Solutions of nonlinear elliptic problems with lower order terms

Journal de Mathématiques Pures et Appliquées, 2006

In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: − p u = g(u)|∇u| p + f, in Ω, u = 0, on ∂Ω. (1) Here Ω is a bounded open set in R N (N 2), p denotes the so-called p-Laplace operator (p > 1) and g is a continuous real function. Given f ∈ L m (Ω) (m > 1), we study under which growth conditions on g problem (1) admits a solution. If m N/p, we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N/p; while if 1 < m < N/p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condition is asked on f. Our methods rely on a priori estimates and compactness arguments and are applied to a large class of equations involving operators of Leray-Lions type. We also make several examples and remarks which give evidence of the optimality of our results.

Nonlinear Analysis: Theory, Methods & Applications, 1997

Nonlinear Analysis: Theory, Methods &amp; Applications, 2000

Advances in Science, Technology and Engineering Systems Journal, 2017

In this work, we shall be concerned with the existence of weak solutions of anisotropic elliptic operators Au + N i=1 g i (x, u, ∇u) + N i=1 H i (x, ∇u) = f − N i=1 ∂ ∂x i k i , where the right hand side f belongs to L p ∞ (Ω) and k i belongs to L p i (Ω) for i = 1, ..., N and A is a Leray-Lions operator. The critical growth condition on g i is the respect to ∇u and no growth condition with respect to u, while the function H i grows as|∇u| p i −1 .

2018

In this paper we are interested in the existence of a solution for the nonlinear degenerate elliptic equations Lu(x) + H(x, u, ∇u) ω 2 = f in the setting of the weighted Sobolev space W 1,p 0 (Ω, ω 1 , ω 2), where H is a nonlinear term with natural growth with respect to ∇u and f ∈ L 1 (Ω) .

Differential and Integral Equations, 2007

In this paper we deal with the problem −div (a(x, u)∇u) + g(x, u, ∇u) = λh(x)u + f in Ω, u = 0 on ∂Ω. The main goal of the work is to get hypotheses on a, g and h such that the previous problem has a solution for all λ > 0 and f ∈ L 1 (Ω). In particular, we focus our attention in the model equation with a(x, u) = (1 + |u| m), g(x, u, ∇u) = m 2 |u| m−2 u|∇u| 2 and h(x) = 1 |x| 2 .

Nonlinear Analysis: Theory, Methods & Applications, 1992

Complex Variables and Elliptic Equations, 2011

We discuss the problem -div(a(x, ru)) ¼ m(x)juj r(x)À2 u þ n(x)juj s(x)À2 u in , where is a bounded domain with smooth boundary in R N (N ! 2), and div(a(x, ru)) is a p(x)-Laplace type operator with 1 < r(x) < p(x) < s(x). We show the existence of infinitely many nontrivial weak solutions in W 1,pðxÞ 0 ð Þ. Our approach relies on the theory of the variable exponent Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory.

Nonlinear Analysis: Theory, Methods & Applications, 2011

In this work we analyze the interaction between the Hardy potential and a lower order term to obtain the existence or nonexistence of positive solution in elliptic problems whose model is

Journal of the London Mathematical Society, 1980