Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorbtion

We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-\Delta u=\lambda (V(x)u-f(u))$ in $\RR^N$, where $V(x)$ is a variable potential that may change sign, $\lambda$ is a real parameter, and $f$ is an absorbtion term such that the mapping $f(t)/t$ is increasing in $(0,\infty)$. We prove that there exists a bifurcation non-negative number $\Lambda$ such that the above problem has exactly one solution if $\lambda>\Lambda$, but no such a solution exists provided $\lambda\leq\Lambda$.


Introduction and the main results
In this paper we are concerned with the existence, uniqueness or the non-existence of positive solutions of the eigenvalue logistic problem with absorbtion where V is a smooth sign-changing potential and f : [0, ∞) → [0, ∞) is a smooth function. Equations of this type arise in the study of population dynamics. In this case, the unknown u corresponds to the density of a population, the potential V describes the birth rate of the population, while the term −f (u) in (1) signifies the fact that the population is selflimiting. In the region where V is positive (resp., negative) the population has positive (resp., negative) birth rate. Since u describes a population density, we are interested in investigating only positive solutions of problem (1). Our results are related to a certain linear eigenvalue problem. We recall in what follows the results that we need in the sequel. Let Ω be an arbitrary open set in R N , N ≥ 3. Consider the eigenvalue problem −∆u = λV (x)u in Ω , u ∈ H 1 0 (Ω).
Problems of this type have a long history. If Ω is bounded and V ≡ 1, problem (2) is related to the Riesz-Fredholm theory of self-adjoint and compact operators (see, e.g., Theorem VI.11 in [4]). The case of a non-constant potential V has been first considered in the pioneering papers of Bocher [3], Hess and Kato [11], Minakshisundaran and Pleijel [16] and Pleijel [18]. For instance, Minakshisundaran and Pleijel [16], [18] studied the case where Ω is bounded, V ∈ L ∞ (Ω), V ≥ 0 in Ω and V > 0 in Ω 0 ⊂ Ω with |Ω 0 | > 0. An important contribution in the study of (2) if Ω is not necessarily bounded has been given by Szulkin and Willem [20] under the assumption that the sign-changing potential V satisfies In order to find the principal eigenvalue of (2), Szulkin and Willem [20] proved that the minimization problem has a solution ϕ 1 = ϕ 1 (Ω) ≥ 0 which is an eigenfunction of (2) corresponding to the eigenvalue λ 1 (Ω) = Ω |∇ϕ 1 | 2 dx. Throughout this paper the sign-changing potential V : R N → R is assumed to be a Hölder function that satisfies We suppose that the nonlinear absorbtion term f : (f 2) the mapping f (u)/u is increasing in (0, +∞).
This assumption implies lim u→+∞ f (u) = +∞. We impose that f does not have a sublinear growth at infinity. More precisely, we assume Our framework includes the following cases: (i) f (u) = u 2 that corresponds to the Fisher equation [9] and the Kolmogoroff-Petrovsky-Piscounoff equation [14] (see also [13] for a comprehensive treatment of these equations); (ii) f (u) = u (N +2)/(N −2) (for N ≥ 6) which is related to the conform scalar curvature equation, cf. [15].
For any R > 0, denote B R = {x ∈ R N ; |x| < R} and set Consequently, the mapping R −→ λ 1 (R) is decreasing and so, there exists We first state a sufficient condition so that Λ is positive. For this aim we impose the additional assumptions there exists A, α > 0 such that and lim x→0 |x| 2(N −1)/N V 2 (x) = 0.
Our main result asserts that Λ plays a crucial role for the nonlinear eigenvalue logistic problem The following existence and non-existence result shows that Λ serves as a bifurcation point in our problem (6). Then the following hold: (i) problem (6) has a unique solution for any λ > Λ; (ii) problem (6) does not have any solution for all λ ≤ Λ.
The additional condition (4) implies that V + ∈ L N/2 (R N ), which does not follow from the basic hypothesis (V ). As we shall see in the next section, this growth assumption is essential in order to establish the existence of positive solutions of (1) decaying to zero at infinity.
In particular, Theorem 1.2 shows that if V (x) < 0 for sufficiently large |x| (that is, if the population has negative birth rate) then any positive solution (that is, the population density) of (1) tends to zero as |x| → ∞.
2 Proof of Theorem 1.1 Since V 1 ∈ L N/2 (R N ), using the Cauchy-Schwarz inequality and Sobolev embeddings we obtain where 2 * = 2N/(N − 2). Fix ǫ > 0. By our assumption (V ), there exists positive numbers δ, R 1 and R such that R −1 < δ < R 1 < R such that for all x ∈ B R satisfying |x| ≥ R 1 we have On the other hand, by (V ), for any x ∈ B R with |x| ≤ δ we have Define Ω : By (8) and Hardy's inequality we find Using now (9) and Hölder's inequality we obtain By compactness and our assumption (V ), there exists a finite covering of ω by the closed balls There exists r > 0 such that, for any Define A := ∪ k j=1 B r (x j ). The above estimate, Hölder's inequality and Sobolev embeddings yield for any j = 1, . . . , k. By addition we find It follows from (12) Now from inequalities (7), (10), (11), (13) and (14) we have and passing to the limit as R → ∞ we conclude that This completes the proof of Theorem 1.1.

An auxiliary result
We show in this section that the logistic equation (1) has entire positive solutions if λ is sufficiently large. However, we are not able to establish that this solution decays to zero at infinity. This will be proved in the next section by means of the additional assumption (4). More precisely, we have has at least one solution, for any λ > Λ.
Proof. For any R > 0, consider the boundary value problem We first prove that problem (16) has at least one solution, for any λ > λ 1 (R). Indeed, the function u(x) = M is a supersolution of (16), for any M large enough. This follows from (f 3) and the boundedness of V . Next, in order to find a positive subsolution, let us consider the problem min Since λ > λ 1 (R), it follows that the least eigenvalue µ 1 is negative. Moreover, the corresponding eigenfunction e 1 satisfies Then the function u(x) = εe 1 (x) is a subsolution of the problem (16). Indeed, it is enough to check that that is, by (17), But So, since f ′ (0) = 0, relation (18) becomes which is true, provided ε > 0 is small enough, due to the fact that µ 1 < 0. Fix λ > Λ and an arbitrary sequence R 1 < R 2 < . . . < R n < . . . of positive numbers such that R n → ∞ and λ 1 (R 1 ) < λ. Let u n be the solution of (16) on B Rn . Fix a positive number M such that f (M )/M > V L ∞ (R N ) . The above arguments show that we can assume u n ≤ M in B Rn , for any n ≥ 1. Since u n+1 is a supersolution of (16) for R = R n , we can also assume that u n ≤ u n+1 in B Rn . Thus the function u(x) := lim n→∞ u n (x) exists and is well-defined and positive in R N . Standard elliptic regularity arguments imply that u is a solution of problem (15).
The above result shows the importance of the assumption (4) in the statement of Theorem 1.2. Indeed, assuming that V satisfies only the hypothesis (V ), it is not clear whether or not the solution constructed in the proof of Proposition 3.1 tends to 0 as |x| → ∞. However, it is easy to observe that if λ > Λ and V satisfies (4) then problem (6) has at least one solution. Indeed, we first observe that is a subsolution of problem (6), for some fixed R > 0, where e 1 satisfies (17). Next, we observe that u(x) = n/(1 + |x| 2 ) is a supersolution of (6). Indeed, u satisfies It follows that u is a supersolution of (6) provided This inequality follows from (f 3) and (4), provided that n is large enough.

Proof of Theorem 1.2
We split the proof of our main result into several steps. We will assume the conditions (V ),  Proof. Let ω N be the surface area of the unit sphere in R N . Consider the function V + u as a Newtonian potential and define A straightforward computation shows that But, by (4) and since u is bounded, So, by Lemma 2.3 in Li and Ni [15], . Hence w(x) → 0 as |x| → ∞. Let us choose C sufficiently large so that w(0) > 0. We claim that this implies Indeed, if not, let x 0 ∈ R N be a local minimum point of w. This means that w(x 0 ) < 0, ∇w(x 0 ) = 0 and ∆w(x 0 ) ≥ 0. But provided that C > λ. This contradiction implies (21). Consequently, So, using again (4),

Lemma 2.3 in [15] yields the improved estimate
provided that 2α < N − 2, and so on. Let n α be the largest integer such that n α α < N − 2.
Repeating n α + 1 times the above argument based on Lemma 2.3 (i) and (iii) in [15] we obtain Proposition 4.2. Let u be a solution of problem (6). Then V + u, V − u, f (u) ∈ L 1 (R N ), and u ∈ H 1 (R N ).
Proof. For any R > 0 consider the average function where ω N denotes the surface area of S N −1 . Then Hence By Proposition 4.1, there exists C > 0 such that |u(r)| ≤ Cr −N +2 , for any r > 0. So, by (4), where C does not depend on r. This implies V + u ∈ L 1 (R N ). By contradiction, assume that V − u + f (u) ∈ L 1 (R N ). So, by (22), u ′ (r) > 0 if r is sufficiently large. It follows that u(r) does not converge to 0 as r → ∞, which contradicts Proposition 4.1. So, V − u + f (u) ∈ L 1 (R N ). Next, in order to establish that u ∈ L 2 (R N ), we observe that our assumption (f 1) implies the existence of some positive numbers a and δ such that f ′ (t) > at, for any 0 < t < δ. This implies f (t) > at 2 /2, for any 0 < t < δ. Since u decays to 0 at infinity, it follows that the set {x ∈ R N ; u(x) ≥ δ} is compact. Hence It remains to prove that ∇u ∈ L 2 (R N ) N . We first observe that after multiplication by u in (1) and integration we find , it follows that the left hand-side has a finite limit as r → ∞. Arguing by contradiction and assuming that ∇u ∈ L 2 (R N ) N , it follows that there exists R 0 > 0 such that Define the functions

Relation (23) can be rewritten as
On the other hand, by the Cauchy-Schwarz inequality, Using now (24) we obtain On the other hand, our assumption |∇u| ∈ L 2 (R N ) implies  Proof. By multiplication with v in (6) and integration on B R we find So, by Proposition 4.2, there exists and is finite lim R→∞ ∂B R u ∂v ∂ν dσ. But, by the Cauchy-Schwarz inequality, Since u, |∇v| ∈ L 2 (R N ), it follows that ∞ 0 Our conclusion now follows by (28) and (29).
Proof of Theorem 1.2. (i) The existence of a solution follows with the arguments given in the preceding section. In order to establish the uniqueness, let u and v be two solutions of (6). We can assume without loss of generality that u ≤ v. This follows from the fact that u = min{u, v} is a supersolution of (6) and u defined in (19) is an arbitrary small subsolution. So, it sufficient to consider the ordered pair consisting of the corresponding solution and v.
We claim that ∇Ψ n → ∇u in L 2 (R N ) N .
Next, we observe that ∇ζ n L N (R N ) = ∇ζ L N (R N ) .