The theory is used to establish the solution corresponding to a concentrated heat source acting in an unbounded continuum.
Since Theorem 4.1 guarantees an unbounded continuum (mathcal{C}^), if we provide the a priori boundedness of solutions for ((P_{lambda})), then the unbounded continuum allows the existence of solutions for all (lambda>0).
For (4.4), from ((frac{mu_{0^{r},theta)) there emanates an unbounded continuum of positive solutions (C_subsetoverline{D}_).
Then from each ( λ k / f 0, 0 ) it bifurcates an unbounded continuum C k of solutions to problem (1.1), with exactly k − 1 simple zeros.
Moreover, there exists an unbounded continuum (mathscr{D}^{nu}) of solutions of problem (5.3), such that (mathscr {D}^{nu}subset((mathbb{R}times P^{nu} cup{(frac{lambda^{nu }}{rf_{0}},0)})).
Moreover, there exists an unbounded continuum (mathscr{D}^{nu}) of solutions of problem (4.3), such that (mathscr {D}^{nu}subset((mathbb{R}times P^{nu} cup{(lambda^{nu},0)})).
The BEM employs frequency-dependent fundamental solutions for a point load in the unbounded continuum, and requires discretization of all external and internal surfaces and interfaces only.
It was proved in [17], Theorem 1.6] that the set T = { ( λ, z ) ∈ R × B : Eq. (A.1) holds }. possesses an unbounded component (continuum) D ⊂ T which meets and satisfies the dichotomy of [17], Theorem 1.6].
We consider the Helmholtz equation in an unbounded periodic media perturbed by an unbounded defect whose structure is compatible with the periodicity of the underlying media.
By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter λ, and especially, we obtain the existence of at least three distinct positive solutions for λ being in a certain interval.
