### Mathematics in Engineering

2021, Issue 6:1-28. doi: 10.3934/mine.2021044
Research article Special Issues

# Gradient Lagrangian systems and semilinear PDE

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

• 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 $p+1\in (2, 2^{*}_{N})$.

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

### Related Papers:

• 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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沈阳化工大学材料科学与工程学院 沈阳 110142

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