Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains

2018 ◽  
Vol 59 (3) ◽  
pp. 031504 ◽  
Author(s):  
Huxiao Luo ◽  
Xianhua Tang ◽  
Zu Gao
Keyword(s):  
Ground State ◽  
Bounded Domains ◽  
Kirchhoff Equations ◽  
Sign Changing Solutions

scholarly journals Ground state sign-changing solutions and infinitely many solutions for fractional logarithmic Schrödinger equations in bounded domains

2021 ◽  
pp. 1-14
Author(s):  
Yonghui Tong ◽  
Hui Guo ◽  
Giovany Figueiredo
Keyword(s):  
Ground State ◽  
State Energy ◽  
Monotonicity Condition ◽  
Infinitely Many Solutions ◽  
Bounded Domains ◽  
Nontrivial Solutions ◽  
Deformation Lemma ◽  
Quantitative Deformation Lemma ◽  
Sign Changing Solutions ◽  
New Inequalities

We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.

scholarly journals Symmetry and concentration behavior of ground state in axially symmetric domains

2004 ◽  
Vol 2004 (12) ◽  
pp. 1019-1030
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Ground State ◽  
Semilinear Elliptic Equation ◽  
Bounded Domains ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Axially Symmetric ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Necessary And Sufficient ◽  
Concentration Behavior

We letΩ(r)be the axially symmetric bounded domains which satisfy some suitable conditions, then the ground-state solutions of the semilinear elliptic equation inΩ(r)are nonaxially symmetric and concentrative on one side. Furthermore, we prove the necessary and sufficient condition for the symmetry of ground-state solutions.

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