Qing-Mei Zhou

Bio: Qing-Mei Zhou is an academic researcher from Northeast Forestry University. The author has contributed to research in topics: Laplace operator & Mountain pass theorem. The author has an hindex of 7, co-authored 10 publications receiving 103 citations.

On the superlinear problems involving p(x)-Laplacian-like operators without AR-condition

Abstract: In the present paper, in view of the variational approach, we discuss the nonlinear eigenvalue problems for p ( x ) -Laplacian-like operators, originated from a capillary phenomenon. Under some suitable conditions, we prove the existence of nontrivial solutions of the system for every parameter λ > 0 .

Eigenvalues of the p ( x )-biharmonic operator with indefinite weight

Abstract: In this article, we consider the nonlinear eigenvalue problem: $$\left\{\begin{array}{ll}\Delta(|\Delta u|^{p(x)-2} \Delta u)=\lambda V(x)|u|^{q(x)-2}u,\quad{\rm in} \,\,\Omega\\ u=\Delta u=0, \qquad\qquad\qquad\quad\,\quad\,{\rm on}\,\,\partial \Omega,\end{array} \right.$$ where $${\Omega}$$ is a bounded domain of $${\mathbb{R}^N}$$ with smooth boundary, $${\lambda}$$ is a positive real number, $${p,\, q: \overline{\Omega} \rightarrow (1,+\infty)}$$ are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekeland’s variational principle.

Multiple solutions for a Robin-type differential inclusion problem involving the p (x )-Laplacian

Abstract: In this paper, we obtain the existence of at least two nontrivial solutions for a Robin-type differential inclusion problem involving p(x)-Laplacian type operator and nonsmooth potentials. Our approach is variational, and it is based on the nonsmooth critical point theory for locally Lipschitz functions. Copyright © 2013 John Wiley & Sons, Ltd.

Infinitely many solutions for a differential inclusion problem in {\mathbb{R}^N} involving p(x)-Laplacian and oscillatory terms

Abstract: In this paper, we consider the differential inclusion in $${\mathbb{R}^N}$$ involving the p(x)-Laplacian of the type $${\begin{array}{lll}-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u(x)),\;\;{\rm in}\;\;\mathbb{R}^N,\quad\quad\quad\quad\quad\quad ({\rm P})\end{array}}$$ where $${p: \mathbb{R}^N \to {\mathbb{R}}}$$ is Lipschitz continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many solutions of (P). We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces.

Existence of at least five solutions for a differential inclusion problem involving the p(x)-Laplacian ✩

Abstract: We studied a nonlinear Dirichlet problem driven by the p ( x ) -Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method combined with suitable truncation techniques based on nonsmooth critical point theory for locally Lipschitz function, we proved the existence of at least five solutions under the suitable conditions.

Extinction behavior of solutions for the p-Laplacian equations with nonlocal sources

Abstract: We investigate the extinction, non-extinction and decay estimates of non-negative nontrivial weak solutions of the initial-boundary value problem for the p -Laplacian equation with nonlocal nonlinear source and interior linear absorption. We show that the critical exponent of extinction for the weak solution is determined by the competition of two nonlinear terms, and decay estimates depend on the choices of initial data, coefficients and domain.

Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions

Abstract: In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving $$p(\cdot )$$ -Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely $$\begin{aligned} \left\{ \begin{array}{rcll} -{\text {div}}(a(| abla u|^{p(x)})| abla u|^{p(x)-2} abla u)&{}=&{}\lambda f(x,u) &{} \text{ in } \Omega ,\\ u&{}=&{}0 &{} \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$ By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter $$\lambda >0$$ small enough, and also that the solution blows up, in the Sobolev norm, as $$\lambda \rightarrow 0^{+}.$$ Finally, by imposing additional hypotheses on the nonlinearity $$f(x,\cdot ),$$ we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii.

On a $$\varvec{p(x)}$$ p ( x ) -biharmonic problem with singular weights

Abstract: In this work, sufficient conditions are given to prove the existence of at least one nontrivial weak solution for a p(x)-biharmonic problem involving Navier boundary conditions and singular weights.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR p(x)-LAPLACIAN EQUATIONS IN R N

Abstract: This article concerns the existence and multiplicity of solutions to a class of p(x)-Laplacian equations. We introduce a revised Ambrosetti- Rabinowitz condition, and show that the problem has a nontrivial solution and innitely many solutions. p x (x) , p (x) = Np(x) N p(x) , and they prove the existence of at least two nontrivial solutions to

Existence of one weak solution for p( x)-biharmonic equations with Navier boundary conditions

Abstract: We study the existence of at least one weak solution for a class of elliptic Navier boundary value problems involving the p(x)-biharmonic operator. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.