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articleNo Access
Regularity and existence for semilinear mixed local–nonlocal equations with variable singularities and measure data
- Sanjit Biswas and
- Prashanta Garain
Communications in Contemporary Mathematics11 Mar 2025
Preview Abstract
This paper proves the existence and regularity of weak solutions for a class of mixed local–nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent.
articleNo Access
ON THE CONNECTION BETWEEN TWO QUASILINEAR ELLIPTIC PROBLEMS WITH SOURCE TERMS OF ORDER 0 OR 1
- HAYDAR ABDEL HAMID and
- MARIE FRANÇOISE BIDAUT-VERON
Communications in Contemporary Mathematics01 Oct 2010
Preview Abstract
We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ℝ^N. The first one, of the form
involves a source gradient term with natural growth, where β is non-negative, λ > 0, f(x) ≧ 0, and α is a non-negative measure. The second one, of the form
presents a source term of order 0, where g is non-decreasing, and μ is a non-negative measure. Here β and g can present an asymptote. The correlation gives new results of existence, non-existence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.
articleNo Access
QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR QUADRATIC GROWTH TERMS
Communications in Contemporary Mathematics01 Aug 2011
Preview Abstract
In this paper, we deal with positive solutions for singular quasilinear problems whose model is
where Ω is a bounded open set of ℝ^N, g ≥ 0 is a function in some Lebesgue space, and γ > 0. We prove both existence and nonexistence of solutions depending on the value of γ and on the size of g.
articleNo Access
A SINGULAR STURM–LIOUVILLE EQUATION INVOLVING MEASURE DATA
- HUI WANG
Communications in Contemporary Mathematics17 Jun 2013
Preview Abstract
Let α > 0 and let μ be a bounded Radon measure on the interval (-1, 1). We are interested in the equation -(|x|^2αu′)′ + u = μ on (-1, 1) with boundary condition u(-1) = u(1) = 0. We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. The cases 0 < α < 1 and α ≥ 1 must be considered separately. We also study the limiting behavior of two different approximation schemes: one is the elliptic regularization and the other is to approximate a measure μ by a sequence of L^∞-functions.
articleNo Access
Evolution equations of p-Laplace type with absorption or source terms and measure data
- Marie-Françoise Bidaut-Véron and
- Quoc-Hung Nguyen
Communications in Contemporary Mathematics29 Oct 2015
Preview Abstract
Let Ω be a bounded domain of ℝ^N, and Q = Ω × (0, T). We consider problems of the type
where Δ_p is the p-Laplacian, μ is a bounded Radon measure, u₀ ∈ L¹(Ω), and ±𝒢(u) is an absorption or a source term. In the model case 𝒢(u) = ±|u|^q-1u (q > p-1), or 𝒢, has an exponential type. We prove the existence of renormalized solutions for any measure μ in the subcritical case, and give sufficient conditions for existence in the general case, when μ is good in time and satisfies suitable capacitary conditions.
articleNo Access
Lorentz–Morrey global bounds for singular quasilinear elliptic equations with measure data
- Minh-Phuong Tran and
- Thanh-Nhan Nguyen
Communications in Contemporary Mathematics29 May 2019
Preview Abstract
The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem:
$\{\begin{matrix} - div (A (x, \nabla u)) = μ & in Ω, \\ u = 0 & on \partial Ω, \end{matrix}$
in Lorentz–Morrey spaces, where $Ω \subset ℝ^{n}$ ( $n \geq 2$ ), $μ$ is a finite Radon measure, $A$ is a monotone Carathéodory vector-valued function defined on $W_{0}^{1, p} (Ω)$ and the $p$ -capacity uniform thickness condition is imposed on the complement of our domain $Ω$ . It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case $2 \leq p \leq n$ , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case $p > 2 - \frac{1}{n}$ . Here in this study, we particularly restrict ourselves to the singular case $\frac{3 n - 2}{2 n - 1} < p \leq 2 - \frac{1}{n}$ . The results are central to generalize our technique of good- $λ$ type bounds in the previous work [M.-P. Tran, Good- $λ$ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.
articleNo Access
Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: $1 < p \leq \frac{3 n - 2}{2 n - 1}$
- Le Cong Nhan and
- Le Xuan Truong
Communications in Contemporary Mathematics28 Aug 2020
Preview Abstract
In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form
$\{\begin{matrix} - div (𝒜 (x, \nabla u)) = μ in Ω, \\ u = 0 on \partial Ω, \end{matrix}$
where $μ$ is a finite signed Radon measure in $Ω$ , $Ω \subset ℝ^{n}$ is a bounded domain such that its complement $ℝ^{n} ∖ Ω$ is uniformly $p$ -thick and $𝒜$ is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case $1 < p \leq \frac{3 n - 2}{2 n - 1}$ . Our result extends the earlier results [19, 22] to the strongly singular case $1 < p \leq \frac{3 n - 2}{2 n - 1}$ and a recent result [18] by considering rough conditions on the domain $Ω$ and the nonlinearity $𝒜$ .

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Regularity and existence for semilinear mixed local–nonlocal equations with variable singularities and measure data

ON THE CONNECTION BETWEEN TWO QUASILINEAR ELLIPTIC PROBLEMS WITH SOURCE TERMS OF ORDER 0 OR 1

QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR QUADRATIC GROWTH TERMS

A SINGULAR STURM–LIOUVILLE EQUATION INVOLVING MEASURE DATA

Evolution equations of p-Laplace type with absorption or source terms and measure data

Lorentz–Morrey global bounds for singular quasilinear elliptic equations with measure data

Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: $1 < p \leq \frac{3 n - 2}{2 n - 1}$