scholarly journals Weak Linking Theorems and Schrödinger Equations with Critical Sobolev Exponent

2003 ◽  
Vol 9 ◽  
pp. 601-619 ◽  
Author(s):  
Martin Schechter ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Sobolev Exponent

A class of asymptotically periodic fractional Schrödinger equations with critical growth

2018 ◽  
Vol 20 (03) ◽  
pp. 1750011
Author(s):  
Manassés de Souza ◽  
Yane Lísley Araújo
Keyword(s):  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Positive Potential ◽  
Fractional Schrödinger Equations ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we study a class of fractional Schrödinger equations in [Formula: see text] of the form [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is the critical Sobolev exponent, [Formula: see text] is a positive potential bounded away from zero, and the nonlinearity [Formula: see text] behaves like [Formula: see text] at infinity for some [Formula: see text], and does not satisfy the usual Ambrosetti–Rabinowitz condition. We also assume that the potential [Formula: see text] and the nonlinearity [Formula: see text] are asymptotically periodic at infinity. We prove the existence of at least one solution [Formula: see text] by combining a version of the mountain-pass theorem and a result due to Lions for critical growth.

WEAK POTENTIAL CONDITIONS FOR SCHRÖDINGER EQUATIONS WITH CRITICAL NONLINEARITIES

2015 ◽  
Vol 100 (2) ◽  
pp. 272-288
Author(s):  
X. H. TANG ◽  
SITONG CHEN
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Measure ◽  
Critical Sobolev Exponent ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities ◽  
Sobolev Exponent ◽  
Image Position ◽  
Potential Conditions

In this paper, we prove the existence of nontrivial solutions to the following Schrödinger equation with critical Sobolev exponent: $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-{\rm\Delta}u+V(x)u=K(x)|u|^{2^{\ast }-2}u+f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N})\end{array}\right.\end{eqnarray}$$ under assumptions that (i) $V(x_{0})<0$ for some $x_{0}\in \mathbb{R}^{N}$ and (ii) there exists $b>0$ such that the set ${\mathcal{V}}_{b}:=\{x\in \mathbb{R}^{N}:V(x)<b\}$ has finite measure, in addition to some common assumptions on $K$ and $f$, where $N\geq 3$, $2^{\ast }=2N/(N-2)$.

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