Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators
Abstract The paper deals with the existence and non-existence of solutions of the following nonlinear non-autonomous boundary value problem governed by the p-Laplacian operator: ( P ) { ( h ( t , x ( t ) ) | x ′ ( t ) | p − 2 x ′ ( t ) ) ′ = g ( t , x ( t ) , x ′ ( t ) ) a.e. t ∈ R , x ( − ∞ ) = a...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-11-01
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| Series: | Boundary Value Problems |
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| Online Access: | http://link.springer.com/article/10.1186/s13661-019-1291-0 |
| _version_ | 1828961004154781696 |
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| author | Milena Petrini |
| author_facet | Milena Petrini |
| author_sort | Milena Petrini |
| collection | DOAJ |
| description | Abstract The paper deals with the existence and non-existence of solutions of the following nonlinear non-autonomous boundary value problem governed by the p-Laplacian operator: ( P ) { ( h ( t , x ( t ) ) | x ′ ( t ) | p − 2 x ′ ( t ) ) ′ = g ( t , x ( t ) , x ′ ( t ) ) a.e. t ∈ R , x ( − ∞ ) = a , x ( + ∞ ) = b $$(P) \quad \textstyle\begin{cases} (h(t,x(t)) \vert x'(t) \vert ^{p-2} x'(t))' = g(t,x(t),x'(t)) \quad \text{a.e. } t\in \mathbb{R}, \\ x(-\infty )=a, \qquad x(+\infty )= b \end{cases} $$ with a < b $a< b$ , where a is a positive, continuous function and g is a Caratheódory nonlinear function. We prove an existence result, underlying the relationship between the behavior of g ( t , x , ⋅ ) $g(t,x,\cdot )$ as y → 0 $y\to 0$ related to that of g ( ⋅ , x , y ) $g(\cdot ,x,y)$ and h ( ⋅ , x ) $h(\cdot ,x) $ as | t | → + ∞ $|t|\to +\infty $ . |
| first_indexed | 2024-12-14T09:37:39Z |
| format | Article |
| id | doaj.art-608a7c247575498e8c9a9da27926df2b |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-14T09:37:39Z |
| publishDate | 2019-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-608a7c247575498e8c9a9da27926df2b2022-12-21T23:07:52ZengSpringerOpenBoundary Value Problems1687-27702019-11-012019111310.1186/s13661-019-1291-0Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operatorsMilena Petrini0Dipartimento di Scienze Matematiche, Università Politecnica delle MarcheAbstract The paper deals with the existence and non-existence of solutions of the following nonlinear non-autonomous boundary value problem governed by the p-Laplacian operator: ( P ) { ( h ( t , x ( t ) ) | x ′ ( t ) | p − 2 x ′ ( t ) ) ′ = g ( t , x ( t ) , x ′ ( t ) ) a.e. t ∈ R , x ( − ∞ ) = a , x ( + ∞ ) = b $$(P) \quad \textstyle\begin{cases} (h(t,x(t)) \vert x'(t) \vert ^{p-2} x'(t))' = g(t,x(t),x'(t)) \quad \text{a.e. } t\in \mathbb{R}, \\ x(-\infty )=a, \qquad x(+\infty )= b \end{cases} $$ with a < b $a< b$ , where a is a positive, continuous function and g is a Caratheódory nonlinear function. We prove an existence result, underlying the relationship between the behavior of g ( t , x , ⋅ ) $g(t,x,\cdot )$ as y → 0 $y\to 0$ related to that of g ( ⋅ , x , y ) $g(\cdot ,x,y)$ and h ( ⋅ , x ) $h(\cdot ,x) $ as | t | → + ∞ $|t|\to +\infty $ .http://link.springer.com/article/10.1186/s13661-019-1291-0Boundary value problemsUnbounded domainsHeteroclinic solutionsNonlinear differential operatorsp-Laplacian operatorΦ-Laplacian operator |
| spellingShingle | Milena Petrini Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators Boundary Value Problems Boundary value problems Unbounded domains Heteroclinic solutions Nonlinear differential operators p-Laplacian operator Φ-Laplacian operator |
| title | Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators |
| title_full | Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators |
| title_fullStr | Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators |
| title_full_unstemmed | Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators |
| title_short | Solvability of some boundary value problems involving p-Laplacian and non-autonomous differential operators |
| title_sort | solvability of some boundary value problems involving p laplacian and non autonomous differential operators |
| topic | Boundary value problems Unbounded domains Heteroclinic solutions Nonlinear differential operators p-Laplacian operator Φ-Laplacian operator |
| url | http://link.springer.com/article/10.1186/s13661-019-1291-0 |
| work_keys_str_mv | AT milenapetrini solvabilityofsomeboundaryvalueproblemsinvolvingplaplacianandnonautonomousdifferentialoperators |