### Abstract

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem(P_{ε}){(- Δ u + u = Q (x) [f (u) + ε g (u)], x ∈ R^{N}, N ≥ 2,; u ≥ 0,; u (x) → 0 as | x | → ∞,) where Q : R^{N} → R is a radial, positive potential, f : [0, ∞) → R is a continuous nonlinearity which oscillates near the origin or at infinity and g : [0, ∞) → R is any arbitrarily continuous function with g (0) = 0. Our aim is to prove that: (a) the unperturbed problem (P_{0}), i.e. ε = 0 in (P_{ε}), has infinitely many distinct solutions; (b) the number of distinct solutions for (P_{ε}) becomes greater and greater whenever | ε | is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent in the sense that they are deducible from each other. Various properties of the solutions are also described in L^{∞}- and H^{1}-norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.

Original language | English |
---|---|

Pages (from-to) | 3849-3868 |

Number of pages | 20 |

Journal | Journal of Differential Equations |

Volume | 245 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 15 2008 |

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### Keywords

- Arbitrarily many solutions
- Perturbed elliptic problem
- Symmetric criticality
- Szulkin-type functional

### ASJC Scopus subject areas

- Analysis

### Cite this

**Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms.** / Kristály, A.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 245, no. 12, pp. 3849-3868. https://doi.org/10.1016/j.jde.2008.05.014

}

TY - JOUR

T1 - Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms

AU - Kristály, A.

PY - 2008/12/15

Y1 - 2008/12/15

N2 - We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem(Pε){(- Δ u + u = Q (x) [f (u) + ε g (u)], x ∈ RN, N ≥ 2,; u ≥ 0,; u (x) → 0 as | x | → ∞,) where Q : RN → R is a radial, positive potential, f : [0, ∞) → R is a continuous nonlinearity which oscillates near the origin or at infinity and g : [0, ∞) → R is any arbitrarily continuous function with g (0) = 0. Our aim is to prove that: (a) the unperturbed problem (P0), i.e. ε = 0 in (Pε), has infinitely many distinct solutions; (b) the number of distinct solutions for (Pε) becomes greater and greater whenever | ε | is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent in the sense that they are deducible from each other. Various properties of the solutions are also described in L∞- and H1-norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.

AB - We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem(Pε){(- Δ u + u = Q (x) [f (u) + ε g (u)], x ∈ RN, N ≥ 2,; u ≥ 0,; u (x) → 0 as | x | → ∞,) where Q : RN → R is a radial, positive potential, f : [0, ∞) → R is a continuous nonlinearity which oscillates near the origin or at infinity and g : [0, ∞) → R is any arbitrarily continuous function with g (0) = 0. Our aim is to prove that: (a) the unperturbed problem (P0), i.e. ε = 0 in (Pε), has infinitely many distinct solutions; (b) the number of distinct solutions for (Pε) becomes greater and greater whenever | ε | is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent in the sense that they are deducible from each other. Various properties of the solutions are also described in L∞- and H1-norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.

KW - Arbitrarily many solutions

KW - Perturbed elliptic problem

KW - Symmetric criticality

KW - Szulkin-type functional

UR - http://www.scopus.com/inward/record.url?scp=54149093120&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54149093120&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.05.014

DO - 10.1016/j.jde.2008.05.014

M3 - Article

AN - SCOPUS:54149093120

VL - 245

SP - 3849

EP - 3868

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -