The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems
In this paper, we consider local Dirichlet problems driven by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>r</mi><mo>(</mo><mi>u</mi><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/2/237 |
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| author | Calogero Vetro |
| author_facet | Calogero Vetro |
| author_sort | Calogero Vetro |
| collection | DOAJ |
| description | In this paper, we consider local Dirichlet problems driven by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>r</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>s</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></semantics></math></inline-formula> are real continuous functions and we have dependence on the solution <i>u</i>. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument. |
| first_indexed | 2024-03-10T01:01:19Z |
| format | Article |
| id | doaj.art-2d81f667ed01491fb8073d41a4d73795 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:01:19Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2d81f667ed01491fb8073d41a4d737952023-11-23T14:34:26ZengMDPI AGMathematics2227-73902022-01-0110223710.3390/math10020237The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-ProblemsCalogero Vetro0Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, ItalyIn this paper, we consider local Dirichlet problems driven by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>r</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>s</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></semantics></math></inline-formula> are real continuous functions and we have dependence on the solution <i>u</i>. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.https://www.mdpi.com/2227-7390/10/2/237(<i>r</i>(<i>u</i>),<i>s</i>(u))-Laplacian operatorPalais-Smale conditionmonotone operatorregularized problemweak solution |
| spellingShingle | Calogero Vetro The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems Mathematics (<i>r</i>(<i>u</i>),<i>s</i>(u))-Laplacian operator Palais-Smale condition monotone operator regularized problem weak solution |
| title | The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems |
| title_full | The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems |
| title_fullStr | The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems |
| title_full_unstemmed | The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems |
| title_short | The Existence of Solutions for Local Dirichlet (<i>r</i>(<i>u</i>),<i>s</i>(<i>u</i>))-Problems |
| title_sort | existence of solutions for local dirichlet i r i i u i i s i i u i problems |
| topic | (<i>r</i>(<i>u</i>),<i>s</i>(u))-Laplacian operator Palais-Smale condition monotone operator regularized problem weak solution |
| url | https://www.mdpi.com/2227-7390/10/2/237 |
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