Existence results for non-autonomous elliptic boundary value problems
$$-Delta u(x) = lambda f(x, u);quad x in Omega$$ $$u(x) + alpha(x) frac{partial u(x)}{partial n} = 0;quad x in partial Omega$$ where $lambda > 0$, $Omega$ is a bounded region in $Bbb{R}^N$; $N geq 1$ with smooth boundary $partial Omega$, $alpha(x) geq 0$, $n$ is the outward unit normal, and $f$ i...
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| Format: | Article |
| Language: | English |
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Texas State University
1994-07-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/1994/04/abstr.html |
| _version_ | 1819199709212835840 |
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| author | V. Anuradha S. Dickens R. Shivaji |
| author_facet | V. Anuradha S. Dickens R. Shivaji |
| author_sort | V. Anuradha |
| collection | DOAJ |
| description | $$-Delta u(x) = lambda f(x, u);quad x in Omega$$ $$u(x) + alpha(x) frac{partial u(x)}{partial n} = 0;quad x in partial Omega$$ where $lambda > 0$, $Omega$ is a bounded region in $Bbb{R}^N$; $N geq 1$ with smooth boundary $partial Omega$, $alpha(x) geq 0$, $n$ is the outward unit normal, and $f$ is a smooth function such that it has either sublinear or restricted linear growth in $u$ at infinity, uniformly in $x$. We also consider $f$ such that $f(x, u) u leq 0$ uniformly in $x$, when $|u|$ is large. Without requiring any sign condition on $f(x, 0)$, thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given $lambda in (lambda_{n}, lambda_{n + 1})$ where $lambda_{k}$ is the $k$-th eigenvalue of $-Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $lambda$ near $lambda_{1}$, and for $lambda$ large when $f$ is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $lambda$ small. |
| first_indexed | 2024-12-23T03:20:39Z |
| format | Article |
| id | doaj.art-15cc3d0fabd9485b92f9b768144c270e |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-23T03:20:39Z |
| publishDate | 1994-07-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-15cc3d0fabd9485b92f9b768144c270e2022-12-21T18:02:00ZengTexas State UniversityElectronic Journal of Differential Equations1072-66911994-07-01199404110Existence results for non-autonomous elliptic boundary value problemsV. AnuradhaS. DickensR. Shivaji$$-Delta u(x) = lambda f(x, u);quad x in Omega$$ $$u(x) + alpha(x) frac{partial u(x)}{partial n} = 0;quad x in partial Omega$$ where $lambda > 0$, $Omega$ is a bounded region in $Bbb{R}^N$; $N geq 1$ with smooth boundary $partial Omega$, $alpha(x) geq 0$, $n$ is the outward unit normal, and $f$ is a smooth function such that it has either sublinear or restricted linear growth in $u$ at infinity, uniformly in $x$. We also consider $f$ such that $f(x, u) u leq 0$ uniformly in $x$, when $|u|$ is large. Without requiring any sign condition on $f(x, 0)$, thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given $lambda in (lambda_{n}, lambda_{n + 1})$ where $lambda_{k}$ is the $k$-th eigenvalue of $-Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $lambda$ near $lambda_{1}$, and for $lambda$ large when $f$ is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $lambda$ small.http://ejde.math.txstate.edu/Volumes/1994/04/abstr.htmlElliptic boundary value problemssemipositone. |
| spellingShingle | V. Anuradha S. Dickens R. Shivaji Existence results for non-autonomous elliptic boundary value problems Electronic Journal of Differential Equations Elliptic boundary value problems semipositone. |
| title | Existence results for non-autonomous elliptic boundary value problems |
| title_full | Existence results for non-autonomous elliptic boundary value problems |
| title_fullStr | Existence results for non-autonomous elliptic boundary value problems |
| title_full_unstemmed | Existence results for non-autonomous elliptic boundary value problems |
| title_short | Existence results for non-autonomous elliptic boundary value problems |
| title_sort | existence results for non autonomous elliptic boundary value problems |
| topic | Elliptic boundary value problems semipositone. |
| url | http://ejde.math.txstate.edu/Volumes/1994/04/abstr.html |
| work_keys_str_mv | AT vanuradha existenceresultsfornonautonomousellipticboundaryvalueproblems AT sdickens existenceresultsfornonautonomousellipticboundaryvalueproblems AT rshivaji existenceresultsfornonautonomousellipticboundaryvalueproblems |