Quasilinear Elliptic Equations Research Articles

Calderón–Zygmund Type Results for a Class of Quasilinear Elliptic Equations Involving the p(x)-Laplacian

Calderón–Zygmund Type Results for a Class of Quasilinear Elliptic Equations Involving the p(x)-Laplacian

Removable singularities in the boundary for quasilinear elliptic equations

Removable singularities in the boundary for quasilinear elliptic equations

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}_{+}} .

Decay estimates for quasilinear elliptic equations and a Brezis–Nirenberg result in D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{1,p}({\\mathbb R}^N)$$\\end{document}

We prove decay estimates for solutions of quasilinear elliptic equations in the whole RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} of the form u∈X:-divA(x,∇u)=a(x)f(x,u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\ ext{ div }\\,A(x,\ abla u)=a(x) f(x,u), \\end{aligned}$$\\end{document}where X=D1,p(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=D^{1,p}({\\mathbb R}^N)$$\\end{document} is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional Φ:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi : X\\rightarrow {\\mathbb R}$$\\end{document} related to the p-Laplacian equation in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^N$$\\end{document} in the form u∈X:-Δpu=a(x)g(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u\\in X: -\\Delta _p u=a(x) g(u), \\end{aligned}$$\\end{document}saying that for the subspace V of bounded continuous functions with weight 1+|x|N-pp,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+|x|^{\\frac{N-p}{p}},$$\\end{document} a local minimizer of Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} in the finer V topology is also a local minimizer in the X-topology. Global L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

Weighted Sobolev spaces and Morse estimates for quasilinear elliptic equations

We establish critical groups estimates for the weak solutions of −Δpu=f(x,u) in Ω and u=0 on ∂Ω via Morse index, where Ω is a bounded domain, f∈C1(Ω‾×R) and f(x,s)>0 for all x∈Ω‾, s>0 and f(x,s)=0 for all x∈Ω‾, s≤0. The proof relies on new uniform Sobolev inequalities for approximating problems. We also prove critical groups estimates when Ω is the ball or the annulus and f is a sign changing function.

Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ(Ωλ∖{0}) pairs (±u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus.

Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data

In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces g − 1 [ M 1 ( μ ) ] ∈ L w , l o c q , r ( Ω ) ⟹ | D u | ∈ L w , l o c q , r ( Ω ) , where g ( t ) = t a ( t ) for t ≥ 0 and M 1 ( μ ) ( x ) is the first-order fractional maximal function M 1 ( μ ) ( x ) := sup r > 0 r | μ | ( B r ( x ) ) | B r ( x ) | , for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case − div ⁡ [ a ( ( A D u ⋅ D u ) 1 2 ) A D u ] = μ in   Ω , whose model cases are the classical elliptic p -Laplacian equation with measure data − div ⁡ ( | D u | p − 2 D u ) = μ for  1 < p < 2 and the elliptic p -Laplacian equation with the logarithmic term and measure data − div ⁡ ( | D u | p − 2 D u ) = μ for  1 < p < 2 It deserves to be specially noted that the subquadratic case is a little different from the superquadratic case since as an example, the modulus of ellipticity in the above-mentioned special cases tends to infinity when | D u | → 0 for 1 < p < 2 .

Existence and uniqueness for anisotropic quasilinear elliptic equations involving singular nonlinearities

Existence and uniqueness for anisotropic quasilinear elliptic equations involving singular nonlinearities

Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

<abstract><p>We studied the modified nonlinear Schrödinger equation</p> <p><disp-formula> <label>0.1</label> <tex-math id="E0.1"> \begin{document}$ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.</p></abstract>

Quasilinear elliptic equations involving measure valued absorption terms and measure data

AbstractLet 1 < p < N and Ω ⊂ ℝN be an open bounded domain. We study the existence of solutions to equation $$(E) - {\Delta _p}u + g(u)\sigma = \mu $$ ( E ) − Δ p u + g ( u ) σ = μ in Ω, where g ∈ C(ℝ) is a nondecreasing function, μ is a bounded Radon measure on Ω and σ is a nonnegative Radon measure on ℝN. We show that if σ belongs to some Morrey space of signed measures, then we may investigate the existence of solutions to equation (E) in the framework of renormalized solutions. Furthermore, imposing a subcritical integral condition on g, we prove that equation (E) admits a renormalized solution for any bounded Radon measure μ. When $$g(t) = |t{|^{q - 1}}t$$ g ( t ) = | t | q − 1 t with q > p − 1, we give various sufficient conditions for the existence of renormalized solutions to (E). These sufficient conditions are expressed in terms of Bessel capacities.

Second order regularity for the [formula omitted]-Laplace equations with [formula omitted] data on the right-hand side

This work considers p(x)-Laplace equations involving the L2 function on the right-hand side. In particular, we extend classical W1,2 regularity estimates for the gradient of a solution to linear equations to nonlinear equations with variable exponents under minimal regularity assumptions on the boundary We focus on the global W1,2 estimate, which is motivated by the L2-coercivity results for quasilinear elliptic equations with Orlicz growth, as demonstrated in Cianchi and Maz’ya (2018).

Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements

Stable determination of the nonlinear term in a quasilinear elliptic equation by boundary measurements

Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction

Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction

Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearities

We consider the following quasi-linear Schrödinger equation−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju+V(x)u=f(u),x∈RN,N≥3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on aij and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f(u)=|u|p−2u, the case p∈(2,4) is also considered.

Liouville-type results for some quasilinear anisotropic elliptic equations

We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

On the Estimate of Integral Norm of Solution of the Dirichlet Problem for Quasilinear Elliptic Equation

Fractional differentiability for elliptic measure data problems with variable exponents

We investigate quasilinear elliptic equations with variable growth exponents having a Borel measure on the right-hand side. We prove fractional differentiability results for a certain nonlinear expression of the gradient of solutions.

The quasi-linear Brezis-Nirenberg problem in low dimensions

We discuss existence results for a quasi-linear elliptic equation of critical Sobolev growth [3,14] in the low-dimensional case, where the problem has a global character which is encoded in sign properties of the “regular” part for the corresponding Green's function as in [9,11].

Structural stability and optimal convergence rates of subsonic Euler flows with large vorticity in infinitely long nozzles

In this paper, we establish the structural stability and optimal convergence rates of non-isentropic subsonic Euler flows with large vorticity in two-dimensional infinitely long nozzles. By applying the stream function formulation for compressible Euler equations, Euler equations are equivalent to a quasilinear second order elliptic equation of the stream function for subsonic flow. Then, the key points to prove the structural stability of subsonic flows with large vorticity under small perturbations of nozzle boundaries are the standard estimates of elliptic equations. Furthermore, using the maximum principle and the choice of compared functions, we obtain the optimal convergence rates of subsonic flows at far fields.