Existence results for a class of nonlocal problems involving p(x)‐Laplacian

This paper is concerned with the existence of solutions to a class of p(x)‐Kirchhoff‐type equations with Dirichlet boundary data as follows: By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish some conditions on the existence of solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and multiplicity results for infinitely many solutions for Kirchhoff‐type problems in

In this paper, we deal with existence and multiplicity of solutions of Kirchhoff‐type problems Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the preceding problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical point theory. Copyright © 2013 John Wiley & Sons, Ltd.     

On a bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus

We study existence and multiplicity of solutions to the following bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus, on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of I R^N, 1 < p(x) < N, M and f are continuous functions, f is an odd function, and r > 0 is a real parameter. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and multiplicity of solutions for Kirchhoff type equations

In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.     

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign‐changing potential by the Nehari method.     

Existence of three solutions for a Neumann problem involving the p(x)‐Laplace operator

In this paper, we study the existence of three solutions to a Neumann problem with nonstandard growth conditions. The technical approach is mainly based on three critical points theorem due to Ricceri. Copyright © 2011 John Wiley & Sons, Ltd.     

Infinitely many solutions for nonlocal elliptic systems of (p1(x),⋯,pn(x))‐Kirchhoff type

In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (p₁(x),?,p_n(x))‐Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of positive solutions for nonlinear Kirchhoff type problems in with critical Sobolev exponent

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.     

含临界指数的p(x)-Laplace方程解的存在性

研究了一含临界指数的p(x)-Laplace方程的Dirichlet边值问题,运用推广的集中紧致性原理,并结合山路引理得到了该问题非平凡弱解的存在性结果.     

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

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.     

带有类p-Laplacian算子的Kirchhoff型方程的多重解

研究一类带有类p-Laplacian算子的Kirchhoff型方程.利用对称山路定理,得到了多重解存在的充分条件,推广和改进了已有结果.     

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.     

Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities

In this paper we study the nonlinear Kirchhoff equations on the whole space. We show the existence, non-existence, and multiplicity of solutions to this problem with asymptotically linear nonlinearities. This result can be regarded as an extension of the result in Li et al. (2012).