-type three critical points theorem is from the results contained in [3], which is the main tool used to obtain our results. Theorem 2.1 ([3, Theorem 3.6]) Let X be a reflexive real Banach space; : X → R be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X * , : X → R be a sequentially weakly upper semicontinuous, continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that Assume that there exist r > 0 andx ∈ X, with r < (x) such that (ii) for each λ ∈ r = ( r r 2 , r r 1 ), the functionalλ is coercive. Then, for each λ ∈ r the functionalλ has at least three distinct critical points in X. Proof By Lemma 2.5 in [5], one has that : W 1,p 0 ( ) → R is convex, sequentially weakly lower semicontinuous, and of class C 1 in W Next, we prove that admits a continuous inverse in W -1,p ( ). For any u ∈ W 1,p A(x, v)), for somec > 0, and assumption (A3), one has thus we have that is uniformly monotone in W 1,p 0 ( ). Taking into account Theorem 26.(A)d of [13], we obtain the conclusion.
, the problem (1.1) has at least three weak solutions.

Funding
Projects ZR2021MA070 and ZR2020MA012 supported by Shandong Provincial Natural Science Foundation.