• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.533-546
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• 2018

The object of this work is to study the existence of solutions for a class of p(x)-Kirchhoff type problem under no-flux boundary conditions with dependence on the gradient. We establish our results by using the degree theory for operators of ($S_+$) type in the framework of variable exponent Sobolev spaces.

• Korean Journal of Mathematics
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• v.22 no.3
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• pp.395-406
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• 2014

We study the following nonlinear problem $-div(w(x){\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u)$ in ${\Omega}$ which is subject to Neumann boundary condition. Under suitable conditions on w and f, we give the existence of a positive infimum eigenvalue for the p(x)-Laplacian Neumann problem.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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• pp.845-860
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• 2010

In this paper, using the Nehari manifold approach and some variational techniques, we discuss the multiplicity of positive solutions for the p(x)-Laplacian problems with non-negative weight functions and prove that an elliptic equation has at least two positive solutions.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.137-152
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• 2014

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1805-1821
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• 2016

In this paper, we are concerned with the nonlinear elliptic equations of the p(x)-Laplace type $$\{\begin{array}{lll}-div(a(x,{\nabla}u))+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u) && in\;{\Omega}\\(a(x,{\nabla}u)\frac{{\partial}u}{{\partial}n}={\lambda}{\theta}g(x,u) && on\;{\partial}{\Omega},\end{array}$$ which is subject to nonlinear Neumann boundary condition. Here the function a(x, v) is of type${\mid}v{\mid}^{p(x)-2}v$ with continuous function $p:{\bar{\Omega}}{\rightarrow}(1,{\infty})$ and the functions f, g satisfy a $Carath{\acute{e}}odory$ condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

• Journal of applied mathematics & informatics
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• v.35 no.1_2
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• pp.11-24
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• 2017

Using variational methods, we study the existence and multiplicity of weak solutions for some p(x)-Laplacian-like problems. First, without assuming any asymptotic condition neither at zero nor at infinity, we prove the existence of a non-zero solution for our problem. Next, we obtain the existence of two solutions, assuming only the classical Ambrosetti-Rabinowitz condition. Finally, we present a three solutions existence result under appropriate condition on the potential F.