In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation: −∆u+ u = λf(x)|u|q−2u in Ω, ∂u ∂ν = g(x)|u|p−2u on ∂Ω, has at least two nontrivial nonnegative solutions for λ is sufficiently small.

In the present paper the Dirichlet problem for semilinear elliptic and parabolic equations in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u|p−1u , p > 1 , is formulated. For sublinear case (for example ln(1+ |u|) or |u|p−1u , p < 1) this condition is automatically fulfilled. Our approac...

We consider the obstacle problem with irregular barriers for semilinear elliptic equations involving measure data and an operator corresponding to a general quasi-regular Dirichlet form. prove existence uniqueness of solution as well its repre

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super an...

An interesting field of modern mathematical research is the study of geometric properties of solutions to elliptic problems. Remarkably, this is often done without any explicit representation of the solution. This paper concentrates on the problem of convexity of level sets for solutions to some elliptic semilinear boundary-value problems in convex rings.More precisely, letΩ0,Ω1 be convex, boun...

We study a class of fractional semilinear elliptic equations and formulate the corresponding Calder\'on problem. determine nonlinearity from exterior partial measurements Dirichlet-to-Neumann map by using first order linearization Runge approximation property.

We prove the existence of at least two positive solutions for the semilinear elliptic boundary-value problem −∆u(x) = λa(x)u + b(x)u for x ∈ Ω; u(x) = 0 for x ∈ ∂Ω on a bounded region Ω by using the Nehari manifold and the fibering maps associated with the Euler functional for the problem. We show how knowledge of the fibering maps for the problem leads to very easy existence proofs.

In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic boundary value problems. We provide a sufficient condition for a solution to an elliptic problem to be positive in the domain of the problem, which can be checked numerically without requiring a complicated computation. Although we focus on the homogeneous Dirichlet case in this paper...