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Combined effects in nonlinear problems arising in the
study of anisotropic continuous media
Dušan Repovš, Vicentiu Radulescu
To cite this version:
Combined effects in nonlinear problems arising
in the study of anisotropic continuous media
Vicent¸iu R˘adulescua,b and Duˇsan Repovˇsc,d
aInstitute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest bDepartment of Mathematics, University of Craiova, 200585 Craiova, Romania
cFaculty of Mathematics and Physics, University of Ljubljana,
Jadranska 19, P. O. Box 2964, 1001 Ljubljana, Slovenia
dFaculty of Education, University of Ljubljana, Kardeljeva ploˇsˇcad 16, 1000 Ljubljana, Slovenia
E-mail: vicentiu.radulescu@imar.ro dusan.repovs@guest.arnes.si
This paper is dedicated with esteem to Professor Marius Iosifescu on his 75th birthday Abstract
We are concerned with the Lane-Emden-Fowler equation −∆u = λk(x)uq± h(x)up
in Ω, subject to the Dirichlet boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in RN, k and h are variable potential functions, and 0 < q < 1 < p.
Our analysis combines monotonicity methods with variational arguments.
Keywords: Lane-Emden-Fowler equation, bifurcation problem, anisotropic continuous media, perturbation, positive solution.
2010 Mathematics Subject Classification: 35A05, 35B40, 35J60, 58E05.
1
Introduction and the main results
In this paper we are concerned with the study of combined effects in a class of nonlinear bifurcation problems that arise in anisotropic continuous media. Bifurcation problems have a long history and their treatment goes back to the XVIIIth century. To the best of our knowledge, the first bifurcation problems is related to the buckling of a thin rod under thrust and was investigated by Daniel Bernoulli and Euler around 1744. In the case in which the rod is free to rotate at both end points, this yields the one-dimensional bifurcation problem
u′′+ λ sin u = 0 in (0, L) 0 ≤ u ≤ π u′(0) = u′(L) = 0 .
Consider the bifurcation problem
−∆u = λf (u) in Ω
u = 0 on ∂Ω , (1)
Cohen [23], who assumed that f is a function of class C1 which is convex and positive, such that f′(0) > 0. They proved the following basic facts:
(i) there exists λ∗ ∈ (0, ∞) such that problem (1) has (resp., has no) classical solution
if λ ∈ (0, λ∗) (resp., λ > λ∗);
(ii) for λ ∈ (0, λ∗), among the solutions of (1) there exists a minimal one, say u(λ);
(iii) the mapping λ 7−→ u(λ) is convex, increasing and of class C1;
(iv) u(λ) is the only solution of problem (1) such that the operator −∆ − λf′(u) is
coercive.
Motivated by a problem raised by Brezis, Mironescu and R˘adulescu studied in [29, 30] the case where f fulfills the above hypotheses and has a linear growth at infinity, that is, there exists limt→∞f (t)/t =: a ∈ (0, ∞). The results in this framework are strongly related
with the sign of b := limt→∞(f (t) − at). For instance , if b ≥ 0 then the following properties
hold: (i) λ∗ = λ
1/a; (ii) u(λ) is the unique solution of problem (1) for all λ ∈ (0, λ∗);
(iii) limλ→λ∗−u(λ) = +∞ uniformly on compact subsets of Ω; and (iv) problem (1) has no
solution if λ = λ∗. A different behaviour holds if b ∈ [−∞, 0).
The perturbed nonlinear problem
−∆u = λf (u) + a(x)g(u) in Ω
u > 0 in Ω
u = 0 on ∂Ω ,
(2)
was studied in [15], where f has either a sublinear or a linear growth at infinity, g is a singular nonlinearity, and a is a nonnegative potential. The framework discussed in [15] includes the case f (t) = tp and g(t) = t−γ, where 0 < p ≤ 1 and 0 < γ < 1. For instance, if f is sublinear and infx∈Ωa(x) > 0 then problem (2) has a unique solution for all λ ∈ R.
From a physical point of view, problem (2) arises in the context of chemical heterogenous catalysis, in the theory of heat conduction in electrically conducting materials, as well as in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids. Such equations are also encountered in glacial advance (see [39]), in transport of coal slurries down conveyor belts (see [12]), and in several other geophysical and industrial contents (see [11] for the case of the incompressible flow of a uniform stream past a semi-infinite flat plate at zero incidence).
The above results are summarized and completed in the recent works [6], [20], [25], and [37].
In this paper we are concerned with the competition between convex and concave nonlin-earities and variable potentials. Such problems arise in the study of anisotropic continuous media. We point out that nonlinear elliptic equations with convex–concave nonlinearities have been studied starting with the seminal paper by Ambrosetti, Brezis and Cerami [4]. They considered the Dirichlet problem
−∆u = λuq+ up, in Ω u > 0, in Ω u = 0, on ∂Ω , (3)
where λ is a positive parameter, Ω ⊂ RN is a bounded domain with smooth boundary, and 0 < q < 1 < p < 2⋆− 1 (2⋆ = 2N/(N − 2) if N ≥ 3, 2⋆ = +∞ if N = 1, 2). Ambrosetti, Brezis and Cerami proved that there exists λ∗> 0 such that problem (3) admits at least two
λ > λ∗. In [2], Alama and Tarantello studied the related Dirichlet problem with indefinite
weights
−∆u − λu = k(x)uq− h(x)up, if x ∈ Ω
u > 0, if x ∈ Ω
u = 0, if x ∈ ∂Ω ,
(4)
where h, k are nonnegative and 1 < p < q. We refer to [33] for a related problem with lack of compactness.
In the present paper we are concerned with the nonlinear elliptic problem
−∆u = λk(x)uq± h(x)up, if x ∈ Ω
u > 0, if x ∈ Ω
u = 0, if x ∈ ∂Ω ,
(Pλ)±
under the basic assumption 0 < q < 1 < p.
If λ = 0, equation (Pλ)± is called the Lane-Emden-Fowler equation and arises in the
boundary–layer theory of viscous fluids (see [40]). This equation goes back to the paper by H. Lane [26] in 1869 and is originally motivated by Lane’s interest in computing both the temperature and the density of mass on the surface of the sun. Equation (Pλ)± describes
the behaviour of the density of a gas sphere in hydrostatic equilibrium and the index p, which is called the polytropic index in astrophysics, is related to the ratio of the specific heats of the gas. Problem (Pλ)± may be also viewed as a prototype of pattern formation
in biology and is related to the steady–state problem for a chemotactic aggregation model introduced by Keller and Segel [24]. Problem (Pλ)± also plays an important role in the
study of activator-inhibitor systems modeling biological pattern formation, as proposed by Gierer and Meinhardt [21]. Problems of this type, as well as the associated evolution equations, describe super-diffusivities phenomena. Such models have been proposed by de Gennes [16] to describe long range Van der Waals interactions in thin films spread on solid surfaces. This equation also appears in the study of cellular automata and interacting particle systems with self-organized criticality (see [13]), as well as to describe the flow over an impermeable plate (see [11]). Problems of this type are obtained from evolution equations of the form
utt = div (um−1∇u) + h(x, u) in Ω × (0, T )
through the implicit discretization in time arising in nonlinear semigroup theory (see [38]). Throughout this paper we assume that the variable weight functions k, h ∈ L∞(Ω)
satisfy
ess infx∈Ωk(x) > 0 and ess infx∈Ωh(x) > 0 .
We are concerned with the existence of weak solutions of problems (Pλ)+ and (Pλ)−,
that is, functions u ∈ H01(Ω) such that
(i) u ≥ 0 a.e. on Ω and u > 0 on a subset of Ω with positive measure; (ii) for all ϕ ∈ H1
0(Ω) the following identity holds
Z Ω ∇u∇ϕ dx = Z Ω [λk(x)uq± h(x)up] ϕ dx .
of a minimal solution, provided that λ > 0 is small enough. This result gives a complete qualitative analysis of the problem and the proof combines monotonicity arguments with variational techniques. We refer to Bartsch and Willem [7] for a related problem that is treated by means of a new critical point theorem, which guarantees the existence of in-finitely many critical values of an even functional in a bounded range. In this framework, Bartsch and Willem proved that there exists a sequence (un) of solutions (not necessarily
positive!) with corresponding energy E(un) < 0 and such that E(un) → 0 as n → ∞.
Theorem 1.1. Assume 0 < q < 1 < p < 2⋆− 1. Then there exists a positive number λ∗
such that the following properties hold: a) for all λ ∈ (0, λ∗) problem (P
λ)+has a minimal solution u(λ). Moreover, the mapping
λ 7−→ u(λ) is increasing.
b) problem (Pλ)+ has a solution if λ = λ∗;
c) problem (Pλ)+ does not have any solution if λ > λ∗.
The next result is concerned with problem (Pλ)− and asserts that there is some λ∗> 0
such that (Pλ)− has a nontrivial solution if λ > λ∗ and no solution exists provided that
λ < λ∗.
Theorem 1.2. Assume 0 < q < 1 < p < 2⋆− 1. Then there exists a positive number λ∗
the following properties hold: a) if λ > λ∗, then problem (P
λ)− has at least one solution;
b) if λ < λ∗, then problem (P
λ)− does not have any solution.
We point out that related results have been established by Il’yasov [22] and Luby-shev [27, 28]. However, the proof techniques of [22, 27, 28] are completely different from the arguments in the present paper and they rely on the global fibering method intro-duced by S.I. Pohozaev [32], which is among the most powerful tools for studying nonlinear differential equations. The fibering scheme allows Il’yasov and Lubyshev to find construc-tively constrained minimization problems which have the property of ground among all constrained minimization problems corresponding to the given energy functional. We also refer to Bozhkov and Mitidieri [8, 9] for relatively closed “convex-concave” settings, both for nonlinear differential equations and for quasilinear elliptic systems with Dirichlet boundary condition. Garcia Azorero and Peral Alonso [18] used the mountain pass theorem to obtain the existence of a sign-changing solution in a related quasilinear setting.
2
Proof of Theorem 1.1
We first prove that if λ > 0 is sufficiently small then problem (Pλ)+ has a solution. For
this purpose we use the method of sub- and super-solutions. Consider the problem
−∆w = λk(x)wq, in Ω w > 0, in Ω w = 0, on ∂Ω . (5)
suffices to show that
ελk(x)wq ≤ λk(x)εqwq+ h(x)εpwp in Ω, which is true for all ε ∈ (0, 1).
Let v be the unique solution of the linear problem
−∆v = 1, in Ω
v = 0, on ∂Ω .
We prove that if λ > 0 is small enough then there is M > 0 such that u := M v is a super-solution of (Pλ)+. Therefore it suffices to show that
M ≥ λk(x)(M v)q+ h(x)(M u)p in Ω. (6)
Set
A := kkkL∞k · kvkq
L∞ and B := khkL∞k · kvkp
L∞.
Thus, by (6), it is enough to show that there is M > 0 such that M ≥ λAMq+ BMp.
or, equivalently,
1 ≥ λAMq−1+ BMp−1. (7)
Consider the mapping (0, ∞) ∋ t 7−→ λAtq−1 + Btp−1. A straightforward computation shows that this function attains its minimum for t = Cλ1/(p−q), where C = [AB−1(1 − q)(p − 1)−1](q−1)/(p−q). Moreover, the global minimum of this mapping is
(ACq−1+ BCp−1)λ(p−1)/(p−q).
This shows that condition (7) is fulfilled for all λ ∈ (0, λ0] and M = Cλ1/(p−q), where λ0> 0
satisfies
(ACq−1+ BCp−1)λ(p−1)/(p−q)0 = 1.
It remain to argue that εw ≤ M v. This is a consequence of the maximum principle (see [34]), provided that ε > 0 is small enough. Thus, problem (Pλ)+ has at least one solution
u(λ) for all λ < λ∗.
Set
λ∗ := sup{λ > 0; problem (Pλ)+ has a solution}.
By the definition of λ∗, problem (P
λ)+ does not have any solution if λ > λ∗. In what
follows we claim that λ∗ is finite. Denote
m := min {ess infx∈Ωk(x), ess infx∈Ωh(x)} > 0.
Let λ′ > 0 be such that m(λ′ + tp−q) > λ
1t1−q for all t ≥ 0, where λ1 stands for the first
eigenvalue of (−∆) in H1
0(Ω). Denote by ϕ1 > 0 an eigenfunction of the Laplace operator
corresponding to λ1. Since u(λ) solves (Pλ)+ we have for all λ > λ′,
This implies that λ∗ ≤ λ′< +∞, which proves our claim.
Let us now prove that u(λ) is a minimal solution of (Pλ)+. Consider the sequence
(un)n≥0 defined by u0 = w (w is the unique solution of (5)) and un is the unique solution
of the problem
−∆un= λk(x)uqn−1+ h(x)upn−1, in Ω
un> 0, in Ω
un= 0, on ∂Ω .
Then, by the maximum principle, un ≤ un+1≤ u(λ). Moreover, by the same argument as
in [4], the sequence (un)n≥0 converges to u(λ). In order to show that u(λ) is a minimal
solution, let U be an arbitrary solution of problem (Pλ)+. Thus, by the maximum principle,
w = u0 ≤ U and, by recurrence, un ≤ U for all n ≥ 1. It follows that u(λ) ≤ U . At this
stage it is easy to deduce that the mapping λ 7−→ u(λ) is increasing. Fix 0 < λ1 < λ2 < λ∗.
Then u(λ2) is a super-solution of problem (Pλ1)+, hence, by minimality, u(λ1) ≤ u(λ2).
The fact that λ1 < λ2, combined with the maximum principle implies that u(λ1) < u(λ2).
It remains to show that problem (Pλ)+ has a solution if λ = λ∗. For this purpose it is
enough to prove that (u(λ)) is bounded in H01(Ω) as λ → λ∗. Thus, up to a subsequence,
u(λ) ⇀ u∗ in H1
0(Ω) as λ → λ∗, which implies that u∗ is a weak solution of (Pλ)+ provided
that λ = λ∗. Moreover, since the mapping λ 7−→ u(λ) is increasing, it follows that u∗ ≥ 0
a.e. on Ω and u∗> 0 on a subset of Ω with positive measure. A key ingredient of the proof
is that all solutions u(λ) have negative energy. More precisely, if E : H1
0(Ω) → R is defined by E(u) := 1 2 Z Ω |∇u|2dx − λ q + 1 Z Ω k(x)|u|q+1dx − 1 p + 1 Z Ω h(x)|u|p+1dx then
E(u(λ)) < 0 for all λ ∈ (0, λ∗). (8)
To deduce (8) we split the proof into the following steps:
(i) the solution u(λ) is semi-stable, that is, the linearized operator −∆−λqk(x)u(λ)q−1− ph(x)u(λ)p−1 is coercive:
Z
Ω
|∇ψ|2− (λqk(x)u(λ)q−1+ ph(x)u(λ)p−1)ψ2 dx ≥ 0 for all ψ ∈ H1 0(Ω).
Therefore Z
Ω
|∇u(λ)|2− (λqk(x)u(λ)q+1+ ph(x)u(λ)p+1) dx ≥ 0. (9)
This follows by the same proof as in [37, Theorem 1.9]. (ii) Since u(λ) is a solution of (Pλ)+ we have
Z Ω |∇u(λ)|2dx = λ Z Ω k(x)u(λ)q+1dx + Z Ω h(x)u(λ)p+1dx. (10)
Combining relations (9) and (10) we deduce that
Next, we observe that relation (10) implies E(u(λ)) = λ 1 2− 1 q + 1 Z Ω k(x)u(λ)q+1dx + 1 2 − 1 p + 1 Z Ω h(x)u(λ)p+1dx = −λ 1 − q 2(q + 1) Z Ω k(x)u(λ)q+1dx + p − 1 2(p + 1) Z Ω h(x)u(λ)p+1dx ≤ −λ 1 − q 2(q + 1) Z Ω k(x)u(λ)q+1dx + λ 1 − q 2(p + 1) Z Ω k(x)u(λ)p+1dx ≤ 0, by (11).
To complete the proof, it remains to argue that supλ<λ∗ku(λ)k
H1
0 < +∞. This follows
after combining relations (8) and (9), Sobolev embeddings, and using the fact that k, h ∈ L∞(Ω). This completes the proof.
3
Proof of Theorem 1.2
The energy functional associated to problem (Pλ)− is Fλ : H01(Ω) → R and it is defined by
Fλ(u) := 1 2 Z Ω |∇u|2dx − λ q + 1 Z Ω k(x)|u|q+1dx + 1 p + 1 Z Ω h(x)|u|p+1dx . Set kuk := Z Ω |∇u|2dx 1/2 ; kukq+1 := Z Ω |u|q+1dx 1/(q+1) ; kukp+1:= Z Ω |u|p+1dx 1/(p+1) .
We first prove that Fλ is coercive. Indeed,
Fλ(u) ≥ 1 2kuk 2− C 1kukq+1q+1+ C2kukp+1p+1, where C1 = λ(q + 1)−1kkkL∞ and C
2 = (p + 1)−1essinfx∈Ωh(x) are positive constants.
Since q < p, a straightforward computation shows that the mapping (0, +∞) ∋ t 7−→ Atp+1− Btq+1 attains its global minimum m < 0 at
t = B(q + 1) A(p + 1) 1/(p−q) . Therefore Fλ(u) ≥ 1 2kuk 2+ m ,
hence Fλ(u) → +∞ as kuk → ∞.
Let (un) be a minimizing sequence of Fλ in H01(Ω). Since Fλ is coercive, it follows that
(un) is bounded. Without loss of generality, we may assume that un is non-negative and
that (un) converges weakly to some u in H01(Ω). Standard arguments based on the lower
semi-continuity of the energy functional show that u is a global minimizer of Fλ, hence a
non-negative solution of problem (Pλ)−.
to prove that the non-negative solution is nontrivial, it suffices to prove that there exists Λ > 0 such that
inf
u∈H1 0(Ω)
Fλ(u) < 0 for all λ > Λ.
For this purpose we consider the constrained minimization problem
Λ := inf 1 2 Z Ω |∇v|2dx + 1 p + 1 Z Ω h(x)|v|p+1dx; v ∈ H01(Ω) and 1 q + 1 Z Ω k(x)|v|q+1dx = 1 . (12) Let (vn) be an arbitrary minimizing sequence for this problem. Then (vn) is bounded, hence
we can assume that it weakly converges to some v ∈ H1
0(Ω) with 1 q + 1 Z Ω k(x)|v|q+1dx = 1 and Λ := 1 2 Z Ω |∇v|2dx + 1 p + 1 Z Ω h(x)|v|p+1dx .
Thus, Fλ(v) = Λ − λ < 0 for all λ > Λ.
Set
λ∗:= inf{λ > 0; problem (Pλ)− admits a nontrivial weak solution} ≥ 0.
The above remarks show that Λ ≥ λ∗ and that problem (P
λ)− has a solution for all
λ > Λ. We now argue that problem (Pλ)− has a solution for all λ > λ∗. Fix λ > λ∗. By
the definition of λ∗, there exists µ ∈ (λ∗, λ) such that F
µ has a nontrivial critical point
uµ∈ H01(Ω). Since µ < λ, it follows that uµ is a sub-solution of problem (Pλ)−. We now
want to construct a super-solution that dominates uµ. For this purpose we consider the
constrained minimization problem
infFλ(v); v ∈ H01(Ω) and v ≥ uµ . (13)
The same arguments as those used to treat (12) show that problem (13) has a solution uλ≥ uµ. Moreover, uλ is a solution of problem (Pλ)−, for all λ > λ∗. With the arguments
developed in [17, p. 712] we deduce that problem (Pλ)−has a solution if λ = λ∗. The same
monotonicity arguments as above show that (Pλ)− does not have any solution if λ < λ∗.
Fix λ > λ∗. It remains to argue that the non-negative weak solution u is, in fact,
positive. Indeed, using similar arguments as in Pucci and Servadei [36], which are based on the Moser iteration, we obtain that u ∈ L∞(Ω). Next, by bootstrap regularity, u is a
classical solution of problem (Pλ)−. Since u is a non-negative smooth weak solution of the
differential inequality ∆u − h(x)up ≤ 0 in Ω, with p > 1, we deduce that u > 0 in Ω. This
follows by applying the methods developed in Section 4.8 of Pucci and Serrin [35] and the comments therein. This completes the proof.
Acknowledgments. The authors are grateful the anonymous referees for the careful reading of the paper and their suggestions. V. R˘adulescu acknowledges the support through Grant CNCSIS PCCE–8/2010 “Sisteme diferent¸iale ˆın analiza neliniar˘a ¸si aplicat¸ii”. D. Re-povˇs acknowledges the support by ARRS grant P1–0292–0101 “Topology and geometry”.
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