Singular Elliptic Problems Research Articles

Regularity of the stress field for degenerate and/or singular elliptic problems

Regularity of the stress field for degenerate and/or singular elliptic problems

A blow-up approach for singular elliptic problems with natural growth in the gradient

A blow-up approach for singular elliptic problems with natural growth in the gradient

Existence of positive solutions for a class of singular elliptic problems with convection term and critical exponential growth

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by {−Δu=λ0uβ0+Λ0|∇u|γ0+f0(u)|x|α0+h0(x),u>0inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} -\\Delta u= \\displaystyle \\frac {\\lambda _{0}}{u^{\\beta _{0}}} + \\Lambda _{0} |\ abla u|^{\\gamma _{0}}+ \\frac{f_{0}(u)}{|x|^{\\alpha _{0}}}+ h_{0}(x), \\ \\ u>0 \\ \\ \ ext{in} \\ \\Omega , \\\\ u=0 \\ \ ext{on} \\ \\ \\partial \\Omega , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset \\mathbb{R}^{2}$\\end{document} is a bounded smooth domain, 0<β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\beta _{0}$\\end{document}, γ0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\gamma _{0} \\leq 1$\\end{document}, α0∈[0,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{0} \\in [0,2)$\\end{document}, h0(x)≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}(x)\\geq 0$\\end{document}, h0≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}\ eq 0$\\end{document}, h0∈L∞(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$h_{0}\\in L^{\\infty}(\\Omega )$\\end{document}, 0<∥h0∥∞<λ0<Λ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0<\\|h_{0}\\|_{\\infty} < \\lambda _{0} < \\Lambda _{0}$\\end{document}, and f0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{0}$\\end{document} are continuous functions. More precisely, f0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{0}$\\end{document} has a critical exponential growth, that is, the nonlinearity behaves like exp(ϒ‾s2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\exp (\\overline{\\Upsilon}s^{2})$\\end{document} as |s|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|s| \ o \\infty $\\end{document}, for some ϒ‾>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline{\\Upsilon}>0$\\end{document}.

Positive solution for a singular and nonlocal problem of the N-Kirchhoff type

We investigate the existence of a positive solution for a singular elliptic problem of the N-Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation consists of two terms: one has a subcritical, critical or supercritical exponential growth governed by the Trudinger–Moser inequality, and the other one presents a singularity at the origin. We also show a result of nonexistence of solutions in dimension 2. Finally, we study the asymptotic behavior of the solutions with respect to the parameter.

Singular elliptic measure data problems with irregular obstacles

We investigate elliptic irregular obstacle problems with p-growth involving measure data. Emphasis is on the strongly singular case 1<p≤2−1/n, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.

A variational approach for some singular elliptic problems with Hardy potential

A variational approach for some singular elliptic problems with Hardy potential

Radial solutions and a local bifurcation result for a singular elliptic problem with Neumann condition

We study the problem − Δ u = λ u − u − 1 with a Neumann boundary condition; the peculiarity being the presence of the singular term − u − 1 . We point out that the minus sign in front of the negative power of u is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution ( u λ , λ ) with u λ approaching the trivial constant solution u = λ − 1 / 2 and λ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions u with λ fixed.

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.

Existence of solutions for singular elliptic problems with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent

In this article, we consider a singular elliptic problem with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent. By using variational methods and Palais-Smale condition, we show the existence of at least two nontrivial solutions. The result depends crucially on the parameters \(a,b,N,\beta,\gamma,\lambda,\mu\). For more information see https://ejde.math.txstate.edu/Volumes/2023/54/abstr.html

On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem

On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem

Existence results for singular elliptic problem involving a fractional p-Laplacian

Existence results for singular elliptic problem involving a fractional p-Laplacian

Singular elliptic problem involving a Hardy potential and lower order term

We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u &gt; 0 in Ω u = 0 on ∂ Ω , where 1 &lt; p &lt; N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 &lt; γ &lt; 1, a ⩾ 0 , b &gt; 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 &lt; m &lt; N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.

Existence of bounded solutions for a class of singular elliptic problems involving the 1-Laplacian operator

Existence of bounded solutions for a class of singular elliptic problems involving the 1-Laplacian operator

A new class of fractional Orlicz-Sobolev space and singular elliptic problems

This paper seeks to introduce a new class of fractional Orlicz-Sobolev space with variable-order. We give basic properties of this space and prove some compactness results. Then, using some techniques of calculus of variations combined with theory of Musielak functions, we prove the existence of a nonnegative weak solution for a singular elliptic type problem in a fractional variable-order Orlicz-Sobolev space with homogeneous Dirichlet boundary conditions.

On a singular elliptic problem with variable exponent

"In the present note we study a semilinear elliptic Dirichlet problem involving a singular term with variable exponent of the following type $$\left\{ \begin{array}{ll} -\Delta u= \frac{f(x)}{u^{\gamma(x)} }, &amp; \hbox{ in } \Omega \\ u&gt;0, &amp; \hbox{ in } \Omega \\ u=0, &amp; \hbox{on } \partial \Omega \end{array} \right.\eqno{(\mathcal{P})}$$ Existence and uniqueness results are proved when $f\geq 0$."

Homogenization of a class of singular elliptic problems in two-component domains

This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 &lt; γ &lt; 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Let \(\Omega\) be a \(C^{2}\) bounded domain in \(\mathbb{R}^{n}\) such that \(\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\), where \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint closed subsets of \(\partial\Omega\), and consider the problem\(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\tau\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=\eta\) on \(\Gamma_{2}\), where \(0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})\), \(\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}\), and \(g:\Omega \times(0,\infty)\rightarrow\mathbb{R}\) is a nonnegative Carath�odory function. Under suitable assumptions on \(g\) and \(\eta\) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow \(g\) to be singular at \(s=0\) and also at \(x\in S\) for some suitable subsets \(S\subset\overline{\Omega}\). The Dirichlet problem \(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\sigma\) on \(\partial\Omega\) is also studied in the case when \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\).

Results of singular Direchelet problem involving the $p(x)$-laplacian with critical growth

In this paper, we study the existence and multiplicity of solutions for Dirichlet singular elliptic problems involving the $p(x)$-Laplace equation with critical growth. The technical approach is mainly based on the variational method combined with the genus theory.

Non-variational radial solutions to a singular elliptic Dirichlet problem on the disk with Leray potential

Non-variational radial solutions to a singular elliptic Dirichlet problem on the disk with Leray potential

Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

We discuss the solvability of Dirichlet problems of the type $- \Delta_{p, w} u = \sigma$ in $\Omega$; $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $\Delta_{p, w}$ is a weighted $(p, w)$-Laplacian and $\sigma$ is a nonnegative locally finite Radon measure on $\Omega$. We do not assume the finiteness of $\sigma(\Omega)$. We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by $L^{p}(w)$-$L^{q}(\sigma)$ trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem $- \Delta_{p, w} u = \sigma u^{- \gamma}$ in $\Omega$; $u = 0$ on $\partial \Omega$ and derive connection with the trace inequalities.