Boundary concentration phenomena for a singularly perturbed elliptic problem

2002 ◽  
Vol 55 (12) ◽  
pp. 1507-1568 ◽  
Author(s):  
Andrea Malchiodi ◽  
Marcelo Montenegro
Keyword(s):  
Elliptic Problem ◽  
Singularly Perturbed ◽  
Boundary Concentration ◽  
Concentration Phenomena

scholarly journals Singularly Perturbed Fractional Schrödinger Equation Involving a General Critical Nonlinearity

2018 ◽  
Vol 18 (3) ◽  
pp. 487-499 ◽  
Author(s):  
Hua Jin ◽  
Wenbin Liu ◽  
Jianjun Zhang
Keyword(s):  
Variational Approach ◽  
Bound State ◽  
Singularly Perturbed ◽  
Critical Nonlinearity ◽  
Bound State Solution ◽  
Critical Nonlinearities ◽  
Concentration Phenomena ◽  
Fractional Schrödinger Equations ◽  
Concentration Result ◽  
Made In

AbstractIn this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem:\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=f(u)\quad\text{in }\mathbb{R}^{N},where{N>2s}and the nonlinearityfhas critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point ofVas{\varepsilon\rightarrow 0}. Our result improves the study made in [X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 91], in the sense that, in the present paper, theAmbrosetti–Rabinowitzcondition and themonotonicitycondition on{f(t)/t}are not required.

Nodal Solutions for a Singularly Perturbed Nonlinear Elliptic Problem on Riemannian Manifolds

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Scalar Curvature ◽  
Positive Solution ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Singularly Perturbed ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic

AbstractLet (M, g) be a smooth compact Riemannian N−manifold, N ≥ 2. We show that if the scalar curvature of g is not constant, the problem−εhas a positive solution with two positive peaks ξ

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