On a quasilinear nonhomogeneous elliptic equation with critical growth in (original) (raw)

A nonhomogeneous elliptic problem involving critical growth in dimension two

Journal of Mathematical Analysis and Applications, 2008

In this paper we study a class of nonhomogeneous Schrödinger equations − u + V (x)u = f (u) + h(x) in the whole two-dimension space where V (x) is a continuous positive potential bounded away from zero and which can be "large" at the infinity. The main difficulty in this paper is the lack of compactness due to the unboundedness of the domain besides the fact that the nonlinear term f (s) is allowed to enjoy the critical exponential growth by means of the Trudinger-Moser inequality. By combining variational arguments and a version of the Trudinger-Moser inequality, we establish the existence of two distinct solutions when h is suitably small.

Existence of nontrivial solutions for quasilinear elliptic equations at critical growth

Applied Mathematics and Computation, 2011

Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, we establish the existence of a nontrivial solution for a class of Dirichlet boundary value problems, with resonance at infinity and zero. Résumé. Par un procédé de minimax et application de la Théorie de Morse, en calculant les groupes critiques en zéro, nousétablissons l'existence d'une solution non triviale pour une classe de problèmes de Dirichlet, avec résonanceà l'infini et en zéro..

Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth

International Journal of Differential Equations

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f in Ω, u=0 on ∂Ω,}, where 0<q<1, γ>0 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation.

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

Communications on Pure & Applied Analysis

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by Φ-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.

Quasilinear elliptic equations with natural growth

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 .

On a nonhomogeneous and singular quasilinear equation involving critical growth inR2

Computers & Mathematics with Applications, 2017

This paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form − ∆u + V (x)u − ∆(u 2)u = g(x, u) |x| a + h(x) in R 2 , where a ∈ [0, 2), V (x) is a continuous positive potential bounded away from zero and which can be ''large'' at infinity, the nonlinearity g(x, s) is allowed to enjoy the critical exponential growth with respect to the Trudinger-Moser inequality and the nonhomogeneous term h belongs to L q (R 2) for some q ∈ (1, 2]. By combining variational arguments in a nonstandard Orlicz space context with a singular version of the Trudinger-Moser inequality, we obtain the existence of two distinct solutions when ∥h∥ q is sufficiently small. Schrödinger equations of this type have been studied as models of several physical phenomena.

Positive and nodal solutions for an elliptic equation with critical growth

Communications in Contemporary Mathematics, 2016

We consider the problem [Formula: see text] where [Formula: see text] is a bounded smooth domain, [Formula: see text], [Formula: see text], [Formula: see text]. Under some suitable conditions on the continuous potential [Formula: see text] and on the parameter [Formula: see text], we obtain one nodal solution for [Formula: see text] and one positive solution for [Formula: see text].

Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

Rendiconti Lincei - Matematica e Applicazioni

We discuss recent advances in the theory of quasilinear equations of the type −∆ p u = σu q in R n , in the case 0 < q < p − 1, where σ is a nonnegative measurable function, or measure, for the p-Laplacian ∆ p u = div(|∇u| p−2 ∇u), as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions u ∈ BMO(R n), u ∈ L r loc (R n), etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems. Contents 1. Introduction 1 2. Nonlinear potentials 9 3. Main lemmas 11 4. Proofs of the main theorems and corollaries 19 References 26

On a class of nonlinear elliptic equations with fast increasing weight and critical growth

Journal of Differential Equations, 2010

We are concerned with the existence of rapidly decaying solutions for the equation −div K (x)∇u = λK (x)|x| β |u| q−2 u + K (x)|u| 2 * −2 u, x ∈ R N , where N 3, 2 q < 2 * := 2N/(N − 2), λ > 0 is a parameter, K (x) := exp(|x| α /4), α 2 and the number β is given by β := (α − 2) (2 * −q) (2 * −2). We obtain a positive solution if 2 < q < 2 * and a sign changing solution if q = 2. The existence results depend on the values of the parameter λ. In the proofs we apply variational methods.