Let Ω be a bounded domain in with a smooth boundary , let , and let . Consider the Dirichlet problem
where , , denotes the r-Laplacian, namely,
the reaction term satisfies Carathéodory’s conditions, while, as usual, if and only if . Elliptic equations involving differential operators of the form
often called -Laplacian, occur in many important concrete situations. For instance, this happens when one seeks stationary solutions to the reaction-diffusion system
which exhibits a wide range of applications in physics and related sciences such as biophysics, quantum and plasma physics, and chemical reaction design; see [3, 6]. Consequently, they have been the subject of numerous investigations, both in bounded domains and on the whole space, mainly concerning the multiplicity of solutions or bifurcation-type results.
This paper falls within the first framework. We show that if, roughly speaking, f has a subcritical growth and, moreover,
uniformly in , where denotes the first eigenvalue of ,
uniformly in ,
for all , where , , while ,
Now, the question of investigating what happens if there is resonance from the right of , i.e., the limit in (1.2) equals , naturally arises. Accordingly, φ turns out to be indefinite and direct methods no longer work. However, via linking arguments and, in place of (ii), via the hypothesis that
where , or and
It should be also noted that, in both settings, due to (ii), the nonlinearity exhibits a concave behavior at the origin. Such a type of growth rate has been widely studied, also combined with further conditions, provided and , i.e., the equation is semilinear. As an example, besides the seminal paper , let us mention [21, 22, 16, 8]. A similar comment holds true also when but , in which case the literature looks to be daily increasing; see for instance the very recent papers [12, 19, 14, 18] and, concerning the nonsmooth framework, [17, 13].
Another meaningful feature of (1.1) is the following. If , then the differential operator turns out to be nonhomogeneous. Hence, standard results for the p-Laplacian not always extend in a simple way to it.
Our approach is variational, based on critical point theory, together with appropriate truncation-comparison arguments and results from Morse theory.
2 Mathematical Background
Let be a real Banach space. Given a set , write for the closure of V and for the boundary of V. If , , then , while . The symbol denotes the dual space of X, indicates the duality brackets for the pair , and (respectively, ) in X means that ‘the sequence converges strongly (respectively, weakly) in X’. An operator is called of type provided
Let and let . Put
We say that φ satisfies the Cerami condition when
every sequence such that is bounded and
admits a strongly convergent subsequence.
This compactness-type assumption turns out to be weaker than the usual Palais–Smale condition. Nevertheless, it suffices to prove a deformation theorem, from which the minimax theory for the critical values of φ follows. In such a framework, the topological notion of linking sets plays a key role.
Suppose are three nonempty closed subsets of a Hausdorff topological space Y with . The pair links E in Y if and, for every such that , one has .
The following general minimax principle is well known; see, e.g., [10, Theorem 5.2.5].
Let X be a Banach space, let , and E be such that the pair links E in X, and let satisfy condition (C). If, moreover, and
then and .
Appropriate choices of linking sets in Theorem 2.2 produce meaningful critical point results. For later use, we state here the famous Ambrosetti–Rabinowitz mountain pass theorem.
If is a Banach space, fulfills (C), , ,
then and .
Let be a topological pair such that and let k be any nonnegative integer. We denote by the k-th relative singular homology group for the pair with integer coefficients. Given an isolated critical point ,
is the k-th critical group of φ at . Here, U indicates any neighborhood of fulfilling . The excision property of singular homology ensures that this definition does not depend on the choice of U. The monographs [5, 11] are general references on this subject.
Hereafter, stands for the -norm, while denotes the N-dimensional Lebesgue measure of . If , then indicates the conjugate exponent of p and is the usual norm of the Sobolev space , namely, thanks to the Poincaré inequality,
Let and let . The symbol means for almost every , , as well as . It is known that provided . Next, define
With the standard norm of , this set is an ordered Banach space whose positive cone
has nonempty interior given by
where denotes the outward unit normal on ; see [10, Remark 6.2.10]. If
then, due to the continuous embedding and the Poincaré inequality, one has
Let be the dual space of and let be the nonlinear operator stemming from the negative p-Laplacian, i.e.,
for all .
There is a unique eigenfunction corresponding to such that
Any other eigenfunction is a scalar multiple of .
The operator is bounded, continuous, strictly monotone, and of type .
Now, with , and f as in Section 1, suppose that
for appropriate , put
and consider the -functional given by
The next result establishes a relation between local -minimizers and local -minimizers of φ. Its proof is the same as that of [1, Proposition 2], with the -Laplacian instead of the differential operator considered therein. This idea goes back to the pioneering works of Brézis and Nirenberg  for and García Azorero, Manfredi, and Peral Alonso  when .
If is a local -minimizer of φ, then lies in for some and turns out to be a local -minimizer of φ.
Finally, we shall write for every . The function
is often called the Nemytskii operator associated with f. Moreover, given and ,
The meaning of etc. is analogous.
3 Resonance from the Left
To avoid unnecessary technicalities, ‘for every ’ will take the place of ‘for almost every ’ and the variable x will be omitted when no confusion may arise. Moreover, if and only if and . We will posit the following assumptions, where F is given by (2.3).
For appropriate , one has
uniformly in .
uniformly in .
There exist and , such that
The energy functional stemming from (1.1) is defined by
Clearly, . Moreover, once
one has , , while the associated truncated functionals
turn out to be as well.
We will verify the conclusion for , the other cases being similar. The space compactly embeds in while the Nemytskii operator turns out to be continuous on . Thus, a standard argument ensures that is weakly sequentially lower semicontinuous. In view of (h3), given any , there exists such that
which clearly means that
After integration, we obtain
Now, suppose by contradiction that there exists a sequence such that
Write . Since , passing to a subsequence when necessary, one has
Fix any and, through (h2), choose fulfilling
Moreover, set . From (3.3) it evidently follows that
because as soon as , while .
We claim that is bounded in . In fact, if the assertion were false, then, up to subsequences, . Dividing (3.4) by gives
Recall next that , but only when . As and , we get
against (3.4). Consequently, the claim holds true.
Finally, also the sequence is bounded in , because and for all . This completes the proof. ∎
Since but if and only if , for sufficiently small , the right-hand side in the above inequality turns out to be negative, which evidently forces , namely, . Proceeding as in [20, Theorem 4.1] then gives . Moreover, is a local -minimizer of φ, because . Now, the conclusion follows from Proposition 2.4. A similar argument yields a function with the asserted properties. ∎
To establish the existence of a third nodal solution, we will first show that there exist two extremal constant-sign solutions, i.e., a smallest positive one and a biggest negative one. In fact, through (h1) and (h4) one has
where . Thus, it is quite natural to compare solutions of (1.1) with those of the auxiliary problem
Let u be a positive solution of (1.1). For every , define the functions
, as well as
Obviously, the functional η belongs to , is coercive, and weakly sequentially lower semicontinuous. So, there exists such that
As in the above proof, for sufficiently small , we have , whence and, a fortiori, . Now, from (3.9) it follows that
By (p5), this evidently forces . Through (3.10) and (3.8) we thus see that the function is a nonnegative nontrivial solution of (3.7). Since, due to [23, Theorem 5.4.1 and Theorem 5.5.1], , while (3.7) possesses a unique positive solution, we get , and the desired inequality follows. A similar argument works for the other conclusion. ∎
Weaker versions of (h4) allow to achieve the last two lemmas, namely,
From now on, Σ will denote the set of all solutions to (1.1), while
Proceeding exactly as in the proof of [20, Lemma 4.2], one obtains the next result.
A mountain pass procedure can now provide a third solution, but in order to exclude that it is the trivial one, we need further information on the critical groups of φ at zero, which will be achieved as in . This is the point where (h4) plays a crucial role.
where , , are suitable constants, while . Consequently,
whenever is sufficiently small, say for some . Thus, in particular, if , , and , then
with appropriate . Hence,
Since , we get
i.e., for small enough. Summing up, given any , either as soon as or
Moreover, if , then and
Let be defined by
We claim that the function is continuous. This immediately follows once one knows that turns out to be continuous on , because, by uniqueness, and evidently imply ; cf. (3.12). Pick . The function belongs to and, on account of (3.13), we have
Since zero turns out to be an isolated critical point for φ, there is no loss of generality in assuming that . So, the implicit function theorem furnishes , , such that
Through , we thus get for all , where denotes a convenient neighborhood of . Consequently,
By (3.12), this results in , from which the continuity of at follows. As was arbitrary, the function turns out to be continuous on .
Next, observe that for all , . Hence, if
then , namely, is contractible in itself. Moreover, the function
is continuous and one has . Since
the set turns out to be a retract of . Being contractible in itself, because is infinite dimensional, we get (see, e.g., [11, p. 389])
This completes the proof. ∎
A careful inspection of the above argument shows that the second inequality in (h4) can be weakened to achieve the same conclusion, requiring instead
for suitable and , .
We are now ready to find a nodal solution of (1.1). Write, provided lie in and ,
For every , define the function
as well as
Moreover, provided , set
The same reasoning as in the proof of [20, Theorem 4.3] guarantees here that
The strict inequality in (3.17) and (3.15) forces . Now, if possesses infinitely many elements, then the conclusion follows at once. Otherwise, , because is a critical point of mountain pass type; see [5, p. 89]. Through , , and , we infer that
Moreover, recalling that turns out to be dense in ,
So, thanks to Theorem 3.6, for all , whence . The solution is nodal by the extremality of and , while standard nonlinear regularity results yield . ∎
4 Resonance from the Right
cf. (3.2). So, under these hypotheses, resonance with respect to from the left occurs and, a fortiori, the energy functional φ turns out to be coercive (Lemma 3.1). Now, the question of investigating what happens when there is resonance from the right of , i.e., the limit in (4.1) equals , naturally arises. In this case, φ turns out to be indefinite and direct methods no longer work. However, the linking structure of suitably defined sets still fits our purpose.
The following assumption will take the place of (h3).
If , then there exist and such that
Since compactly embeds in , the Nemytskii operator is continuous on , and enjoys property (p5), it suffices to show that every sequence fulfilling
turns out to be bounded. If the assertion were false, then, along a subsequence when necessary, . Let . We may evidently assume
because . Inequality (4.2) gives
Proceeding exactly as in the proof of Lemma 3.1, one obtains , which forces for appropriate . Therefore, and thus
Through (4.3), we easily have , whence
where . From (4.2) it follows that
i.e., after an elementary calculation,
However, since , dividing (4.8) by and letting produces
which contradicts (4.7). Therefore, the sequence turns out to be bounded in , as required.∎
Consider first the case . Without loss of generality, we may suppose in (h3’). Thus, there exist such that
After integration, this results in
Letting , on account of (h2) we have
which clearly implies that
Hence, for any ,
namely, as . The proof for is analogous.
Now, let . By (h3’) again, to every corresponds such that
The same argument as before yields here
and observe that
provided . Since , letting leads to
As was arbitrary, we actually have . The case is quite similar. ∎
Obviously, E turns out to be nonempty and closed.
Pick . The hypotheses give such that
Consequently, for any ,
Since , the assertion follows. ∎
The pair links E in .
One evidently has . Moreover, if
then , because . Let us verify that and lie in different pathwise connected components of U. Arguing by contradiction, there exists such that . On the other hand, (p4) forces
which leads to for some . However, this is impossible. Hence, any such that must satisfy the condition . Since , the proof is complete. ∎
We are now in a position to treat the existence of solutions to (1.1) when resonance from the right of occurs. To the best of our knowledge, multiplicity is still an open question.
Moreover, , because is a critical point of mountain pass type; see [5, p. 89]. On the other hand, due to Lemma 4.3 and Theorem 3.6, one has . Therefore, . Standard results from nonlinear regularity theory then ensure that . ∎
Aizicovici S., Papageorgiou N. S. and Staicu V., On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 2, 151–175. Google Scholar
Ambrosetti A., Brézis H. and Cerami G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. Google Scholar
Benci V., Fortunato D. and Pisani L., Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), no. 3, 315–344. Google Scholar
Brézis H. and Nirenberg L., versus local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465–472. Google Scholar
Chang K.-C., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993. Google Scholar
Cherfils L. and Il’yasov Y., On the stationary solutions of generalized reaction diffusion equations with -Laplacian, Comm. Pure Appl. Anal. 3 (2005), no. 1, 9–22. Google Scholar
Cuesta M., de Figueiredo D. and Gossez J.-P., The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), no. 1, 212–238. Google Scholar
de Paiva F. O. and Massa E., Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal. 66 (2007), no. 12, 2940–2946. Google Scholar
García Azorero J. P., Manfredi J. J. and Peral Alonso I., Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math. 2 (2000), no. 3, 385–404. Google Scholar
Gasiński L. and Papageorgiou N. S., Nonlinear Analysis, Ser. Math. Anal. Appl. 9, Chapman and Hall/CRC Press, Boca Raton, 2006. Google Scholar
Granas A. and Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003. Google Scholar
Hu S. and Papageorgiou N. S., Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J. (2) 62 (2010), no. 1, 137–162. Google Scholar
Iannizzotto A., Marano S. A. and Motreanu D., Positive, negative, and nodal solutions to elliptic differential inclusions depending on a parameter, Adv. Nonlinear Stud. 13 (2013), no. 2, 431–445. Google Scholar
Iannizzotto A. and Papageorgiou N. S., Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations, Osaka J. Math. 51 (2014), no. 1, 179–203. Google Scholar
Lê A., Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), no. 5, 1057–1099. Google Scholar
Li S., Wu S. and Zhou H.-S., Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations 185 (2002), no. 1, 200–224. Google Scholar
Marano S. A., On a Dirichlet problem with p-Laplacian and set-valued nonlinearity, Bull. Aust. Math. Soc. 86 (2012), no. 1, 83–89. Google Scholar
Marano S. A., Motreanu D. and Puglisi D., Multiple solutions to a Dirichlet eigenvalue problem with p-Laplacian, Topol. Methods Nonlinear Anal. 42 (2013), no. 2, 277–291. Google Scholar
Marano S. A. and Papageorgiou N. S., Multiple solutions to a Dirichlet problem with p-Laplacian and nonlinearity depending on a parameter, Adv. Nonlinear Anal. 1 (2012), no. 3, 257–275. Google Scholar
Marano S. A. and Papageorgiou N. S., Constant sign and nodal solutions of coercive -Laplacian problems, Nonlinear Anal. 77 (2013), 118–129. Google Scholar
Moroz V., Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal. 10 (1997), no. 2, 387–397. Google Scholar
Perera K., Multiplicity results for some elliptic problems with concave nonlinearities, J. Differential Equations 140 (1997), no. 1, 133–141. Google Scholar
Pucci P. and Serrin J., The Maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Basel, 2007. Google Scholar
About the article
Published Online: 2015-12-04
Published in Print: 2016-02-01
The first two authors acknowledge the support of GNAMPA of INdAM, Italy.