The purpose of this paper is to study the singular problem involving the p(x)-Laplace operator: (Section.Display) where be a bounded domain with boundary, is a positive parameter and p(x), and f(x, u) are assumed to satisfy the assumptions (H0)–(H4) in the Introduction. We employ variational techniques in order to show the existence of a number such that problem has two solutions for one solution for and no solutions for To obtain multiple (at least two distinct, positive) solutions of problem , we need to prove two new results: a regularity result for solutions to problem in with some and a strong comparison principle.

The purpose of this work is to study the following singular problem: where , be a bounded smooth domain, is a positive parameter, such that and where and r is the homogeneity degree of the function f. Under appropriate assumptions on the function f, we employ the method of Nehari manifold combined with the fibering maps in order to show the existence of such that for all , problem has at least two positives solutions.

In this paper, we prove the existence of multiple weak solutions for a nonlinear nonlocal elliptic partial differential equation involving a singularity and a power nonlinearity, which is given as ${\left(-{\mathrm{\Delta }}_p\right)}^{s}u=\frac{\lambda }{u^{\gamma }}+u^q;u>0\text{\hspace{0.17em}in\hspace{0.17em}}\mathrm{\Omega }$ with zero Dirichlet boundary condition. Here, Ω is an open bounded domain in ${\mathbb{R}}^N$ with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, and $p-1<q\le p_s^{*}=\frac{Np}{N-ps}$ . We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some α ∈ (0, s], the weak solution to the problem is in $C^{1,\alpha }\left(\overline{\mathrm{\Omega }}\right)$ .

In this paper, we study the following singular elliptic problem: where be a bounded smooth domain, denotes the -Laplace operator, is a parameter and is a constant. We require to satisfy assumptions (g1)–(g2) and to satisfy assumptions (h1)–(h4) in Section 1. We employ variational methods in order to show the existence of such that admits at least two solutions for all one solution for and no solution for all

We study the following singular problem involving the p(x)-Laplace operator ${\mathrm{\Delta }}_{p(x)}u=div(|\mathrm{\nabla }u{|}^{p(x)-2}\mathrm{\nabla }u)$, where $p(x)$ is a nonconstant continuous function, $({\mathrm{P}}_{\lambda })\{\begin{array}{ll}{\displaystyle -{\mathrm{\Delta }}_{p(x)}u=a(x)|u{|}^{q(x)-2}u(x)+\frac{\lambda b(x)}{u^{\delta (x)}}}& \text{in}\mathrm{\Omega },\\ {\displaystyle u>0}& \text{in}\mathrm{\Omega },\\ {\displaystyle u=0}& \text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$ Here, Ω is a bounded domain in ${\mathbb{R}}^{N\ge 2}$ with $C^2$-boundary, λ is a positive parameter, $a(x),b(x)\in C(\overline{\mathrm{\Omega }})$ are positive weight functions with compact support in Ω, and $\delta (x),$ $p(x),$ $q(x)\in C(\overline{\mathrm{\Omega }})$ satisfy certain hypotheses ( $A_0$) and ( $A_1$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $({\mathrm{P}}_{\lambda })$.