Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials
| Autor: | Marino Badiale, Sergio Rolando, Michela Guida |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Continuous function Computer Science::Information Retrieval Applied Mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Singular and vanishing potentials Singular and vanishing potentials sum of Lebesgue spaces double-power nonlinearity Elliptic curve TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Compact space Mathematics - Analysis of PDEs sum of Lebesgue spaces ComputingMethodologies_DOCUMENTANDTEXTPROCESSING FOS: Mathematics Computer Science::General Literature double-power nonlinearity ComputingMilieux_MISCELLANEOUS Analysis Mathematics Analysis of PDEs (math.AP) |
| Popis: | Given $N\geq 3$, $1 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$. Comment: arXiv admin note: text overlap with arXiv:1609.05556, arXiv:1510.03879, arXiv:1403.3803 |
| Databáze: | OpenAIRE |
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