Multiple positive solutions of some elliptic equations in bold R^N

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If , then Q is too low. By using a bisection-type technique, one can find an approximate solution for Q to the equation. In the last step of this iterative procedure, one adds to 32 the equation. The counter intuitive conclusion is that an increase in c may lead to an increase of the total population size whenever the unsuitable area outside the favorable core area is not too harsh. This is due to a lag effect in the left tail: the decay of the population in the region that was favorable until recently may be slow while meanwhile, the rise of the population in the right region that just became favorable is relatively fast.

This possibility of increases in both range and population size was not shown in the related work of Potapov and Lewis In conclusion of this section, we formulate an insight deriving from 23 : for small c , an increase of D entails an increase of the minimal interval length, since diffusion creates a net loss over the boundary of the favorable region.

For larger c , however, the influence of D on the minimal interval length may be opposite, since increased mobility helps to track the moving climate.

A = P + Prt for r, solve for the specified value

In this section, we investigate formally the stability of the extinct state. We find that the principal eigenvalue switches sign exactly at the codimension one manifold in parameter space that separates the domain of existence of a nontrivial solution from the domain of nonexistence. In the next section, we shall see that the principal eigenvalue for a general equation characterizes the existence and nonexistence of nontrivial solutions and determines as well the large time dynamics of this model. Note that within the domain of non-existence this eigenvalue further yields information about the rate of decay to zero, i.

In the next section, we will see that the sign of the principal or dominant eigenvalue of this operator, when properly defined, yields the long term dynamics in Eq. Its sign gives a criterion for either extinction or persistence. Therefore, methods to determine the sign of the dominant eigenvalue are of great interest, as are methods to give more quantitative estimates in case it is negative at which time scale will the extinction happen? For the caricatural case of Section 2 , we can derive an explicit equation for the dominant eigenvalue.

Non radial positive solutions for the Hénon equation with critical growth

The derivation follows the same pattern as the analysis leading to This equation is the analogue of It reads. In terms of the unscaled time and parameters, this means that. In other words, the dominant eigenvalue is a quadratically decreasing function of c , with a coefficient of the quadratic term which is inversely proportional to D but independent of all other parameters.

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So far, we have considered a particular type of heterogeneity, that which is obtained by juxtaposing two homogeneous media—the favorable and unfavorable ones—with an abrupt transition at the two end points of the favorable interval. Are the results which we have derived previously robust? And is the co-moving nontrivial solution stable if it exists? Here we give very strong affirmative answers to both these questions in a rather general setting.

The motivation for considering general types of nonlinearities is twofold. First, the assumptions made in Section 2 are rather contrived from a modeling point of view and one would like to consider more complex transitions e. Second, a general mathematical theory sheds much more light on the underlying mechanisms, since the proofs reveal the role that various assumptions play in yielding the conclusions.

Here, for instance, the linearization at the trivial steady state, in particular the sign of the associated principal eigenvalue, will be seen to fully account for the ability to keep pace with a shifting climate.

Download Multiple Positive Solutions Of Some Elliptic Equations In Bold Rn 1999

In this section, we consider Eq. The functions f and g related through 2 will be assumed to satisfy the following set of conditions. Allow for multiple discontinuities, e. The properties formulated in a — e above are the standing hypotheses on the function g throughout this section. The last one means that everywhere the population declines when it exceeds some level M, i.

Note that the values of the limit can be changed by a scaling of the time variable t. Since g may have discontinuities with respect to x , we consider generalized solutions. For studies of solutions of 1 on bounded domains , without an imposed translation speed i. Recently, the effect of a heterogeneous but spatially periodic environment has been studied by Berestycki et al.

Lastly, periodic stochastic environments are considered by Roques and Stoica The problem we study here involves a lack of compactness the problem is set on the whole real line as well as the difficulty deriving from the fact that c is imposed. We shall find that such a solution exists if and only if the zero steady state of the equation. Next, we settle the uniqueness issue by showing that there is at most one traveling wave solution. Concerning the large time asymptotic behavior of solutions of the initial value problem for 1 , we then formulate a dichotomy:.

Indeed, note that no conditions at infinity, other than being bounded are imposed here on solutions. More general results in this direction can be found in Berestycki and Nirenberg, But, in general, it is not so. For instance,. Proposition 4.

On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system

Let w be a bounded positive solution of It should be noted here, also for future use, that even though f may be discontinuous, the maximum principle still applies to the C 1 solutions that we consider. In the present one-dimensional setting, this can be verified rather directly. More general statements in Gilbarg and Trudinger also cover the multi-dimensional situation.

Next, we want to derive lower bounds. We first formulate an auxiliary result that will also be used later. Then there exists a positive constant K such that. We omit the easy proof, since it follows exactly the same line of argumentation that we used to prove the preceding proposition.

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Now let w be a bounded positive solution of Thus, as a corollary of Proposition 4. To conclude this subsection, we formulate an estimate for the derivative of w. Proof: We restrict our attention to large positive x , the case of negative x being the same. From 48 , we get. The result now follows from the properties of g and the estimates for w obtained in Proposition 4.


Since we consider solutions defined on the whole real line, some special care is needed. So, it is meaningful to formulate the following definition. As a special case of the results established in Berestycki et al. We refer to Berestycki et al. The regularity theory of elliptic equations next guarantees that any sequence has a converging subsequence and that we can pass, for such a subsequence, to the limit in the differential equation.

We only state the properties that we shall use in the next subsection.

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Proof: Again, we restrict our attention to large positive x. So the inequality 57 follows from Proposition 4. Sketch of the proof: The proof is essentially identical to the proofs of Propositions 4. Theorem 4. We shall prove that a solution exists by constructing both a sub- and a supersolution. Now define. Assumption e guarantees that the constant function taking the value M is a supersolution. We conclude that a solution exists. We claim that. So, this minimum must be zero, i.

The estimates presented in Propositions 4. But the right-hand side is an increasing function of A which takes positive values, so is bounded away from zero for large A. Equation 48 has at most one bounded positive solution.