Tornado Solutions for Semilinear Elliptic Equations in R: Regularity

A two-phase free boundary problem for a semilinear elliptic equation

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 ...

Existence and multiplicity of positive solutions for a class of semilinear elliptic system with nonlinear boundary conditions

This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.

Optimal Control of Semilinear Elliptic Equations in Measure Spaces

Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. ...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

Partial Regularity for Weak Solutions of Semilinear Elliptic Equations with Supercritical Exponents

Let Ω be an open subset in R (n ≥ 3). In this paper, we study the partial regularity for stationary positive weak solutions of the equation (1.1) ∆u + h1(x)u + h2(x)u = 0 in Ω. We prove that if α > n+2 n−2 , and u ∈ H(Ω) ∩ L(Ω) is a stationary positive weak solution of (1.1), then the Hausdorff dimension of the singular set of u is less than n−2α+1 α−1 , which generalizes the main results in Pa...