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Register a new userThe moving plane method for doubly singular elliptic equations involving a first order term
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In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov-Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
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In this paper we obtain symmetry and monotonicity results for positive solutions to some $p$-Laplacian cooperative systems in bounded domains involving first order terms and under zero Dirichlet boundary condition.
Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schrodinger term: $-Delta_p u + mathbb{V}|u|^{p-2}u$ with bound constraints $psi_1 le u le psi_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calderon-Zygmund theory for nonlinear Schrodinger type equations and variational inequalities for double obstacle problems.
In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) abla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse H{o}lders regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $alpha$ in the H{o}lder $C^alpha$-regularity theory on the textup{BMO}-semi norm of the skew-symmetric part of $mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $mathbf{A}$ is assumed to be zero.
The existence and multiplicity of solutions for a class of non-local elliptic boundary value problems with superlinear source functions are investigated in this paper. Using variational methods, we examine the changes arise in the solution behaviours as a result of the non-local effect. Comparisons are made of the results here with those of the elliptic boundary value problem in the absence of the non-local term under the same prescribed conditions to highlight this effect of non-locality on the solution behaviours. Our results here demonstrate that the complexity of the solution structures is significantly increased in the presence of the non-local effect with the possibility ranging from no permissible positive solution to three positive solutions and, contrary to those obtained in the absence of the non-local term, the solution profiles also vary depending on the superlinearity of the source functions.
Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short time.
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