A dichotomy result for a modified Schrödinger equations on unbounded domains
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2025
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| author | Candela, Anna Maria Palmieri, Giuliana Salvatore, Addolorata |
| author_facet | Candela, Anna Maria Palmieri, Giuliana Salvatore, Addolorata |
| contents | This article aims to investigate the existence of bounded positive solutions of problem \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $Ω$,}\\ u\ = \ 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] with $A_t(x,t,ξ) = \frac{\partial A}{\partial t}(x,t,ξ)$, $a(x,t,ξ) = \nabla_ξA(x,t,ξ)$ for a given $A(x,t,ξ)$ which grows as $|ξ|^p + |t|^p$ , $p > 1$, where $Ω\subseteq \mathbb{R}^N$, $N \ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $Ω= \mathbb{R}^N$, which generalizes the modified Schrödinger equation \[ - {\rm div} ((A^*_1(x) + A^*_2(x)|u|^{s}) \nabla u) + \frac{s}2 A^*_2(x)\ |u|^{s - 2} u\ |\nabla u|^2 + u\ =\ |u|^{μ-2}u \quad\hbox{in $\mathbb{R}^3$.} \] Under suitable assumptions on $A(x,t,ξ)$ and $g(x,t)$, problem $(P)$ has a variational structure. Then, even in lack of radial symmetry hypotheses, one bounded positive solution of $(P)$ can be found by passing to the limit on a sequence $(u_k)_k$ of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant $\barλ > 0$ and a sequence of points $(y_k)_k \subset \mathbb{R}^N$ exist such that \[ |y_k| \to +\infty\qquad \hbox{and}\qquad \int_{B_1(y_k)} |u_k|^p dx \ge \barλ\quad \hbox{for all $k \ge 1$.} \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_18528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A dichotomy result for a modified Schrödinger equations on unbounded domains Candela, Anna Maria Palmieri, Giuliana Salvatore, Addolorata Analysis of PDEs 35J62, 35J92, 47J30, 35Q55, 58E30 This article aims to investigate the existence of bounded positive solutions of problem \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $Ω$,}\\ u\ = \ 0 & \hbox{on $\partialΩ$,} \end{array}\right.\] with $A_t(x,t,ξ) = \frac{\partial A}{\partial t}(x,t,ξ)$, $a(x,t,ξ) = \nabla_ξA(x,t,ξ)$ for a given $A(x,t,ξ)$ which grows as $|ξ|^p + |t|^p$ , $p > 1$, where $Ω\subseteq \mathbb{R}^N$, $N \ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $Ω= \mathbb{R}^N$, which generalizes the modified Schrödinger equation \[ - {\rm div} ((A^*_1(x) + A^*_2(x)|u|^{s}) \nabla u) + \frac{s}2 A^*_2(x)\ |u|^{s - 2} u\ |\nabla u|^2 + u\ =\ |u|^{μ-2}u \quad\hbox{in $\mathbb{R}^3$.} \] Under suitable assumptions on $A(x,t,ξ)$ and $g(x,t)$, problem $(P)$ has a variational structure. Then, even in lack of radial symmetry hypotheses, one bounded positive solution of $(P)$ can be found by passing to the limit on a sequence $(u_k)_k$ of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant $\barλ > 0$ and a sequence of points $(y_k)_k \subset \mathbb{R}^N$ exist such that \[ |y_k| \to +\infty\qquad \hbox{and}\qquad \int_{B_1(y_k)} |u_k|^p dx \ge \barλ\quad \hbox{for all $k \ge 1$.} \] |
| title | A dichotomy result for a modified Schrödinger equations on unbounded domains |
| topic | Analysis of PDEs 35J62, 35J92, 47J30, 35Q55, 58E30 |
| url | https://arxiv.org/abs/2507.18528 |