Singular Elliptic Equations Research Articles

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

Radially symmetric solutions of a nonlinear singular elliptic equation

Radially symmetric solutions of a nonlinear singular elliptic equation

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

Some Infinite Expansions of the Lauricella Functions and Their Application in the Study of Fundamental Solutions of a Singular Elliptic Equation

Some Infinite Expansions of the Lauricella Functions and Their Application in the Study of Fundamental Solutions of a Singular Elliptic Equation

Existence and uniqueness of solution for a singular elliptic differential equation

Abstract In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: − Δ u − 1 2 ( x ⋅ ∇ u ) = μ h ( x ) u q − 1 + λ u − u p , x ∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , \left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N ⩾ 3 N\geqslant 3 , 0 < q < 1 0\lt q\lt 1 , λ > 0 \lambda \gt 0 , p > 1 p\gt 1 , μ > 0 \mu \gt 0 is a parameter and the function h ( x ) h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ > N 2 \lambda \gt \frac{N}{2} . In addition, there is no solution as λ ⩽ N 2 \lambda \leqslant \frac{N}{2} . Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space

Comparison results for a nonlocal singular elliptic problem

We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are presented, depending on the summability of the right-hand side of the equation. The maximum principle arguments employed in the core of the proofs of the main results offer a nonstandard, flexible alternative to the ones described in (Arch. Ration. Mech. Anal. 239 (2021) 1733–1770, Theorem 31). Some interesting consequences are L p regularity results and nonlocal energy estimates for solutions.

Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity

Abstract In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p-Laplacian-type diffusion operator with 1 < p < ∞ {1<p<\infty} . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi’s truncation and freezing coefficient methods.

Modulus of continuity of weak solutions to a class of singular elliptic equations

Modulus of continuity of weak solutions to a class of singular elliptic equations

Doubly Singular Elliptic Equations Involving a Gradient Term: Symmetry and Monotonicity

Doubly Singular Elliptic Equations Involving a Gradient Term: Symmetry and Monotonicity

Gradient estimates of general nonlinear singular elliptic equations with measure data

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations −divA(x,u,Du)=μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1,2−1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.

Boundary regularity for viscosity solutions of nonlinear singular elliptic equations

Boundary regularity for viscosity solutions of nonlinear singular elliptic equations

Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations

<abstract><p>In this paper, we consider the existence of positive solutions to mixed local and nonlocal singular quasilinear singular elliptic equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \left\{\begin{array}{rl} -\Delta_{\vec{p}}u(x)+\left(-\Delta\right)_{p}^{s}u(x) = \frac{f(x)}{u(x)^{\delta}}, &x\in\Omega, \\ u(x)>0, \; \; \; \; \; \; &x\in\Omega, \\ u(x) = 0, \; \; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{array} \right. \end{align*} $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^{N}(N > 2) $, $ -\Delta_{\vec{p}}u $ is an anisotropic $ p $-Laplace operator, $ \vec{p} = (p_{1}, p_{2}, ..., p_{N}) $ with $ 2\leq p_{1}\leq p_{2}\leq\cdot\cdot\cdot\leq p_{N} $, $ \left(-\Delta \right)_{p}^{s} $ is the fractional $ p $-Laplace operator. The major results shows the interplay between the summability of the datum $ f(x) $ and the power exponent $ \delta $ in singular nonlinearities.</p></abstract>

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

Nonlinear Singular Elliptic Equations of p-Laplace Type with Superlinear Growth in the Gradient

A new existence result for some nonlocal problems involving Orlicz spaces and its applications

This paper studies some quasilinear elliptic nonlocal equations involving Orlicz–Sobolev spaces. On the one hand, a new sub-supersolution theorem is proved via the pseudomonotone operator theory; on the other hand, using the obtained theorem, we present an existence result on the positive solutions of a singular elliptic nonlocal equation. Our work improves the results of some previous researches.

Singular elliptic equations with nonlinearities of subcritical and critical growth

AbstractIn this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

A Problem with Mixed Boundary Conditions for a Singular Elliptic Equation in an Infinite Domain

A Problem with Mixed Boundary Conditions for a Singular Elliptic Equation in an Infinite Domain

Steady states of thin-film equations with van der Waals force with mass constraint

We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation, \begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*} we prove the existence of infinitely many radially symmetric solutions. Also, we perform rigorous asymptotic analysis to identify the blow-up limit when the steady state is close to a constant solution and the blow-down limit when the maximum of the steady state goes to the infinity.

On some singular semilinear elliptic equation with doubly exponential growth at infinity in $${\mathbb {R}}^2 $$

On some singular semilinear elliptic equation with doubly exponential growth at infinity in $${\mathbb {R}}^2 $$