Infinitely many solutions for elliptic problems in RN\mathbb{R}^N involving the p(x)p(x)-Laplacian

We consider the $p(x)$-Laplacian equations in $\mathbb{R}^N$. The potential function does not satisfy the coercive condition. We obtain the existence of infinitely many solutions of the equations, improving a recent result of Duan--Huang [L. Duan, L. H. Huang, Electron. J. Qual. Theory Differ. Equ. 2014, No. 28, 1--13]

Multiplicity of solutions for p-Laplacian equation in R^N with indefinite weight

In this article, we study the existence of infinitely many nontrivial solutions for a class of superlinear $p$-Laplacian equations $$-Delta_p u+V(x)|u|^{p-2}u=f(x,u),$$ where the primitive of the nonlinearity $f$ is of subcritical growth near $infty$ in $u$ and the weight function $V$ is allowed to be sign-changing. Our results extend the recent results of Zhang and Xu [Q. Y. Zhang, B. Xu, {em Multiplicity of solutions for a class of semilinear Schr"{o}dinger equations with sign-changing potential}, J. Math. Anal. Appl {bf 377}(2011), 834--840]

On a class of superlinear nonlocal fractional problems without Ambrosetti–Rabinowitz type conditions

In this note, we deal with the existence of infinitely many solutions for a problem driven by nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions −LKu = λ f(x, u), in Ω, u = 0, in Rn\Ω, where Ω is a smooth bounded domain of Rn and the nonlinear term f satisfies superlinear at infinity but does not satisfy the the Ambrosetti–Rabinowitz type condition. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions by using abstract critical point results for an energy functional satisfying the Cerami condition

Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by the p

A class of nonlinear Neumann problems driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions

DETERMINATION OF SERUM ALBUMIN WITH TRIBROMOARSENAZO BY SPECTROPHOTOMETRY

The reaction of tribromoarsenazo(TB-ASA) with serum albumin in the presence of emulgent OP was studied by spectrophotometry. In a Britton-Robinson buffer solution at pH 2.9, tribromoarsenazo and bovine serum albumin can immediately form a red compound in the presence of emulgent OP with a maximum absorption wavelength at 354 nm. The presence of emulgent OP can increase the reaction sensitivity and the compound stability. The molar absorptivity of the compound is ε354 nm = 6.13 x 105 M-1•cm-1. Beer's law is obeyed over the range of 5.0-75.0 mg•L-1 for bovine serum albumin. The present method was applied to the determination of the total proteins in human serums with satisfactory results. KEY WORDS: Serum albumin, Tribromoarsenazo, Emulgent OP, Human serums, Spectrophotometry Bull. Chem. Soc. Ethiop. 2007, 21(2), 291-296

Infinitely many weak solutions for p(x)-Laplacian-like problems with sign-changing potential

This study is concerned with the p(x)-Laplacian-like problems and arising from capillarity phenomena of the following type −div ��1 + |∇u| p(x) 1+|∇u| 2p(x) |∇u| p(x)−2∇u = λ f(x, u), in Ω, u = 0, on ∂Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω, p ∈ C(Ω), and the primitive of the nonlinearity f of super-p + growth near infinity in u and is also allowed to be sign-changing. Based on a direct sum decomposition of a space W 1,p(x) 0 (Ω), we establish the existence of infinitely many solutions via variational methods for the above equation. Furthermore, our assumptions are suitable and different from those studied previously

Existence of solutions for a class of quasilinear degenerate p(x)p(x)-Laplace equations

We study the existence of weak solutions for a degenerate p(x)-Laplace equation. The main tool used is the variational method, more precisely, the Mountain Pass Theorem

Infinitely many weak solutions for p(x)p(x)-Laplacian-like problems with sign-changing potential

This study is concerned with the $p(x)$-Laplacian-like problems and arising from capillarity phenomena of the following type $$\begin{cases} -{\rm{div}}\left(\left(1+\tfrac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\right)|\nabla u|^{p(x)-2}\nabla u\right)=\lambda f(x,u),\quad {\rm in}~\Omega, &\\ u=0,\quad {\rm{on}} ~\partial \Omega,& \end{cases}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$, $p\in C(\overline{\Omega})$, and the primitive of the nonlinearity $f$ of super-$p^+$ growth near infinity in $u$ and is also allowed to be sign-changing. Based on a direct sum decomposition of a space $W_0^{1,p(x)}(\Omega)$, we establish the existence of infinitely many solutions via variational methods for the above equation. Furthermore, our assumptions are suitable and different from those studied previously