On a Class of Nonlinear Schrödinger Equations in R2 Involving Critical Growth (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.

Positive Solutions for a Quasilinear Schrödinger Equation with Critical Growth

Journal of Dynamics and Differential Equations, 2011

We consider the quasilinear problem −ε p div(|∇u| p−2 ∇u) + V (z)u p−1 = f (u) + u p * −1 , u ∈ W 1,p (R N), where ε > 0 is a small parameter, 1 < p < N , p * = N p/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory.

Infinitely many solutions for the Schrödinger equations in with critical growth

Journal of Differential Equations, 2012

We consider the following nonlinear problem in RN(0.1){−Δu+V(|y|)u=uN+2N−2,u&amp;amp;amp;gt;0, in RN;u∈H1(RN), where V(r) is a bounded non-negative function, N⩾5. We show that if r2V(r) has a local maximum point, or local minimum point r0&amp;amp;amp;gt;0 with V(r0)&amp;amp;amp;gt;0, then (0.1) has infinitely many non-radial solutions, whose energy can be made arbitrarily large. As an application, we show that the solution set of the

On a quasilinear nonhomogeneous elliptic equation with critical growth in

Journal of Differential Equations, 2009

In this paper, Ekeland variational principle, mountain-pass theorem and a suitable Trudinger-Moser inequality are employed to establish sufficient conditions for the existence of solutions of quasilinear nonhomogeneous elliptic partial differential equations of the form − N u + V (x)|u| N−2 u = f (x, u) + εh(x) in R N , N 2, where V : R N → R is a continuous potential, f : R N × R → R behaves like exp(α|u| N/(N−1)) when |u| → ∞ and h ∈ (W 1,N (R N)) * = W −1,N , h ≡ 0. As an application of this result we have existence of two positive solutions for the following elliptic problem involving critical growth − u + V (x)u = λu e u 2 − 1 + εh(x) in R 2 , where λ > 0 is large, ε > 0 is a small parameter and h ∈ H −1 (R 2), h 0.

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.

Kirchhoff–Schrödinger equations in ℝ2 with critical exponential growth and indefinite potential

Communications in Contemporary Mathematics, 2020

We prove the existence of ground state solution for the nonlocal problem [Formula: see text] where [Formula: see text] is a Kirchhoff type function, [Formula: see text] may be negative and noncoercive, [Formula: see text] is locally bounded and the function [Formula: see text] has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case [Formula: see text]. In the proofs, we apply Variational Methods besides a new Trudinger–Moser type inequality.