Singular Quasilinear Elliptic Equations Research Articles

Weak positive solutions to singular quasilinear elliptic equation

Abstract In this paper, we study the existence of multiple solutions for the singular problem { a ⁢ ( x , u , ∇ ⁡ u ) - div ⁢ ( b ⁢ ( x , u , ∇ ⁡ u ) ) = u - α + λ ⁢ c ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , u = 0 on ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle{}a(x,u,\nabla u)-{\rm div}(b(x,u,\nabla u% ))&\displaystyle=u^{-\alpha}+\lambda c(x,u)&&\displaystyle\phantom{}\text{in }% \Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }{\mathbb{R}}% ^{n}\setminus\Omega,\end{aligned}\right. where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} ( n ≥ 3 ) {(n\geq 3)} is a bounded domain with C 1 {C^{1}} boundary, λ is a positive parameter, 0 < α ≤ 1 < p ≤ n {0<\alpha\leq 1<p\leq n} and p * = n ⁢ p n - p {p^{*}=\frac{np}{n-p}} is the critical exponent for Sobolev embedding. Using the fibering maps and the Nehari manifold, we prove the existence of at least two positive solutions for all values of the parameter λ belonging to an open bounded interval of ℝ + {\mathbb{R}_{+}} .

Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p -Laplace equation -\Delta_{p} u=\mu with p\in (1,2) . The cases when p\in (2-\frac{1}{n},2) and p\in (\frac{3n-2}{2n-1},2-\frac{1}{n}] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379–418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p\in(1,\frac{3n-2}{2n-1}] . Interior and global modulus of continuity estimates of the gradients of solutions are also established.

On Singular Quasilinear Elliptic Equations in $$\mathbb {R}^N$$

On Singular Quasilinear Elliptic Equations in $$\mathbb {R}^N$$

Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data

This paper continues the development of regularity results for quasilinear elliptic equations −div(A(x,∇u))=μinΩ,andu=0on∂Ω,in Lorentz and Lorentz–Morrey spaces, where Ω⊂Rn (n≥2); A is a monotone Carathéodory vector valued operator acting between W01,p(Ω) and its dual W−1,p′(Ω); and μ is a datum in some Lebesgue space Lm(Ω), for m<p′. It emphasizes that in this paper, we restrict our study to the case of ‘very singular’ when 1<p≤3n−22n−1, and under mild assumption that the p-capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the ‘very singular’ case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces.

A multiplicity results for a singular quasilinear elliptic equation

The purpose of this work is to study the following singular problem: [Formula: see text] where [Formula: see text], [Formula: see text] be a bounded smooth domain, [Formula: see text] is a positive parameter, [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] We employ the Nehari manifold approach and the fibering maps in order to show the existence of [Formula: see text] such that for all [Formula: see text], problem [Formula: see text] has at least two solutions.

On singular quasilinear elliptic equations with data measures

AbstractThe aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document} -Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1 , where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document} . Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds

In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from the variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.

Existence of solutions for a class of singular quasilinear elliptic equations

ABSTRACT In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems where , the potential and . By Nehari manifold approach and the technique of splitting a minimizing sequence, we prove the existence of positive solutions for the above problem.

Singular quasilinear elliptic problems with changing sign datum: existence and homogenization

We study singular quasilinear elliptic equations whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\,\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) &{} \quad \hbox {in}\,\, \Omega , \\ u=0 &{}\quad \hbox {on} \,\, \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\) (\(N\ge 3\)), \(\lambda \in {\mathbb {R}}\), \(1 \frac{N}{2}\), may change sign. We prove existence of solution and we deal with the homogenization problem posed in a sequence of domains \(\Omega ^\varepsilon \) obtained by removing many small holes from a fixed domain \(\Omega \).

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

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: [Formula: see text] in Lorentz–Morrey spaces, where [Formula: see text] ([Formula: see text]), [Formula: see text] is a finite Radon measure, [Formula: see text] is a monotone Carathéodory vector-valued function defined on [Formula: see text] and the [Formula: see text]-capacity uniform thickness condition is imposed on the complement of our domain [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text]. Here in this study, we particularly restrict ourselves to the singular case [Formula: see text]. The results are central to generalize our technique of good-[Formula: see text] type bounds in the previous work [M.-P. Tran, Good-[Formula: see text] 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.

Quasilinear elliptic problems with singular and homogeneous lower order terms

We deal with singular quasilinear elliptic equations, namely −Δu=λu+μ(x)|∇u|quq−1+f(x)inΩ,u>0inΩ,u=0on∂Ω,where Ω is a bounded smooth domain of RN (N≥3), λ∈R, 1<q≤2, 0≤μ∈L∞(Ω) and 0≤f∈Lp(Ω) for some p>N2. We completely describe the set of values of the parameter λ for which the problem admits solution. Thus, we study existence, nonexistence and uniqueness of bounded weak solutions in both cases f⪈0 and f≡0.

Singular nonhomogeneous quasilinear elliptic equations with a convection term

In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub-super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.

Existence results for a singular quasilinear elliptic equation

Let \(\Omega \subset \mathbb R^N\) be a bounded domain with smooth boundary. Existence of a positive solution to the quasilinear equation $$\begin{aligned} -\text {div}\left[ \left( a(x)+|u|^\theta \right) \nabla u\right] +\frac{\theta }{2}|u|^{\theta -2}u|\nabla u|^2=|u|^{p-2}u \quad \text {in}\ \Omega \end{aligned}$$ with zero Dirichlet boundary condition is proved. Here \(\theta >0\) and a(x) is a measurable function satisfying \(0<\alpha \le a(x)\le \beta \). The equation involves singularity when \(0<\theta \le 1\). As a main novelty with respect to corresponding results in the literature, we only assume \(\theta +2<p<\frac{2^*}{2}(\theta +2)\). The proof relies on a perturbation method and a critical point theory for E-differentiable functionals.

Resonance problem of a class of singular quasilinear elliptic equations

In this paper, we study the resonance problem of a class of singular quasilinear elliptic equations with respect to its higher near-eigenvalues. Under a generalized Landesman–Lazer condition, it is proved that the resonance problem admits at least one nontrivial solution in weighted Sobolev spaces. The proof is based upon applying the Galerkin-type technique, the Brouwer’s fixed-point theorem and a compact embedding theorem of weighted Sobolev spaces by Shapiro.

Critical Points for a Functional Involving Critical Growth of Trudinger–Moser Type

We establish the existence of critical points of leat energy for functionals associated with a class of singular quasilinear elliptic equations involving critical growth of Trudinger-Moser type. For this, in line with the Concentration–Compactness Principle due to P. -L. Lions (I. Rev. Mat. Iberoamericana 1, 145–201, 1985), we study the lack of compactness of Sobolev imbedding in Orlicz space defined in ℝn(n ≥ 2).

Nonexistence of solutions for singular quasilinear elliptic equation in $\mathbb{R}^N$

In this paper, we study the nonexistence of solutions for singular quasilinear elliptic equation div(|x| ap|—u|p 2—u)+V (x)|u|p 2u = h(x)|u|s 2u+H(x)|u|r 2u, x 2 RN , (P) where 1 1. When the weight functions V (x),h(x) and H(x) are radially symmetric with respect to x in RN , we establish a variational identity for solution of (P) and then obtain some sufficient conditions on the nonexistence of nontrivial solutions for (P). When V (x)⌘ 0 and the functions h(x),H(x) are not necessarily radially symmetric with respect to x in RN , we get conditions sufficient to ensure the nonexistence of nonnegative entire solutions for (P) via the test function method developed by Mitidieri and Pohozaev.

Point Rupture Solutions of a Singular Quasilinear Elliptic Equation Arise from Non-Newtonian Fluids

We consider the quasilinear elliptic equation in a finite ball in , where Motivated by the thin film equations, a solution is said to be a point rupture solution if for some , and for any . Our main result is a sufficient condition on for the existence of radial point rupture solutions.

Positive Radial Solutions for Singular Quasilinear Elliptic Equations in a Ball

We establish the existence of positive radial solutions for the boundary value problems \left\{ \begin{array}{rcll} -\Delta _{p}u&amp;=&amp;\lambda f(u)&amp;\text{ in }B, \\ u&amp;=&amp;0&amp;\text{ on }\partial B, \end{array} \right. where \Delta _{p}u=\textbf{div}(|\nabla u|^{p-2}\nabla u),p\geq 2 , B is the open unit ball \mathbb{R}^{N} , \lambda is a positive parameter, and f:(0,\infty )\rightarrow \mathbb{R} is p -superlinear at \infty and is allowed to be singular at 0 .

Existence and multiplicity of weak solutions for a singular quasilinear elliptic equation

In this paper, we consider the singular quasilinear elliptic equation (0.1)−div(|x|−ap|Du|p−2Du)+V(x)|u|p−2u=h(x)|u|q−2u+λH(x)|u|r−2u,x∈RN∖{0}, where λ>0,−∞<a<N−pp,1<p<N,1<q,r<p∗=NpN−p. The function V(x) is continuous and nonnegative; the functions h(x),H(x)∈L∞(BR+1∖Bτ) with 0<τ<R0<R<+∞. Using variational methods, we obtain a number of results on existence and multiplicity of solutions by considering all possible orderings of exponents p,q,r.