We study fundamental solutions of elliptic operators of order $$2m\ge 4$$ 2 m ≥ 4 with constant coefficients in large dimensions $$n\ge 2m$$ n ≥ 2 m , where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as $$n\ge 3$$ n ≥ 3 , the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m may no longer serve as a prototype for the general elliptic operator. It is known from examples of Maz’ya and Nazarov (Math. Notes 39:14–16, 1986; Transl. of Mat. Zametki 39, 24–28, 1986) and Davies (J Differ Equ 135:83–102, 1997) that in dimensions $$n\ge 2m+3$$ n ≥ 2 m + 3 fundamental solutions of specific operators of order $$2m\ge 4$$ 2 m ≥ 4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to $$+\infty $$ + ∞ and $$-\infty $$ - ∞ respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a “positive” direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for $$n=2m$$ n = 2 m , $$n=2m+2$$ n = 2 m + 2 and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order 2m in dimension $$n=2m+2$$ n = 2 m + 2 . On the other hand we show that in the dimensions $$n=2m$$ n = 2 m and $$n=2m+1$$ n = 2 m + 1 the fundamental solution of any such elliptic operator is always positive around its singularity.
We consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter (“doubly” nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the linear problem using a Lyapunov–Schmidt reduction and provide an expansion of both the nonlinear eigenvalue and the solution. We further prove that if the linear eigenvalue is real and the nonlinear problem $${\mathcal {PT}}$$ PT -symmetric, then the bifurcating nonlinear eigenvalue remains real. These general results are then applied in the context of surface plasmon polaritons (SPPs), i.e. localized solutions for the nonlinear Maxwell’s equations in the presence of one or more interfaces between dielectric and metal layers. We obtain the existence of transverse electric SPPs in certain $${\mathcal {PT}}$$ PT -symmetric configurations.
We study the ground states of the following generalization of the Kirchhoff-Love functional,where Ω is a bounded convex domain in R 2 with C 1,1 boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on σ. Positivity of ground states is proved with different techniques according to the range of the parameter σ ∈ R and we also provide a convergence analysis for the ground states with respect to σ. Further results concerning positive radial solutions are established when the domain is a ball.
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