2020/03/30 by Denis Bonheure, Ederson Moreira dos Santos, Bonheure, Denis +7
Computer Science · Mathematics · #35B07 #35J15 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2003.13587
openalex publication_date 2020/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider nonlinear second order elliptic problems of the type \-
Deltaν=f(u)
text in
Omega,
qquad u=0
text on
partial
Omega, where\n\Ω is an open C1,1-domain in \ℝN, N\≥ 2, under some\ngeneral assumptions on the nonlinearity that include the case of a sublinear\npure power f(s)=|s|p-1s with 0<p<1 and of Allen-Cahn type\nf(s)=\λ(s-|s|p-1s) with p>1 and \λ>\λ2(\Ω) (the\nsecond Dirichlet eigenvalue of the Laplacian). We prove the existence of a\nleast energy nodal (i.e. sign changing) solution, and of a nodal solution of\nmountain-pass type. We then give explicit examples of domains where the\nassociated levels do not coincide. For the case where \Ω is a ball or\nannulus and f is of class C1, we prove instead that the levels coincide,\nand that least energy nodal solutions are nonradial but axially symmetric\nfunctions. Finally, we provide stronger results for the Allen-Cahn type\nnonlinearities in case \Ω is either a ball or a square. In particular we\ngive a complete description of the solution set for \λ\∼\n\λ2(\Ω), computing the Morse index of the solutions.\n