An existence theorem for a perturbed singular elliptic problem

The existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents

We prove the existence of exact solutions and convergence of numerical (Galerkin) approximations for the semilinear degenerate parabolic problem ut+Lu = F(t,u) and for the degenerate elliptic problem Lu = f where L is a highly singular linear elliptic operator. The class of operators L contains in particular the operators of generalized bi-axially symmetric potential theory.

In this paper, we prove the existence of solutions to nonlinear elliptic equations, which present first-order terms with natural growth with respect to the gradient and lower order terms singular in the variable that represents the solution. The problems are considered on a domain Ω, which may have infinite Lebesgue measure.

We study the existence of multiple positive solutions of $ -\Delta u = \lambda u^{-q} +u^p $ in $\Omega$ with homogeneous Dirichlet boundary condition, where $\Omega$ is a bounded domain in $\mathbb R^N$, $\lambda >0$, and $0 < q \leq 1 < p \leq (N+2)/(N-2)$. We show by a variational method that if $\lambda$ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.

In this paper, we are going to show existence of branches of bifurcation for positive $W^{1,p}_{loc}(\Omega)$-solutions for the very singular non-local $\lambda$-problem $$ -{\Big(\int_\Omega g(x,u)dx\Big)^r}\Delta_pu={\lambda } \Big(a(x)u^{-\delta} + b(x)u^{\beta}\Big) \ \ \mbox{in} \ \ \Omega, \ \ \ \ u > 0 \ \ \ \mbox{in} \ \Omega \ \ \ \mbox{and} \ \ u=0 \ \ \mbox{on} \ \partial \Omega, $$ where $\Omega \subset \mathbb{R}^N $ is a smooth bounded domain, $\delta >0$, $0 < \beta < p-1$, $a $ and $b$ are non-negative measurable functions and $g$ is a positive continuous function. Our approach is based on sub-supersolutions techniques, fixed point theory, in the study of $ W^{1,p}_{loc}(\Omega)$-topology of a solution application and a new comparison principle for sub-supersolutions in $W^{1,p}_{loc}(\Omega)$ to a problem with $p$-Laplacian operator perturbed by a very singular term at zero and sublinear at infinity.

In this work we prove a uniqueness result for a quasilinear singular elliptic problem posed in a two-component domain. We prescribe a Dirichlet condition on the exterior boundary, while we prescribe a continuous flux and a jump of the solution proportional to the flux on the interface.

Existence of a positive solution for a singular elliptic problem with general exponential growth

Existence results for time scale boundary value problem

This paper investigates the existence of positive solutions of nonresonant singular boundary value problem of second order differential equations. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C1[0, 1] positive solutions is given by means of the method of lower and upper solutions with the fixed point theorems.

Let T be a time scale. Let T be a time scale. We study the existence of positive solutions for the nonlinear three-point singular boundary value problem with higher-order p-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.

In this paper, the existence of positive solutions for a singular second-order three-point boundary value problem is investigated. By using Krasnoselskii’s fixed point theorem, several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exist positive solutions are obtained. Finally, two examples are given to illustrate the importance of results obtained.

Existence of positive solutions for singular boundary value problems involving the one-dimensional p-Laplacian

Positive solutions of nonlinear singular third-order two-point boundary value problem

Positive solutions for a singular second-order three-point boundary value problem

Presently a direct analytical method is available for the digit-by-digit extraction of the square root of a given positive real number. To calculate the nthroot of a given positive real number one may use trial and error method, iterative method, etc. When one desires to determine the nth root, it is found thatsuch methods are inherent with certain weaknesses like the requirement of an initial guess, a large number of arithmetic operations and several iterativesteps for convergence, etc. There has been no direct method for the determination of the nth root of a given positive real number. This paper focusesattention on developing a numerical algorithm to determine the digit-by-digit extraction of the nth root of a given positive real number up to any desiredaccuracy. Examples are provided to illustrate the algorithm.