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 $N\geq 3$, $2^{\ast }=2N/(N-2)$.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Large Energy Bubble Solutions for Schrödinger Equation with Supercritical Growth

2021 ◽  
Vol 21 (2) ◽  
pp. 421-445
Author(s):  
Yuxia Guo ◽  
Ting Liu
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Nonlinear Schrodinger Equation ◽  
Large Energy ◽  
Nonlinear Schrödinger ◽  
Sobolev Exponent

Abstract We consider the following nonlinear Schrödinger equation involving supercritical growth: { - Δ ⁢ u + V ⁢ ( y ) ⁢ u = Q ⁢ ( y ) ⁢ u 2 * - 1 + ε in  ⁢ ℝ N , u > 0 , u ∈ H 1 ⁢ ( ℝ N ) , \left\{\begin{aligned} &\displaystyle{-}\Delta u+V(y)u=Q(y)u^{2^{*}-1+% \varepsilon}&&\displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,\quad u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.{} where 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} is the critical Sobolev exponent, N ≥ 5 {N\geq 5} , and V ⁢ ( y ) {V(y)} and Q ⁢ ( y ) {Q(y)} are bounded nonnegative functions in ℝ N {\mathbb{R}^{N}} . By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions V and Q, we prove that for ε > 0 {\varepsilon>0} is small enough, problem ( * ) {(*)} has large number of bubble solutions whose functional energy is in the order ε - N - 4 ( N - 2 ) 2 . {\varepsilon^{-\frac{N-4}{(N-2)^{2}}}.}

scholarly journals Existence of Nontrivial Solutions for Periodic Schrödinger Equations with New Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shaowei Chen ◽  
Dawei Zhang
Keyword(s):  
Schrödinger Equation ◽  
Nontrivial Solution ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions

We study the Schrödinger equation:-Δu+Vxu+fx,u=0,  u∈H1(RN), whereVis1-periodic andfis1-periodic in thex-variables;0is in a gap of the spectrum of the operator-Δ+V. We prove that, under some new assumptions forf, this equation has a nontrivial solution. Our assumptions for the nonlinearityfare very weak and greatly different from the known assumptions in the literature.

Standing waves of modified Schrödinger equations coupled with the Chern–Simons gauge theory

2019 ◽  
Vol 150 (4) ◽  
pp. 1915-1936 ◽  
Author(s):  
Pietro d'Avenia ◽  
Alessio Pomponio ◽  
Tatsuya Watanabe
Keyword(s):  
Gauge Theory ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Chern Simons ◽  
Constraint Minimization

AbstractWe are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.

scholarly journals Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan Meng ◽  
Xianjiu Huang ◽  
Jianhua Chen
Keyword(s):  
Schrödinger Equation ◽  
Variational Methods ◽  
Schrodinger Equation ◽  
Infinitely Many Solutions ◽  
Type Condition ◽  
Nontrivial Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Sobolev Exponent

AbstractIn this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ − div ( a ( x , ∇ u ) ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( x , u ) in  R N , where $N\geq 3$ N ≥ 3 , $1< p< N$ 1 < p < N , $-\infty <\alpha <\frac{N-p}{p}$ − ∞ < α < N − p p , $\alpha \leq e\leq \alpha +1$ α ≤ e ≤ α + 1 , $d=1+\alpha -e$ d = 1 + α − e , $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$ p ∗ : = p ∗ ( α , e ) = N p N − d p (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.

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