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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2207.10831 |
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| _version_ | 1866917999429025792 |
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| author | Rendón, Fiorella Soares, Mayra |
| author_facet | Rendón, Fiorella Soares, Mayra |
| contents | We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations
\begin{align*}\tag{$P_λ$}
-\mathrm{div}(A(x)Du)=c_λ(x)u+( M(x)Du,Du)+h(x),\qquad
u\in H_0^1(Ω)\cap L^\infty(Ω),
\end{align*}
where $Ω\subset\mathbb{R}^n$, $n\geq 3$, is a bounded domain with a low-regularity boundary
$\partialΩ$.
The coefficients $c, h \in L^p(Ω)$ for some $p > n$, with $c^\pm \geq 0$ and $c_λ(x) := λc^+(x) - c^-(x)$ for a real parameter $λ$. The matrix $A(x)$ is uniformly positive definite and bounded, while $M(x)$ is positive definite and bounded.
Under suitable assumptions, we characterize the solution continuum of $(P_λ)$, including its bifurcation points. We establish existence and uniqueness results in the coercive case ($λ\leq 0$) and prove multiplicity results in the non-coercive case ($λ> 0$).
\bigskip
\textbf{Keywords}: Quasilinear elliptic equations, quadratic growth on the gradient,
sub and super solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_10831 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient Rendón, Fiorella Soares, Mayra Analysis of PDEs We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_λ$} -\mathrm{div}(A(x)Du)=c_λ(x)u+( M(x)Du,Du)+h(x),\qquad u\in H_0^1(Ω)\cap L^\infty(Ω), \end{align*} where $Ω\subset\mathbb{R}^n$, $n\geq 3$, is a bounded domain with a low-regularity boundary $\partialΩ$. The coefficients $c, h \in L^p(Ω)$ for some $p > n$, with $c^\pm \geq 0$ and $c_λ(x) := λc^+(x) - c^-(x)$ for a real parameter $λ$. The matrix $A(x)$ is uniformly positive definite and bounded, while $M(x)$ is positive definite and bounded. Under suitable assumptions, we characterize the solution continuum of $(P_λ)$, including its bifurcation points. We establish existence and uniqueness results in the coercive case ($λ\leq 0$) and prove multiplicity results in the non-coercive case ($λ> 0$). \bigskip \textbf{Keywords}: Quasilinear elliptic equations, quadratic growth on the gradient, sub and super solutions. |
| title | Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2207.10831 |