Geodesic
Eigenvalues of −Δp − Δq Under Neumann Boundary Condition
Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 606-616

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The eigenvalue problem $-{{\Delta }_{p}}u-{{\Delta }_{q}}u=\lambda {{\left| u \right|}^{q-2}}u$ with $p\,\in \,\left( 1,\,\infty\right),\,q\,\in \,\left( 2,\,\infty\right),\,p\ne \,q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ${{\mathbb{R}}^{N}}$ with $N\,\ge \,2$ . A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $\left( {{\lambda }_{1,}}+\infty\right)$ plus an isolated point $\lambda \,=\,0$ . This comprehensive result is strongly related to our framework, which is complementary to the well-known case $p\,=\,q\,\ne \,2$ for which a full description of the set of eigenvalues is still unavailable.
DOI : 10.4153/CMB-2016-025-2
Mots-clés : 35J60, 35J92, 46E30, 49R05, eigenvalue problem, Sobolev space, Nehari manifold, variational methods 
Mihăilescu, Mihai; Moroşanu, Gheorghe. Eigenvalues of −Δp − Δq Under Neumann Boundary Condition. Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 606-616. doi: 10.4153/CMB-2016-025-2
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     author = {Mih\u{a}ilescu, Mihai and Moro\c{s}anu, Gheorghe},
     title = {Eigenvalues of {\ensuremath{-}\ensuremath{\Delta}p} \ensuremath{-} {\ensuremath{\Delta}q} {Under} {Neumann} {Boundary} {Condition}},
     journal = {Canadian mathematical bulletin},
     pages = {606--616},
     year = {2016},
     volume = {59},
     number = {3},
     doi = {10.4153/CMB-2016-025-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-025-2/}
}
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