Existence of three nontrivial solutions for asymptotically -linear noncoercive -Laplacian equations

2011 ◽  
Vol 74 (16) ◽  
pp. 5314-5326 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Eugénio M. Rocha
Keyword(s):  
Nontrivial Solutions ◽  
Asymptotically Linear ◽  
Laplacian Equations

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu
Keyword(s):  
Bounded Domain ◽  
Exponential Growth ◽  
Nonnegative Solution ◽  
Nontrivial Solutions ◽  
Critical Exponential Growth ◽  
Trudinger Inequality ◽  
Laplacian Equations

AbstractLet Ω be a bounded domain in ℝwhen f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

scholarly journals Multiple Results to Some Biharmonic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xingdong Tang ◽  
Jihui Zhang
Keyword(s):  
Elliptic Problem ◽  
Degree Theory ◽  
Critical Groups ◽  
Biharmonic Operator ◽  
Nonlinear Elliptic Problem ◽  
Topological Degree Theory ◽  
Nontrivial Solutions ◽  
Asymptotically Linear ◽  
Nonlinear Elliptic ◽  
Biharmonic Problems

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

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