Ireneo Peral: Forty Years as Mentor

1 Introduction

In the following pages we, former students of Ireneo (to whom we owe so much of our achievements), will present a few lines about the contents of our Ph.D. theses.They were driven by Ireneo’s tireless enthusiasm and permanent scientific curiosity, always looking for the deep understanding of the problems, trying to reach limit or extreme cases in which interesting phenomena occur.In this brief tour, we will see problems with critical Sobolev exponents, concave-convex reaction terms, potentials with singularities of Hardy type, or related with Caffarelli–Kohn and Nirenberg inequalities, variational, distributional, viscosity, or entropy solutions, semi linearand fully nonlinear problems and in the most recent works, nonlocal operators such as the fractional Laplacian.In all cases, for brevity and to unify the presentation, we will focus on the contents of the thesis, so much of our later work with Ireneo falls out of the scope of this article, which in no way can be considered as a complete picture of his long curriculum.Instead of this viewpoint, we would like to emphasize the huge work of Ireneo as advisor of the initial research of many young students.For us, in addition to the very consolidated researcher, Ireneo is the mentor who has guided us in the beginning of our research career and, since then, the friend to ask for advice.

In this article our Ph.D. theses are presented as consecutive sections in chronological order.Moreover, each one of us has chosen some words that remind us of Ireneo and that we dedicate him with much affection.

We all feel very grateful to Ireneo; thanks to his ability and his work during these years we have formed a mathematical family. We also thank Magdalena Wallias, for taking care of all of us with great tenderness.

On the other hand, we would like this to also serve as a remembrance and tribute to one of us, Juan Antonio Aguilar, prematurely deceased. We all miss him very much.

2 The p-Laplace Operator, Nonlinear Eigenvalues and Critical Problems

Ph.D. Jesús García Azorero, 1989:

“… un río de aguas diáfanas que se precipitaban por un lecho de piedras pulidas, blancas y enormes como huevos prehistóricos. El mundo era tan reciente, que muchas cosas carecían de nombre …”

Gabriel García Márquez – Cien años de soledad

Our work started in 1985 (that is, a few years ago …), trying to understand and explore some quasilinear elliptic problems around the p-Laplace operator.At this time, the famous paper by Brezis–Nirenberg about semilinear problems with critical exponent [42] was one of the main reasons for a growing interest of the PDE community in the variational analysis of this kind of problems. A natural question was to extend the results obtained in the Hilbertian setting in H01,2 to the general W01,p framework, that is, to pass from the Laplacian to the p-Laplacian.The starting point was to study

where we consider the simplest case,f⁢(x,u)=|u|α-2⁢u.The first step was to study the subcritical case, α<p*=N⁢pN-p, p* being the critical Sobolev exponent, since in this case the Palais–Smale compactness condition follows easily from the compactness in the Sobolev embedding.Moreover, the symmetry on the right-hand side allows us to deal with an even energy functional, and minimax methods can be used to get infinitely many solutions when 1<α<p and when p<α<p*. In both cases, the difference between the homogeneity in the differential operator (Δp) and the nonlinear right-hand side implies that one can get solutions for f⁢(u)=λ⁢|u|α-2⁢u for all λ>0, if one knows the solutions for λ=1.This difference of homogeneities fails when α=p, and for this reason this case was much more interesting. In fact, the problem was to find the nonlinear eigenvalues for

that is, to find the values of λ that allow us to get nontrivial solutions for that problem. We use the Ljusternik–Schnirelman theory to analyze the non-quadratic Raleigh quotient

proving the existence of a sequence of nonlinear eigenvalues λk→∞.

For this sequence of eigenvalues (as of now, it is not known yet if the p-Laplacian admits some other eigenvalues, different from the Ljusternik–Schnirelman sequence), we were able to prove the asymptotic behavior

This asymptotic estimate was the key to apply some techniques introduced by Rabinowitz [80] to study a non-symmetric problem with f⁢(x,u)=g⁢(x,u)+h⁢(x).The problem is easy if h⁢(x) is small in a suitable norm, but it becomes much more involved if we try to avoid this restriction.The result says that if Ω is a bounded set in ℝN, which is the product of N bounded intervals in ℝ, and h∈Lq′⁢(Ω),with q′<p⁢qN⁢(q-p)-1, then the problem has infinitely many solutions.That is, a p-Laplace counterpart of a result by Rabinowitz in the case p=2 that uses in an essential way the asymptotic estimate for the eigenvalues that we proved before.

Also, we analyze a different problem with non-symmetric energy functional with a fast-growing nonlinearity F⁢(x,u)=λ⁢eu, which is admissible if N⩽p.That is, the p-Laplace version of the Emden–Fowler problem.We showed the following results:

1. There exists a positive constant λ1 such that if λ>λ1 and the domain is star-shaped, then there is no positive solution (Pohozaev-type result).

2. There exists a constant λ0⩽λ1 such that if 0<λ<λ0, then the problem has at least two positive solutions.

The next step was to study problems with a critical term. The idea was to adapt the Brezis–Nirenberg program to the non-hilbertian case.The lack of compactness in the Sobolev embedding was harder to handle in the absence of a scalar product that fits well with the weak convergence.The Palais–Smale condition analysis was harder, and the key was the use of the concentration-compactness lemma by Lions; more precisely, the version that was published in the Revista Matemática Iberoamericana in 1985 [74], which has become since then an essential tool when dealing with these critical problems.With the help of that lemma, we were able to analyze critical problems with f⁢(u)=|u|p*-2⁢u+λ⁢|u|q-2⁢u, getting the following results:

1. If p<q<p*, then there exists a constant λ0 such that the critical problem with f⁢(u)=|u|p*-2⁢u+λ⁢|u|q-2⁢u has at least a positive solution.

2. If max⁡{p,p*-pp-1}<q<p*, then the problem has at least a positive solution for any λ>0.

3. If q=p, N⩾p2 and 0<λ<λ1 (first eigenvalue), then the problem has at least a positive solution.

Up to this point, these results were counterparts of some of the Brezis–Nirenberg ones for p=2. But we also studied the case 1<q<p that was not covered in the original work when p=2 (see [42]). In this case, we proved the existence of a constant λ0 such that if 0<λ<λ0, then the problem has infinitely many solutions (at least one of them being positive).The idea was to take advantage of the geometry of the functional, using a suitable truncation to localize critical points with negative energy.The theorem proved multiplicity of solutions for a critical problem, but it was far from being optimal, and the so-called concave-convex problems have generated a rich literature over the last years.In fact, the solutions that we found were small perturbations of the trivial one, and the problem of finding solutions with large energy, via some application of the mountain pass lemma remained open.A fundamental step forward in this direction, in the critical case when p=2, was the beautiful paper by Ambrosetti, Brezis and Cerami [25], where the second positive solution was found, jointly with a sharp estimate of the interval of existence for the parameter λ. The extension of this result to the case p≠2 was obtained later on in 2000 [64].

So far we have included the contents of the Ph.D. thesis. The details can be seen in [65, 66].There were further developments and researches, but its presentation is not the aim of these lines.Nevertheless, it is important to point out that after the completion of the thesis, we had the honor to meet Prof. Antonio Ambrosetti, and we started a fruitful collaboration.Since then he has been a true friend.

3 Quasilinear Problems with Nonlinear Diffusion

Ph.D. Juan Antonio Aguilar Crespo, 1998:

“One man come in the name of love“One man come and go.“One man come he to justify“One man to overthrow. “In the name of love“What more in the name of love.“In the name of love“What more in the name of love.”

U2 – Pride (In The Name Of Love)

His thesis dealt with quasilinear parabolic problems with nonlinear diffusion and some stationary related problems, namely,

where p>1, λ>0, f⁢(u) is a nonlinear reaction term and V⁢(x) satisfies certain integrability conditions.

The common point of all the problems was the critical and supercritical character, in the sense that they are borderline for the classical techniques.

The first chapter studied the Dirichlet elliptic problem with f⁢(u)=eu, which appears in chemical reactions whose reaction rate follows the Arrhenius law.For a constant V⁢(x)=V, this Emden–Fowler problem was very well known, and the goal was to understand the influence of the potential V⁢(x):

where Ω is a bounded domain in ℝN and V⁢(x)∈Lq⁢(Ω) can be a sign-changing function.The main results are the following:

1. For λ small, there exists at least one solution to (3.1) if q>Np>1, and there exist at least two solutions if p⩾N.

2. A sufficient condition assures the existence of positive solutions when V is a sign-changing function.

3. For λ large enough and V⩾0, the problem has no solution.Moreover, the set of λ such that (3.1) has a solution is an interval whose supremum is denoted by λ*.

4. If V⁢(x)∈Lq⁢(Ω), q>Np>1 and λn↗λ*, then there exists a sequence of regular solutions un corresponding to λn such that:

   1. If N⩾p⁢q⁢(3+p)(4+q⁢(p-1)), then limn→∞⁡un=u* is a singular (non-variational) solution corresponding to λ*.

   2. If N<p⁢q⁢(3+p)(4+q⁢(p-1)), then limn→∞⁡un=u*∈L∞ is a regular solution corresponding to λ*.

5. For N>p and V⁢(x)=|x|l, l>-p, Ω=B1, there exists a unique radial singular solution corresponding to

   λ ~ = ( l + p ) p - 1 ⁢ ( N - p ) , S ⁢ ( r ) = log ⁡ ( 1 | x | l + p ) ,

   and it follows that:

   1. If N⩾p+4⁢(l+p)p-1, then λ*=λ~, and for every λ<λ*, there exists a unique regular solutionu¯⁢(λ) and limλ→λ*⁡u¯⁢(λ)=S⁢(x).

   2. If p<N<p+4⁢(l+p)p-1, then λ~<λ*, and for λ=λ~, there exist infinite radial regular solutions, that tend to infinity at the origin and satisfy limλ→λ*⁡u¯⁢(λ)=u*∈L∞.

In the case p⩾N, variational techniques were used, while for 1<p<N, a classical iteration method was used to get the existence of solutions when λ is small.

In the second chapter of the thesis, the problem was to study the lack of compactness in sequences uk of singular solutions to

where Ω is a bounded domain, 1<q⩽∞, 1q+1q′=1 and

The main result says that there exists a subsequence ukl satisfying one of the following statements:

1. u k l is bounded in Lloc∞⁢(Ω).

2. u k l → - ∞ uniformly in compact subsets.

3. The set S of the points x∈Ω such that there exists a sequence xkl∈Ω, xkl→x satisfying ukl⁢(xkl)→∞ is finite, not empty, and ukl⁢(xkl)→∞ uniformly in compact subsets in Ω∖S.

This is an extension of the results for N=2 that can be found in [41].

The third chapter studied a concave-convex and supercritical elliptic problem with f⁢(u)=λ⁢uq-1+uα-1 in the unit ball B1, where λ>0,1<q⩽p<N and α>p*.For an extremal value of the supercritical power,

the following hold:

1. There exists a value λ* such that this problem has a singular radial solution S.

2. If wj denotes a sequence of radial regular solutions of the corresponding problem with λj such that ∥wj∥∞→∞ when j→∞, then λj→λ*.

3. If α<α~, then there exists a sequence of regular radial solutions wj of the problem such that λj=λ* and wj⁢(0)→∞ when j→∞.

4. If α⩾α~, then for every sequence wj of radial regular solutions of the problem with wj⁢(0)→∞ when j→∞, we have λj→λ* for j⩾j0 large enough.

5. If wj denotes a sequence of regular radial solutions corresponding to λj such that ∥wj∥∞→∞ when j→∞ and S is as in (a), then wj→S when j→∞ uniformly in compact subsets not containing the origin.Moreover, wj→S∈Lβ⁢(B1) and in W01,p⁢(B1) with β<N⁢(α-p)p.

6. Analogous results are obtained for |x|l,l>-p.

Finally, the fourth chapter was devoted to critical parabolic problems with Hardy type potentials, namely,

where 1<p<N and λ>0, getting the following results:

1. There exists a self-similar solution S for 1<p<2⁢NN+1, with u0≡0 if λ>μN,p=(p2-p)p-1⁢(N-p2-p).

2. If 1<p<N, 0<λ<ΛN,p=(N-pp)p and u0∈L2⁢(B), then there exists a global solution of the problem.

3. If 1<p<2⁢N(N+1), λ>μN,p and u0∈L2⁢(ℝN)∩Ls⁢(ℝN), with s=N⁢(2-p)p, then there exists a finite time T* (depending on N,p,λ and ∥u0∥s) for the solution S found in (a).

All these results can be seen in the references [18, 19, 20, 21, 22].

4 Elliptic and Parabolic Problems Related to Caffarelli–Kohn–Nirenberg Inequalities

Ph.D. Boumediene Abdellaoui, 2003:

“The scientist is the man by whom the distinction is easily madebetween frankness and falsehood in words,between truth and error in convictions,between beauty and ugliness in acts.”

Emir Abd el Kader

This thesis focused on elliptic and parabolic problemsrelated with the well-known Caffarelli–Kohn–Nirenberg inequalities (CKN for brevity), see [46].

The connection of our problems with CKN inequalities is that they were either the Euler equation for some minimization problems associated to some of these inequalities, or perturbations of these, or parabolic problems whose elliptic part was in one of the previous classes.

The common factor of the family of problems under consideration was the critical character, which can be assimilated to some extreme cases ofwell known results.

The thesis is divided into two main parts: Part (I) elliptic problems, Part (II) parabolic problems.

A particular and interesting case of the CKN inequalities are the well-known weighted Hardy–Sobolev inequalities for admissible weights.

The family of equations associated to the above inequality is

for the elliptic case, and

for the parabolic case.These equations and some of their variants were the object of our study.It is clear that the natural framework to study the elliptic problems from the variational point of view is the weighted Sobolev space 𝒟0,γ1,p.Moreover, if γ is very negative, the elements of 𝒟0,γ1,p are very singular at the origin, not necessarily distributions.

Some surprising results were found in relation with the previous results given by Baras–Goldstein [29] and Brezis–Cabré [40].

In the first part of the thesis we treated semilinear problems of the form

where N⩾3, λ⩾0 and 0<q⩽1<p⩽N+2N-2.We suppose that Ω is a regular bounded domain that contains theorigin or that Ω=ℝN and h and g are non-negative functions.

Under suitable hypotheses on h and g, we proved the existence or non-existence of positive solutions.In particular, in the concave case (q<1), existence of a second positive solution was obtained for all λ<λ*, where

In the case where q=1, p=2*-1 and Ω=ℝN, using the concentration-compactness lemmas of Lions and some qualitative properties of the coefficients h and g, we were able to get the existence and multiplicity of positive solutions. If the set where g reaches it maximum is infinite and bounded, then using the category theory of Ljusternik–Schnirelmann, we get the existence of infinitely many positive solutions.Details can be seen in [9, 7].

Next we treated quasilinear problems with general admissible weights, namely, we have considered the family of problems

with λ⩾0, -∞<γ<N-pp, 1<p<N and 0∈Ω, which is the nontrivial case.

We take as a model the function f⁢(λ,x,u)≡λ⁢uα⁢|x|-p⁢(γ+1). As we said before, these equations are related with the CKN inequalities obtained in [46].

When f⁢(λ,x,u)=g⁢(x) with g∈L1⁢(Ω), the variationalframework is not valid and it is necessary to find a new concept of solution, insuch a way that we get existence and uniqueness of solutions.

If -(p-1)⁢N+pp⩽γ<N-pp, it is possible to use the concept of entropy solutions.If γ⩽-(p-1)⁢N+pp, the weight is very degenerate at the origin and it was necessary to extend the sense of entropy solutions in order to get existence and uniqueness for problem (4.2), where g∈L1⁢(Ω).Notice that, in this case, if u∈𝒟0,γ1,p⁢(Ω), then we do not necessarily have u∈𝒟′⁢(Ω).

The main results obtained for this problem can be summarized in the following points:

1. If α>p-1, then, independently of λ, problem (4.2) does not have any entropy positive local solution.

2. If α=p-1, then there is no entropy positive local solution when λ>ΛN,p,γ.

3. As consequence of the non-existence results, a complete blow-up was obtained for a family of approximated problems.

4. For p=2, we have proved a generalization of a Weylregularity result for our operator when γ≠0.

These results were published in [11, 10].

In the parabolic case we dealt mainly with the following problem:

In this type of problems, solutions are understood in the entropy sense.The results in this case are part of the papers [3, 12, 13].

In the elliptic case we have obtained a non-existence and a blow-up result independently of the value of γ.For p=2, since the elliptic problem has similar properties of non-existence and instantaneous blow-up as the case γ=0, it could be conjectured that, roughly, the parabolic problem has the same instantaneous and complete blow-up behavior as the Baras–Goldstein case [29].However, this conjecture is not true.

In the case (1+γ)⩽0, problem (4.3), with p=2 and α=1, has a weak global solution for all λ∈ℝ, namely, there is no blow-up phenomenon.Moreover, the corresponding optimal Hardy–Sobolev constant is not achieved and it is the same for any domain containing the origin.

The main difference between the case (1+γ)>0 and (1+γ)⩽0 is that when (1+γ)>0, the associated homogeneous equation verifies the parabolic Harnack inequality; see, for instance, [54].

For the case p=2, α>1 and γ+1>0, a simple analysis of the homogeneity of the operator shows that the instantaneous and complete blow-up behavior is independent of λ.

After establishing our results, one can say that the Harnack inequality and the optimal behavior of the Hardy–Sobolev constant are the main properties to get instantaneous and complete blow-up.

For the quasilinear general problem, we have proved the following results:

1. For p>2 and γ+1>0, we established a weak version of the Harnack inequality inspired by the work of Di Benedetto.

2. As a consequence, we extended the non-existence and blow-up results as in the case p=2.

3. For p<2, the weak version of the Harnack inequality does not hold (including for γ+1>0) and we proved existence of solutions for all λ.

4. For γ+1<0, existence of a local solution was obtained for all p and for all α>1.

Let us remark that the arguments that we used in our proofs were quite different from the techniques used by Baras–Goldstein, based on a representation formula.

5 Elliptic and Parabolic Problems with Dirichlet–Neumann Boundary Conditions

Ph.D. Eduardo Colorado, 2004:

“Intuition is the source of scientific knowledge.”

Aristotle

I started my Ph.D. in the summer of 1999 after I finished my degree thesis with Ireneo as advisor in the spring that year.Ireneo had in mind a problem proposed by professor G. I. Barenblatt, dealing with a boundary value problem for the Laplacian operator.More precisely, the boundary conditions were of mixed Dirichlet–Neumann type, and the idea consisted on moving the boundary conditions. One can think of it as passing to the limit as the (N-1)-Hausdorff dimensional measure of the Dirichlet part of the boundary decreases to zero in some appropriate way, and try to understand what happens in the limit problem, which, a priori, should be a Neumann-type problem.The first difficulty we found was that there was not so much bibliography about mixed Dirichlet–Neumann problems, and even worse, we did not know any reference for the moving boundary conditions.

We started by studying the regularity results of Miranda and Stampacchia [78, 84].In those papers it is proved that the highest regularity one can expect for solutions is 𝒞α with 0<α⩽1/2, as Shamir [82] pointed out.Those results are established for fixed Dirichlet–Neumann boundary conditions, hence one should track the constants appearing in the regularity results (that depend on the boundary conditions) in order to find uniform bounds in terms of the boundary conditions.In this way the uniform bound allows passing to the limit when the boundary conditions are moving, for example, passing from Dirichlet to Neumann boundary conditions through the mixed Dirichlet–Neumann conditions in some way to be specified.Those three papers, [78, 82, 84], including [59], by De Giorgi, were the starting point of my Ph.D.As a common fact in the theses supervised by Ireneo, my Ph.D. thesis has two main parts, the first one is elliptic, while the second one is parabolic.

First, we present the results corresponding to elliptic problems.In the first paper of my Ph.D. [55], we studied the following problem:

where Ω⊂ℝN (N⩾3) is a smooth bounded domain,1<r<2*-1=N+2N-2, 0<q<r, and the boundary conditions are of Dirichlet–Neumann type as in the whole thesis. Also

where Σi, i=1,2, are smooth (N-1)-dimensional submanifolds of ∂⁡Ω such that

The natural energy space for looking solutions as critical points of the Euler–Lagrange functional associated to (5.1) is

which is defined as the closure of𝒞c1⁢(Ω∪Σ2) with the norm of the Sobolev space W1,2⁢(Ω).

Problem (5.1) is called a concave-convex problem. In this case, we extend the results of the famous paper by Ambrosetti, Brezis and Cerami [25], to the case of mixed boundary conditions.

For fixed Dirichlet–Neumann boundary conditions, we proved a complete result for (5.1) as in [25], i.e., existence and multiplicity of positive solutions. Precisely, we proved the following:

1. There exists Λ>0 such that for 0<λ<Λ, problem (5.1) has at least two solutions, and one of them is a minimal solution.

2. In the extremal λ=Λ, there is at least one solution that is obtained as limit of the minimal one when λ→Λ.Furthermore, there is no solution of (5.1) for λ>Λ.

Also we proved a Liouville-type result for mixed Dirichlet–Neumann boundary conditions.That result allowed us to show a uniform L∞ bound for solutions of (5.1), inspired by the famous result by Gidas and Spruck [67], i.e., supx∈Ω⁡uλ⁢(x)⩽C<∞ for any solution uλ of problem (5.1).Even more, we introduced and studied the behavior of solutions to these problems moving the boundary conditions in a regular way.Precisely, we considered a family of smooth (N-1)-dimensional submanifolds of∂⁡Ω, {Σ1⁢(α)}α∈Iε, where Iε=[ε,|∂⁡Ω|] (and |⋅| means the measure in the appropriate dimension), such that

Thus, assuming the family {Σα}α∈Iεto satisfy (5.3) and letting

be the family of all solutions to the problems (5.1), with α∈Iε, we proved the existence of a positive constant Mε such that

Following the study of the mixed Dirichlet–Neumann boundary conditions, we introduced the Caffarelli–Kohn–Nirenberg potentials [46] into the operator -Δ2,γ⁢u:=-div⁡(|x|-2⁢γ⁢∇⁡u), with -∞<γ<N-22. We also considered the associated (p,γ)-Laplace operator to the corresponding Caffarelli–Kohn–Nirenberg potentials, i.e.,-Δp,γ⁢u=-div⁡(|x|-p⁢γ⁢|∇⁡u|p-2⁢∇⁡u), where 1<p<N, -∞<γ<N-pp.Here the model problem we studied is

where the boundary conditions are given by

with Σi, i=1,2, being smooth (N-1)-dimensional submanifolds of ∂⁡Ω satisfying (5.2).

The study of (5.4) with different types of nonlinearities fλ,γ was done in [56, 57, 4, 5].To be more precise, in [56] we analyzed the subcritical nonlinearities and proved bifurcation results.Also we extended the classical results about global Hölder continuity by using the De Giorgi and Stampacchia techniques [59, 84] adapted to our framework.

With respect to the boundary conditions, we assumed that the (N-1)-dimensionalHausdorff measure of Σ1 is α, hence 0⩽α⩽ℋN-1⁢(∂⁡Ω).Then we proved that λ1⁢(α)→λ1Das α→ℋN-1⁢(∂⁡Ω), with λ1D being the first eigenvalue of the Dirichlet problem.Also, a corresponding eigenfunction sequence converges to an eigenfunction of the Dirichlet problem associated to λ1D.Even more, λ1⁢(α)→0 as α→0 and a sequence of eigenfunctions converges to a constant, i.e., we obtain the convergence to the Neumann Problem.

Next, in [4], we studied problem (5.4) with p=2, where the right-hand side of the equation given by fλ,γ⁢(x,u)=λ⁢|x|-2⁢(γ+1)⁢u+g⁢(x,u).Here, we were mainly interested in the behavior of the solutions for λ∈[0,ΛN,γ⁢(Ω,Σ1)], and according to the relation of thevalues of ΛN,γ⁢(Ω,Σ1) and ΛN,γ, defined as the best Hardy–Sobolev constant in Ω for mixed Dirichlet–Neumann boundary conditions and the best Hardy–Sobolev constant with Dirichlet boundary conditions, respectively.

The last work in the elliptic part of my dissertation, see [5], dealt with the mixed Dirichlet–Neumann problem (5.4) with p=2 and

Again, Ω is a bounded regular domain in ℝN with N⩾3 that contains the origin, 0<q⩽1, r⩽2*-1, -∞<γ<N-22 and 0⩽λ⩽ΛN,γ, the critical value defined above. In this framework, we showed the following:

1. The attainability of constants ΛN,γ⁢(Ω,Σ1) and SN,γ⁢(Ω,Σ1), where the latter is defined as the corresponding Sobolev constant with mixed boundary conditions.Precisely, on the one hand, we showed geometrical conditions on Ω, and on the other hand, for Ω fixed, we gave geometrical conditions on Σ1,Σ2 for which the constants Sγ⁢(Ω,Σ1) and ΛN,γ⁢(Ω,Σ1) are attained or not.

2. We also studied the corresponding concave-convex problem, i.e., when 0<q<1 and 1<r⩽2*-1, which includes the critical problem.Here we showed the general existence and multiplicity result as in [25, 55].For the doubly critical problem, i.e., when q=1 and r=2*-1, we proved geometrical sufficient conditions in order to have existence of solutions.Finally, for

   f λ , γ ⁢ ( x , u ) = λ ⁢ u q | x | 2 ⁢ β + u r | x | ( r + 1 ) ⁢ γ

   with 0<q⩽1<r<2*-1 and β⩽γ+1, i.e., with subcritical and also critical potential, we proved a uniform L∞ bound for the solutions when 0⩽γ<N-22, via a Liouville-type result that we proved. Also we analyzed the regularity of solutions as well as concentration-compactness in the spirit of [74], and trace results.

Next, we present the parabolic part.In this part we published two papers.The first one, dealing with Dirichlet boundary conditions, has been previously commented in Section 4, see [3]. In the second one, dealing with Dirichlet–Neumann boundary conditions, see [6], we studied the following parabolic problem:

where Ω⊂ℝN is a smooth bounded domain with 0∈Ω as before,-∞<γ<N-pp and

In particular, we considered the case where

with a⩾p-1.The main points under analysis were existence, non-existence and complete blow-up results related to some Hardy–Sobolev inequalities and a weak version of a Harnack inequality that holds for p⩾2 and γ+1>0.To finish, we gave some additional information on the behavior of the solution, and existence and non-existence of finite time extinction was also discussed.

6 Influence of Hardy Potential and Gradient Terms

Ph.D. Ana Primo, 2008:

“Decíamos ayer …”

Fray Luis de León

The protagonists in this thesis were the Hardy potential and terms that were either potentials of the unknown quantity or of the modulus of its gradient.The question was to study if these protagonists were competing or cooperating, and how.

The powers of the unknown quantity, that we assume to be positive, when appearing on the right-hand side of the equation, are known to represent reaction terms.On the other hand, the terms depending on the gradient, also on the right-hand side, describe certain growth problems; on the left-hand side, with a suitable structure, they are Euler equations of functionals in the calculus of variations.

The thesis starts with the following problem:

with Ω⊂ℝN a bounded open domain, λ∈ℝ, N⩾3, 0∈Ω and f a positive measurable function.

On the one hand, if the gradient term is on the right-hand side of the equation, i.e.,

it was proved that for all λ>0, a solution does not exist even in the weakest possible sense.This fact strongly emphasizes the influence of the Hardy term. Without such term, the problem

has a solution for a suitable datum f (see [23, 68] for more details). Moreover, if we change the Hardy term |x|-2 by a weight g∈Lm⁢(Ω) with m>N2, then there exists 0<λ0<λ1⁢(g) such that for all 0<λ<λ0, the problem has a weak solution for a suitable datum f.

Consequently, using this strong result about non-existence, it was proved in addition that the interaction between the Hardy term and the quadratic term in the gradient produces a complete blow-up.

On the other hand, if the gradient term appears on the left-hand side of the equation, i.e.,

it was proved that if λ⩾0, then there exists a solution for each function f∈L1⁢(Ω), f⩾0, without any restriction on the size of λ.In fact, an existence result for a wide class of weights was proved (admissible weights). In particular, Hardy’s term belongs to this class of weights.

The importance of this result follows from the strong regularizing effect of the gradient on the Hardy term. In [39], it is proved that for all λ>0, the equation

has no solution in general for a positive function f∈L1⁢(Ω).

These results appeared in [14].

Inspired on these results, we studied the following problem:

where 1⩽q⩽2, f,g are positive measurable functions and Ω is a bounded domain. We found conditions on g with respect to q in order to have a positive solution to the previous problem for all λ>0 and for all f∈L1, f⩾0.In particular, we focused our attention mainly on Hardy’s potential g⁢(x)=1|x|2 and we proved that the hypotheses on g are optimal.

The effect of the gradient is quite surprising. Without its presence, the linear problem

satisfies that if λj⁢(g) denotes the j-eigenvalue associated to the weight g with associated eigenvector φj, then the following hold:

1. If λ<λ1⁢(g), there exists a weak positive solution.

2. If λ1⁢(g) is reached and λ=λ1⁢(g), there exists a positive solution if ∫Ωf⁢φ1⁢𝑑x=0.

3. If λj⁢(g)<λ<λj+1⁢(g), there exist solutions that necessarily change sign. Thus, there does not exist a positive solution.

This result emphasizes that the term |∇⁡u|q on the left-hand side of the equation is enough to break any effect of resonance of the zero order linear term g⁢(x)⁢u.

On the other hand, we proved that if u is a positive solution to the problem, then u∈W01,p⁢(Ω), for all p⩽q if q⩾NN-1, and p<NN-1 otherwise.Comparing this with the regularity obtained by the renormalized solutions, we obtained that |∇⁡u|q produces a regularizing effect provided that q⩾NN-1.

These results appeared in [15].A result about general operators without any restriction on the sign of the function f appeared in [2].

In the parabolic case, we analyzed the influence of the presence of Hardy potential with semilinear terms:

where p>1, and u0⁢(x)⩾0 and f⩾0 are measurable functions in a suitable class.

The main difference with the heat equation is the existence of a critical exponent p+⁢(λ) such that for p⩾p+⁢(λ), there is no positive solution for any nontrivial datum f, while for p<p+⁢(λ), it is proved the existence under some additional conditions on the initial data f and u0.Moreover, we analyzed the Cauchy problem, which corresponds to the case Ω=ℝN, when p<p+⁢(λ), obtaining a Fujita-type exponent depending on λ.This result was published in [16].

The thesis finishes studying the joint influence on the heat equation of the Hardy potential and a power of the gradient on the right-hand side of the equation

where Ω⊂ℝN is a bounded open domain with N⩾3, 0∈Ω, p>1 and λ>0.

In the case λ=0 and p=2, this model appears in the physical theory of growth and roughening of surfaces, where it is known as a Kardar–Parisi–Zhang model, see [69]. A modification of the Kardar–Parisi–Zhang problem has been considered by Berestycki, Kamin and Sivashinsky in [37], as a flame propagation model. The authors studied the existence of a critical exponent p+⁢(λ) in order to have existence or non-existence of solutions. The corresponding Cauchy problem is studied too (see [17]).

7 Fully Nonlinear Equations with Improper Zero-Order Terms

Ph.D. Fernando Charro, 2009:

“Take everything you like seriously, except yourselves.”

Rudyard Kipling