A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

Abstract

The existence of at least one nontrivial solution to a class of semilinear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable reflection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity.

Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

“…We record the result of Proposition 2.2 of [16] along with its extension using the argument of Theorem 2.2 of [19] in the following example. ðOÞ; the unique generalized solution to problem (LT) satisfies u50ð40Þ a.e.…”

“…[20]); that is, the existence of a positive eigenvalue of minimum modulus with an associated positive eigenfunction. Moreover, by reexamining the proof of the solvability theorems in [16,19] we will now show that all real eigenvalues associated to generalized eigenfunctions must be positive. This spectral information is summarized in the following theorem.…”

“…Moreover, we record the main operator theoretic properties of T AC[s;r in the second part of the theorem whose similar proof for r ¼ 0 and 1 appears in [16,19] (ii) There exists an e > 0 such that the elliptic boundaries s and s 0 of O and O 0 coincide in a strip fðx; yÞ: 04y4eg:…”

“…The results we present here are a companion to our study of a nonlocal version of (ST) on symmetric domains in which the nonlinearity is composed of a linear reflection operator and the dual variational method has been employed to give results on existence and multiplicity of nontrivial solutions (cf. [15][16][17][18]). …”

A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow

“…The Tricomi equation [1] is a mixed type of linear partial differential equation of the second order, which has been used to narrate the theory of plane transonic flow [6][7][8][9]. The Tricomi equation was used to recount differentiable problems for the theory of plane transonic flow.…”