Natural Growth Terms Research Articles

Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

Abstract The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L 1(Ω). Furthermore the weight function θ(·) is in W̊ 1,p→(·) (Ω), with θ(·) > 0 and connected with the coefficient b(·) ∈ L 1(Ω) of the lower order term.

Strongly nonlinear parabolic problems with natural growth terms and L1 data in Musielak-Orlicz-Sobolev spaces

Strongly nonlinear parabolic problems with natural growth terms and L1 data in Musielak-Orlicz-Sobolev spaces

Existence of Solution for a General Class of Strongly Nonlinear Elliptic Problems Having Natural Growth Terms and $L^1$ -Data

Existence of Solution for a General Class of Strongly Nonlinear Elliptic Problems Having Natural Growth Terms and $L^1$ -Data

Existence of solution for nonlinear degenerate elliptic unilateral problems having natural growth terms

Existence of solution for nonlinear degenerate elliptic unilateral problems having natural growth terms

Some Nonlinear Parabolic Problems with Singular Natural Growth Term

Some Nonlinear Parabolic Problems with Singular Natural Growth Term

Existence of $W_0^{1,1}(\Omega)$ solutions to nonlinear elliptic equation with singular natural growth term

In this paper, we investigate the existence of $W_0^{1, 1}(\Omega)$ solutions to the following elliptic equation with principal part having noncoercivity and singular quadratic term $ \begin{equation*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{\nabla u}{(1+|u|)^{\gamma}}\right)+\frac{|\nabla u|^2}{u^{\theta}} = f,&x\in\Omega, u = 0,&x\in\partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N(N\geq3)$, $\gamma \gt 0$, $\frac{N}{N-1}\leq\theta \lt 2$, $f\in L^m(\Omega)(m\geq1)$ is a nonnegative function.

Asymptotic behavior of solutions for nonlinear parabolic operators with natural growth term and measure data

We are interested in the asymptotic behavior, as t tends to $$+\infty $$ , of finite energy solutions and entropy solutions $$u_{n}$$ of nonlinear parabolic problems whose model is 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\Delta _{p}u+g(u)|\nabla u|^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$ where $$\Omega \subseteq \mathbb {R}^{N}$$ is a bounded open set, $$N\ge 3$$ , $$u_{0}\in L^{1}(\Omega )$$ is a nonnegative initial data, while $$g:\mathbb {R}\mapsto \mathbb {R}$$ is a real function in $$C^{1}(\mathbb {R})$$ which satisfies sign condition with positive derivative and $$\mu $$ is a nonnegative measure independent on time which does not charge sets of null p-capacity.

Hölder continuity up to the boundary of solutions to nonlinear fourth-order elliptic equations with natural growth terms

Hölder continuity up to the boundary of solutions to nonlinear fourth-order elliptic equations with natural growth terms

Existence Result to a Parabolic Equation with Quadratic Gradient Term and an L 1 $L^{1}$ Source

In this paper, we mainly consider the existence of bounded solutions to nonlinear parabolic problem involving natural growth term and $L^{1}$ source. Parabolic regularization method and iterative technique are applied to the existence result with necessary estimates and compactness argument.

Hölder continuity of bounded generalized solutions for some degenerated quasilinear elliptic equations with natural growth terms

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation $\sum_{i =1}^{m} \frac{\partial}{\partial x_i} a_i (x, u, \nabla u) - c_0 |u|^{p-2} u = f(x, u, \nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda (|u|) \sum_{i=1}^m a_i (x,u, \eta) \eta_i \geq \nu(x) |\eta|^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

Formula omitted]-partial regularity for sub-elliptic systems with Dini continuous coefficients in Carnot groups

In this paper, we consider nonlinear sub-elliptic systems with Dini continuous coefficients for the case 1<m<2 in Carnot groups. Based on a generalization of the technique of A-harmonic approximation introduced by Duzaar–Grotowski–Kronz, and a Sobolev–Poincaré type inequality established in Carnot groups, a C1-partial regularity result for weak solutions of sub-elliptic systems with natural growth terms is established. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients it yields directly the optimal Hölder exponent for horizontal gradients of weak solutions on its regular set.

Non-linear elliptic unilateral problems in Musielak-Orlicz spaces with L1 data

We prove an existence result of solutions for nonlinear elliptic unilateral problems having natural growth terms and L1 data in Musielak-Orlicz-Sobolev space W1Lφ, under the assumption that the conjugate function of φ satisfies the ∆2-condition.

Existence result for nonlinear elliptic unilateral problems in Musielak framework with $$L^1$$ L 1 data

We give an existence result for nonlinear elliptic unilateral problems having natural growth terms and \(L^1\) data in Musielak–Orlicz spaces.

Existence of solutions for some doubly degenerate parabolic equations with natural growth terms

This paper deals with a class of doubly degenerate parabolic equations of the type ∂b(u)∂t−div(a(x,t,u,∇u))+F(x,t,u,∇u)=f, where b is a strictly increasing C1-function. Under some suitable hypothesis, we prove that there exists at least a bounded solution to this problem in standard weak sense.

A quasilinear elliptic system with natural growth terms

In this paper, we prove existence of solutions for an elliptic system of the type $$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z) \nabla u) = f, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z)+ h(x,z)|\nabla u|^2 = g, &{}\quad \text{ in } \varOmega \text{; } \\ \,u = 0 = z, &{}\quad \text{ on } \partial \varOmega \text{, } \end{array}\right. } \end{aligned}$$under various assumptions on the functions \(a(x,s)\) and \(h(x,s)\), and on the data \(f\) and \(g\) (in Lebesgue spaces).

Existence of positive solutions for a class of parabolic equations with natural growth terms and L1 data

We study a class of nonlinear parabolic equations of the type: ∂b(u)∂t−div (a(x,t,u)∇u)+g(u)|∇u|2=f,where the right hand side belongs to L1(Q), b is a strictly increasing C1-function and −div (a(x,t,u)∇u) is a Leray-Lions operator. The function g is just assumed to be continuous on ℝ and to satisfy a sign condition. Without any additional growth assumption on u, we prove the existence of a renormalized solution.

Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group

We consider nonlinear sub-elliptic systems with Dini continuous coefficients in divergence form in the Heisenberg group. Based on a generalization of the method of A-harmonic approximation introduced by Duzaar and Steffen, partial regularity of weak solutions for sub-elliptic systems under natural growth conditions is established. In particular, our result is optimal in the sense that in the case of Hölder continuous coefficients we obtain directly the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set.

Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, natural growth terms and L1 data

We give an existence result of a renormalized solution for a class of nonlinear parabolic equations ∂b(x,u)∂t-div(a(x,t,u,∇u))+g(u)|∇u|p=f, where the right side belongs to L1(Ω×(0,T)), b(x,u) is an unbounded function of u and −div(a(x,t,u,∇u)) is a Leray–Lions type operator with growth ∣∇u∣p−1 in ∇u, but without any growth assumption on u. The function g is just assumed to be continuous on R and satisfying a sign condition.

The fundamental solution of nonlinear equations with natural growth terms

We find bilateral global bounds for the fundamental solutions associated with some quasilinear and fully nonlinear operators perturbed by a nonnegative zero order term with natural growth under minimal assumptions. Important model problems involve the equations −!pu = σ |u|p−2 u + δx0 , for p > 1, and Fk(−u) = σ |u|k−1 u + δx0 , for k ≥ 1. Here !p and Fk are the p-Laplace and k-Hessian operators respectively, and σ is an arbitrary positive measurable function (or measure). We will in addition consider the Sobolev regularity of the fundamental solution away from its pole. Mathematics Subject Classification (2010): 35J60 (primary); 42B37, 31C45, 35J92, 42B25 (secondary).

Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

In this paper we study the Dirichlet problem $$\left\{\begin{array}{lll}-\Delta_p{u} = \sigma |u|^{p-2}u + \omega \quad {\rm in}\;\Omega,\\ u = 0 \qquad\quad\qquad\quad\;\qquad{\rm on}\;\partial\Omega,\end{array}\right.$$ , where σ and ω are nonnegative Borel measures, and $${\Delta_p{u} = \nabla \cdot (\nabla{u} \, |\nabla{u}|^{p-2})}$$ is the p-Laplacian. Here $${\Omega \subseteq \mathbf{R}^n}$$ is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff’s potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.