Abstract
This paper is concerned with the existence of infinitely many positive solutions to a class of double phase problem. By variational methods and the theory of the Musielak–Orlicz–Sobolev space, we establish the existence of infinitely many positive solutions whose W_{0}^{1,H}(varOmega )-norms and L^{infty }-norms tend to zero under suitable hypotheses about nonlinearity.
Highlights
1 Introduction and main results The study of differential equations and variational problems with double phase operator is a new and interesting topic. Such interest is widely justified by many physical examples, such as elasticity, strongly anisotropic materials and Lavrentiev’s phenomenon
Their research is related to the following energy functional: u → |∇u|p + a(x)|∇u|q dx, (1.1)
In the past the problem of existence and multiplicity of nontrivial solutions for double phase problems driven by the double phase operator was studied in the context of Dirichlet boundary value problems
Summary
We know that J ∈ C1(E, R) and double phase operator – div(|∇u|p–2∇u + a(x)|∇u|q–2∇u) is the derivative operator of J in the weak sense. We observe that problem (P) has a variational structure, and as a matter of fact, its solutions can be searched as critical points of the energy functional φ : E → R defined as follows:. In [14], it is shown that Φ(u) is a Gâteaux differentiable functional in E whose derivative is given by. Φ(u) is weakly lower semi-continuous and coercive. Standard arguments show that Ψ is a well defined and continuously Gâteaux differentiable functional whose Gâteaux derivative. |∇u|p–2∇u · ∇v + a(x)|∇u|q–2∇u · ∇v dx = f (x, u)v dx, we will prove Theorem 1.1 by virtue of some idea due to Kristaly, Morosanu and Tersian [24], where the infinitely many homoclinic solutions for a p-Laplace equation was obtained. By (3.3), it is easy to see that φ is well defined, weakly sequentially lower semi-continuous and Gâteaux differentiable in E.
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