Singular quasilinear elliptic equations and Hölder regularity

Giacomoni, Jacques; Schindler, Ian; Takáč, Peter

doi:10.1016/j.crma.2012.04.007

Cited by 19 publications

(25 citation statements)

References 8 publications

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“…(Ω) if p + q ≥ 1 for some α, β ∈ (0, 1) has been obtained in [3,18,20]. Problems of type (1.5) have been studied in [2,4,5] and the semilinear case m = 2 has been studied in [9,11,14,15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weak solutions for singular quasilinear elliptic systems

Singh

2016

Complex Variables and Elliptic Equations

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We investigate the quasilinear elliptic systemwhere Ω ⊂ R N (N ≥ 1) is a bounded and smooth domain, 1 < m < ∞, p, q, r, s > 0. Under certain conditions imposed on the exponents we obtain the existence and uniqueness of a weak solution (u, v) with u, v ∈ W 1,m 0(Ω) ∩ C(Ω). We also investigate the W 1,τ 0 (Ω) regularity of solution and determine the optimal range of τ ≥ m for such regularity.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Weak solutions for singular quasilinear elliptic systems

Singh

2016

Complex Variables and Elliptic Equations

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“…and the equality holds if and only if u = tφ for some t ∈ ℝ. Next, we consider the singular problem A similar result to Theorem 2.1 was discussed in [10] for the case of a bounded domain, and was recently extended to a coupled system of equations in [11]. For the case p = , we refer the reader to the book by Ghergu and Rădulescu [9], where singular elliptic problems are studied in detail, and a rich list of references is provided.…”

Section: Auxiliary Problemsmentioning

confidence: 95%

Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

Chhetri

Drábek

Shivaji

2016

Advances in Nonlinear Analysis

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We consider the problem\left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}} ({N>2}) is a simply connected bounded domain containing the origin with {C^{2}} boundary {\partial\Omega}, {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, {1<p<N} and {0\leq\delta<1}. In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution {u\in C^{1}(\overline{\Omega^{e}})} with {u=0} pointwise on {\partial\Omega}. Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in {\mathbb{R}^{N}}, for certain classes of {K(x)=K(|x|)}, we observe that our solution must also be radial.

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“…The literature on singular problems for equations of the form  = − has primarily focused on local operators as the Laplacian,  = − div ∇ (see [2,4,7,10,17,18,21]), or the -Laplacian,  = − div ( |∇ | −2 ∇ ) , > 1 (see [1,9,[13][14][15]20]). As for singular problems involving nonlocal (fractional) operators, the literature is quite recent and more restricted to  = ( −Δ ) (see [3,8]).…”

Section: Introductionmentioning

confidence: 99%

“…The literature on singular problems for equations of the form