Existence results for a borderline case of a class of p-Laplacian problems
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2024
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| author | Candela, Anna Maria Perera, Kanishka Salvatore, Addolorata |
| author_facet | Candela, Anna Maria Perera, Kanishka Salvatore, Addolorata |
| contents | The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically ``linear'' problem \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\\ \qquad\qquad\qquad =\ μ|u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $Ω$,}\\ u = 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(Ω)$ and far away from 0, $μ\in \mathbb{R}$, and the ``perturbation'' term $g(x,t)$ is a Carathéodory function on $Ω\times \mathbb{R}$ which grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that $g(x,t) \approx ν|t|^{p-2} t$ as $t \to 0$. By introducing suitable thresholds for the parameters $ν$ and $μ$, which are related to the coefficients $A_0(x)$, respectively $A(x)$, under suitable hypotheses on $g(x,t)$, the existence of a nontrivial weak solution is proved if either $ν$ is large enough with $μ$ small enough or $ν$ is small enough with $μ$ large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_12954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence results for a borderline case of a class of p-Laplacian problems Candela, Anna Maria Perera, Kanishka Salvatore, Addolorata Analysis of PDEs 35J62, 35J92, 47J30, 35Q55, 58E30 The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically ``linear'' problem \[ \left\{ \begin{array}{ll} - {\rm div} \left[\left(A_0(x) + A(x) |u|^{ps}\right) |\nabla u|^{p-2} \nabla u\right] + s\ A(x) |u|^{ps-2} u\ |\nabla u|^p &\\ \qquad\qquad\qquad =\ μ|u|^{p (s + 1) -2} u + g(x,u) & \hbox{in $Ω$,}\\ u = 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N \ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\infty(Ω)$ and far away from 0, $μ\in \mathbb{R}$, and the ``perturbation'' term $g(x,t)$ is a Carathéodory function on $Ω\times \mathbb{R}$ which grows as $|t|^{r-1}$ with $1\le r < p (s + 1)$ and is such that $g(x,t) \approx ν|t|^{p-2} t$ as $t \to 0$. By introducing suitable thresholds for the parameters $ν$ and $μ$, which are related to the coefficients $A_0(x)$, respectively $A(x)$, under suitable hypotheses on $g(x,t)$, the existence of a nontrivial weak solution is proved if either $ν$ is large enough with $μ$ small enough or $ν$ is small enough with $μ$ large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used. |
| title | Existence results for a borderline case of a class of p-Laplacian problems |
| topic | Analysis of PDEs 35J62, 35J92, 47J30, 35Q55, 58E30 |
| url | https://arxiv.org/abs/2408.12954 |