Existence of a ground state solution for a nonlinear scalar field equation with critical growth

2011 ◽  
Vol 43 (3-4) ◽  
pp. 537-554 ◽  
Author(s):  
Claudianor O. Alves ◽  
Marco A. S. Souto ◽  
Marcelo Montenegro
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Ground State ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Scalar Field Equation ◽  
Nonlinear Scalar Field

scholarly journals Existence of a Positive Solution to a Nonlinear Scalar Field Equation with Zero Mass at Infinity

2018 ◽  
Vol 18 (4) ◽  
pp. 745-762 ◽  
Author(s):  
Mónica Clapp ◽  
Liliane A. Maia
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Positive Solution ◽  
Ground State ◽  
Bound State ◽  
Zero Mass ◽  
Scalar Field Equation ◽  
Nonlinear Scalar Field ◽  
Decay Assumption

AbstractWe establish the existence of a positive solution to the problem-\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}),for {N\geq 3}, when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.

scholarly journals Positive multipeak solutions to a zero mass problem in exterior domains

2019 ◽  
pp. 1950062
Author(s):  
Mónica Clapp ◽  
Liliane A. Maia ◽  
Benedetta Pellacci
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Mass Formula ◽  
Exterior Domains ◽  
Mass Problem ◽  
Zero Mass ◽  
Scalar Field Equation ◽  
Nonlinear Scalar Field

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text], and the nonlinearity [Formula: see text] is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as [Formula: see text].

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