An Introduction to Nonlinear Functional Analysis and Elliptic Problems

Asymptotic bifurcation and second order elliptic equations on \( R^{N} \)

“…Krasnoselskii [13,14] in the context of equations of the form (1.1) where G : X → Y is asymptotically linear. In 1973 Rabinowitz [15] and Toland [23] introduced independently the use of the inversion u → v = u/ u 2 X to deal with asymptotic bifurcation for asymptotically linear problems and since then it has become the standard tool for dealing with such problems, see [1,3,27] for example. Setting G * (v) = v 2 X G(v/ v 2 X ) for v = 0 and G * (0) = 0, there is asymptotic bifurcation at μ for the equation G(u) = λu if and only if μ is a bifurcation point for the equation G * (v) = λv in the usual sense.…”

Large gyres as a shallow-water asymptotic solution of Euler’s equation in spherical coordinates

“…On the other hand, if F (0)≤0, then ψ − ≡0 is a sub-solution in the Southern Hemisphere because (7.1) now ensures x 2 + y 2 <1; more sophisticated sub/super-solutions are provided by the functions ψ associated with the special solutions (4.9) and (4.11), or by seeking radial functions (in which case one has to solve a second-order ordinary differential equation). If F is smooth and if one can find sub/super-solutions ψ ± such that ψ − ≤ ψ + throughout $\mathrm{scriptO}$ and such that ψ − ≤0≤ ψ + on $\mathrm{normal\partial }\mathrm{scriptO}$, then a functional-analytic approach (see [12]) shows that there exists a solution ψ to (7.4) such that ψ − ≤ ψ ≤ ψ + throughout $\mathrm{scriptO}$.(ii)  The comparison method and uniqueness : If the function F is non-decreasing, using the strong maximum principle (see [13]), we can reduce the above requirements that ψ − ≤ ψ + throughout $\mathrm{scriptO}$ and ψ − ≤0≤ ψ + on $\mathrm{normal\partial }\mathrm{scriptO}$ to merely ψ − ≤0≤ ψ + on $\mathrm{normal\partial }\mathrm{scriptO}$ by taking advantage of the comparison principle: if ψ is a solution and ψ + a supersolution with 0≤ ψ + on $\mathrm{normal\partial }\mathrm{scriptO}$, then the mean-value theorem yields $0\le \mathrm{normal\Delta }false(\psi -{\psi }^{+}false)-\frac{4false[Ffalse(\psi false)-Ffalse({\psi }^{+}false)false]}{{false(1+x^2+y^{2}false)}^2}=\mathrm{normal\Delta }false(\psi -{\psi }^{+}false)-\frac{4F^{\prime }false(\stackrel{false~}{\psi }false)}{{false(1+x^2+y^{2}false)}^2}false(\psi -{\psi }^{+}false),$and the strong maximum principle ensures ψ ≤ ψ + in $\mathrm{scriptO}$, if this inequality holds on $\mathrm{normal\partial }\mathrm{scriptO}$; a similar comparison can be made with a sub-solution. In particular, in this setting, we therefore have uniqueness.…”

“…The above approach is also valid without the monotonicity assumption on F , but only if the domain $\mathrm{scriptO}$ is sufficiently narrow or of sufficiently small area—see [14]. Moreover, if there exist constants M >0 and q ≥1 with $|Ffalse(sfalse)|\le Mfalse(1+|s{false|}^{q}false),s\in \mathrm{double-struckR},$then the solution between the sub- and super-solution is a local minimum of the functional $\mathrm{scriptE}false(\psi false)=\frac{1}{2}{\int }_{\mathrm{scriptO}}|\mathrm{normal\nabla }\psi {false|}^2\hspace{0.17em}\mathrm{normald}x\hspace{0.17em}\mathrm{d}y-{\int }_{\mathrm{scriptO}}[8\omega \frac{1-(x^2+y^2)}{{(1+x^2+y^2)}^3}\psi -\frac{4\mathcal{F}(\psi )}{{(1+x^2+y^2)}^2}]\mathrm{d}x\hspace{0.17em}\mathrm{d}y,$where $\mathrm{scriptF}false(tfalse)={\int }_0^{t}Ffalse(sfalse)\hspace{0.17em}\mathrm{d}s$; see [12]. …”

“…On the other hand, if F (0)≤0, then ψ − ≡0 is a sub-solution in the Southern Hemisphere because (7.1) now ensures x 2 + y 2 <1; more sophisticated sub/super-solutions are provided by the functions ψ associated with the special solutions (4.9) and (4.11), or by seeking radial functions (in which case one has to solve a second-order ordinary differential equation). If F is smooth and if one can find sub/super-solutions ψ ± such that ψ − ≤ ψ + throughout $\mathrm{scriptO}$ and such that ψ − ≤0≤ ψ + on $\mathrm{normal\partial }\mathrm{scriptO}$, then a functional-analytic approach (see [12]) shows that there exists a solution ψ to (7.4) such that ψ − ≤ ψ ≤ ψ + throughout $\mathrm{scriptO}$.…”

Validated bounds on embedding constants for Sobolev space Banach algebras

“…Since functions in H n (Ω) are only determined up to sets of measure zero, it usually does not make sense to talk about their boundary values. However, the requirement u = 0 on the boundary Ω can be replaced by the condition that the sought-after solution can be approximated arbitrarily well by classical smooth functions that do vanish on the boundary, with respect to the norm in (3). In other words, functions that satisfy the boundary condition u = 0 should be precisely the functions in the space H n 0 (Ω) defined in (4).…”