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Here, we prove the Haraux-Weissler equation has no solution on a domain which is strictly starshped with respect to 0. Finally, we present some questions. [ABSTRACT FROM AUTHOR]

A variational principle is applied to examine a Muckenhoupt weighted p(·)-Laplacian equation on the Heisenberg groups. The existence of at least one positive radial solution to the problem under the Dirichlet boundary condition belongs to the first order Heisenberg-Sobolev spaces is proved. [ABSTRACT FROM AUTHOR]

Here, a singular boundary value problem involving the p-biharmonic operator in a smooth bounded domain of RN is considered. Using the variational method and critical point theory, the existence of at least three weak solutions is proved. [ABSTRACT FROM AUTHOR]

This paper proposes an analytical procedure for the nonlinear singular boundary value problem that arises in biology and in the study of some diseases. As a first step, we present a constructive proof of the existence and uniqueness of solution. Then, we apply the Picard iterative sequence by constructing an integral equation whose Green's function is not negative. The convergence of this iterative sequence is then controlled by an embedded parameter so that it tends to the unique solution. [ABSTRACT FROM AUTHOR]

In this paper, a constructive proof under some conditions is presented for existence and uniqueness of the solutions to the singular problem with the boundary conditions In general,f(x,y(x)) is assumed to be nonsingular with respect to the independent variablexbut it is allowed be singular with respect toyand moreover,f(x,y(x)) can be sign changing as well. The Picard iterative sequence is then constructed based on integral equation with the help of positive Green’s function. The convergence of this iterative sequence is controlled by an adjustable parameter so that it converges to the unique solution in the finite region in whichis supposed to be nonnegative and bounded. Some examples also are given to demonstrate the trustworthiness of our constructive theory. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic expansion of the Dirichlet to Neumann operator (DNO) for the Dirichlet problem on perturbations of the unit disk. We write our result in terms of pseudodifferential operators which themselves have expansions in the perturbation parameter. For a given power of the perturbation parameter, m > 0, and a given order, n < 0, we give an algorithm which allows for the expansion of the symbol of the DNO up to mth power in the perturbation parameter, with error terms belonging to symbols of order n. [ABSTRACT FROM AUTHOR]

In this article we consider the existence of infinitely many solutions to the fourth-order boundary-value problem uiv + αu'' + β(x)u = λf(x, u) + h(u), x ∈]0, 1[ u(0) = u(1) = 0, u''(0) = u''(1) = 0. Our approach is based on variational methods and critical point theory. [ABSTRACT FROM AUTHOR]