2025/07/24 by Anna María Candela, G. Palmieri, Candela, Anna Maria +3
Computer Science · Mathematics · #35J62 #35J92 #35Q55 #47J30 #58E30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2507.18528
openalex publication_date 2025/07/24 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
This article aims to investigate the existence of bounded positive solutions of problem (P) \ - \rm div (a(x,u,∇ u)) + At(x,u,∇ u) = g(x,u) · amp;\hboxin Ω,
u = 0 · amp; \hboxon ∂Ω,. with At(x,t,ξ) = (∂ A)/(∂ t)(x,t,ξ), a(x,t,ξ) = ∇ξA(x,t,ξ) for a given A(x,t,ξ) which grows as |ξ|p + |t|p , p > 1, where Ω⊆ ℝN, N ≥ 2, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually Ω= ℝN, which generalizes the modified Schrödinger equation - \rm div ((A^*1(x) + A^*2(x)|u|s) ∇ u) + \fracs2 A^*2(x) |u|s - 2 u |∇ u|2 + u = |u|μ-2u \hboxin ℝ3. 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 (uk)k of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant λ > 0 and a sequence of points (yk)k ⊂ ℝN exist such that |yk| → +∞ \hboxand ∫B1(yk) |uk|p dx ≥ λ \hboxfor all k ≥ 1.