Gradient Lagrangian systems and semilinear PDE

Francesca G. Alessio; Piero Montecchiari; Francesca G. Alessio; Piero Montecchiari

doi:10.3934/mine.2021044

Mathematics in Engineering

2021, Volume 3, Issue 6:1-28. doi: 10.3934/mine.2021044

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Gradient Lagrangian systems and semilinear PDE

Francesca G. Alessio ¹,
Piero Montecchiari ^{2
,
,}

1.
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, I-60131, Italy
2.
Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, I-60131, Italy

Received: 23 August 2020 Accepted: 20 October 2020 Published: 06 November 2020
MSC : Primary 35B08, 35J60; Secondary 34C37, 35B40, 35J20

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We survey some results about multiplicity of certain classes of entire solutions to semilinear elliptic equations or systems of the form $-\Delta u = F_{u}(x, u)$, $x\in\mathbb{R}^{N+1}$, including the Allen Cahn or the stationary Nonlinear Schr\"odinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional $V(u) = \tfrac 12\|\nabla u\|_{H^{1}(\mathbb{R}^{N})}^{2}-\tfrac 1{p+1}\| u\|_{L^{p+1}(\mathbb{R}^{N})}^{p+1}$ in relation to the value of the exponent $p+1\in (2, 2^{*}_{N})$.
- semilinear elliptic equations,
- variational methods,
- energy constraints,
- Jacobi distance
Citation: Francesca G. Alessio, Piero Montecchiari. Gradient Lagrangian systems and semilinear PDE[J]. Mathematics in Engineering, 2021, 3(6): 1-28. doi: 10.3934/mine.2021044

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Abstract

We survey some results about multiplicity of certain classes of entire solutions to semilinear elliptic equations or systems of the form $-\Delta u = F_{u}(x, u)$, $x\in\mathbb{R}^{N+1}$, including the Allen Cahn or the stationary Nonlinear Schr\"odinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional $V(u) = \tfrac 12\|\nabla u\|_{H^{1}(\mathbb{R}^{N})}^{2}-\tfrac 1{p+1}\| u\|_{L^{p+1}(\mathbb{R}^{N})}^{p+1}$ in relation to the value of the exponent $p+1\in (2, 2^{*}_{N})$.

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