Class Of 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

Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries

AbstractIn this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating(N+1)\left(N+1)-dimensional thin domains (i.e., a family of bounded open sets fromRN+1{{\mathbb{R}}}^{N+1}, with corrugated bounder, which degenerates to an open bounded set inRN{{\mathbb{R}}}^{N}). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits inNN-dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.

On a Class of Quasilinear Elliptic Equations

On a Class of Quasilinear Elliptic Equations

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

AbstractWe establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz–Sobolev spaces and under general growth conditions on the convection term. The sub‐ and supersolutions method is a key tool in the proof of the existence results.

Global Gradient Estimates of Very Weak Solutions for a General Class of Quasilinear Elliptic Equations

Global Gradient Estimates of Very Weak Solutions for a General Class of Quasilinear Elliptic Equations

Weak Harnack inequalities for eigenvalues and a constant rank theorem for level sets

We consider convex level sets of solutions of a class of quasilinear elliptic equations. We prove a weak Harnack inequality for the eigenvalues of the Weingarten tensor of level sets and then obtain a quantitative version of constant rank theorem.

Existence of positive solutions for a class of quasilinear elliptic equations with parameters

Existence of positive solutions for a class of quasilinear elliptic equations with parameters

Existence of solutions to a class of quasilinear elliptic equations

This paper is concerned with the existence and nonexistence of nontrivial solutions for a class of quasilinear elliptic equations in RN involving the p-Laplacian. By utilizing a change of variables, the quasilinear equations are reduced to semilinear equations, whose associated functionals satisfy the geometric hypotheses of the mountain pass theorem. We establish the existence of nontrivial solutions via Mountain-Pass theorem. Furthermore, by using a Pohozaev type identity we prove the nonexistence of nontrivial solution under certain conditions.

Existence of solutions for a class of quasilinear elliptic equations involving the p-Laplacian

This paper is concerned with the existence of solutions for the quasilinear elliptic equations − Δ p u − Δ p ( | u | 2 α ) | u | 2 α − 2 u + V ( x ) | u | p − 2 u = | u | q − 2 u , x ∈ R N , where α ≥ 1 , 1<p<N, p ∗ = Np / ( N − p ) , Δ p is the p-Laplace operator and the potential 0 $ ]]> V ( x ) > 0 is a continuous function. In this work, we mainly focus on nontrivial solutions. When 2 αp < q < p ∗ , we establish the existence of nontrivial solutions by using Mountain-Pass lemma; when q ≥ 2 α p ∗ , by using a Pohozaev type variational identity, we prove that the equation has no nontrivial solutions.

Hardy inequality for domains with a geometric boundary condition and applications

In this paper, we state a Hardy inequality for domains with a geometric boundary condition. As a consequence, we prove a weighted Trudinger–Moser inequality. After that, we apply our results to investigate the existence of solutions for a class of quasilinear elliptic equations with Neumann boundary condition and non-linearities with critical exponential growth.

Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div(A(x,∇u))=div(f)inΩ,andu=gon∂Ω,that related to Riesz potentials. Here, in our setting, Ω⊂Rn, n≥2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

A class of quasilinear elliptic equations with singular and exponential terms

We use a scheme of approximation together with Brouwer's theorem to establish the existence of positive solutions for a class of quasilinear elliptic problems. More precisely, we study the class of equations: (1) { − Div ( a ( | ∇u | p ) | ∇u | p − 2 ∇u ) = η u − γ + λ u s + f ( u ) in Ω , u = 0 on ∂Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary, 2 ≤ p < N , 0 < γ < 1 , 0<s<N−1, η and λ are positive parameters, a : R + → R + is a function of class C 1 ( R + ) . The nonlinear term f involves three different types of Trudinger–Moser growth: subcritical, critical or supercritical. We also study the asymptotic behavior of the solutions with respect to the parameters.

A new class of double phase variable exponent problems: Existence and uniqueness

In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S+)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.

Existence of a Least Energy Nodal Solution for a Class of Quasilinear Elliptic Equations with Exponential Growth

Existence of a Least Energy Nodal Solution for a Class of Quasilinear Elliptic Equations with Exponential Growth

Regularity of solutions for a class of quasilinear elliptic equations related to the Caffarelli-Kohn-Nirenberg inequality

This paper is concerned with a class of quasilinear elliptic equations involving some potentials related to the Caffarelli-Korn-Nirenberg inequality. We prove the local boundedness and Hölder continuity of weak solutions by using the classical De Giorgi techniques. Our result extends the results of Serrin (1964) [17] and Colorado and Peral (2004) [2].

Global gradient estimates for a general class of quasilinear elliptic equations with Orlicz growth

We provide an optimal global Calderón-Zygmund theory for quasilinear elliptic equations of a very general form with Orlicz growth on bounded nonsmooth domains under minimal regularity assumptions of the nonlinearity A = A ( x , u , D u ) A=A(x,u,Du) in the first and second variables ( x , z ) (x,z) as well as on the boundary of the domain. Our result improves known regularity results in the literature regarding nonlinear elliptic operators depending on a given bounded weak solution.

Ground state solutions for quasilinear scalar field equations arising in nonlinear optics

In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space (Hirata et al. in Topol Methods Nonlinear Anal 35:253–276, 2010; Jeanjean in Nonlinear Anal Theory Methods Appl 28:1633–1659, 1997), we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki–Lions’ type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.

Maximal Differentiability for a General Class of Quasilinear Elliptic Equations with Right-hand Side Measures

Abstract We prove maximal differentiability for the gradient of solutions to a certain type of nonlinear elliptic equations with a measure on the right-hand side. Our results generalize the limiting case of Calderón–Zygmund theory for an elliptic equation with $p$-growth to the case of Orlicz growth.

Infinitely many nodal solutions for a class of quasilinear elliptic equations

Infinitely many nodal solutions for a class of quasilinear elliptic equations