Strauss’s Radial Compactness and Nonlinear Elliptic Equation Involving a Variable Critical Exponent

Hashizume, Masato; Sano, Megumi

doi:https://doi.org/10.1155/2018/5497172

Journal of Function Spaces

On this page

Abstract Introduction Appendix Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 5497172 | https://doi.org/10.1155/2018/5497172

Strauss’s Radial Compactness and Nonlinear Elliptic Equation Involving a Variable Critical Exponent

Masato Hashizume¹and Megumi Sano²

Academic Editor: Dumitru Motreanu

Received04 Apr 2018

Accepted05 Aug 2018

Published17 Sept 2018

Abstract

We study existence of a nontrivial solution of , under some conditions on , especially, . Concerning this problem, we firstly consider compactness and noncompactness for the embedding from to . We point out that the decaying speed of at infinity plays an essential role on the compactness. Secondly, by applying the compactness result, we show the existence of a nontrivial solution of the elliptic equation.

1. Introduction and Main Results

In this article, we consider the following nonlinear elliptic equation:for , where is Laplacian and variable exponent is a measurable function satisfying , . -Laplacian type elliptic equation is one of the problems with variable exponent and this type equation on is studied by many researchers in both bounded domain case and unbounded domain case. In this paper, we refer the paper studying on , in several subjects: multiplicity of solutions (see, e.g., [1, 2]), existence of solutions of equations involving several nonlinearities (see, e.g., [3, 4]), equations under periodic assumptions (see e.g. [5, 6]), and so on. Moreover, existence of solutions of (1) involving variable exponent touching the critical exponent, that is, , is studied by [7, 8].

Concerning the classical Sobolev embedding in , it is well known that the embedding from to is not compact. And the embedding from to is not compact, either. In the viewpoint of the lack of compactness, consider the case where is natural as another critical case.

However, even for -Laplace equation there are no results in this case. Thus we study problem (1) at the opening of this article. In this case, unlike the subcritical case, we need to overcome some difficulties to show the existence of a nontrivial solution of (1). We will explain them more precisely after Remark 5. Thus in advance of study of (1), we consider the related embedding to the equation. Namely, we study the embedding from to .

We define the generalized Sobolev spaces with variable exponents according to [9]. For a domain and a function with we set These and are Banach spaces with the following norms:

The Sobolev embedding theorem of and related subjects have been well studied so far; see, e.g., [10–19], and we refer to the book [20]. For example, in bounded domain case, the Sobolev best constant with a variable critical exponent and the existence of extremals were studied in [21]; see also [22] in the Sobolev trace embedding case. One of the results in [18] is the existence of the compact embedding. They consider the situation when is uniformly continuous on and . Under this situation there exists a compact embedding from to for satisfying a.e. in and , where . On the other hand, for Kurata and Shioji [17] consider the critical case, that is, . They showed that if there exist and such that and then the embedding from to is compact. Conversely, if then the embedding from to is not compact.

When , Strauss [23] and Lions [24] showed that the radial Sobolev space can be embedded to compactly for . In addition, related results are in [25, 26] and so on. In case, under the same conditions as those of bounded domain case the compact embedding from to is obtained for satisfying and by [27]. On the other hand, the critical case, that is, or , has not been treated so far even if . In this paper, we fix . Our first study is to obtain a sufficiently condition of compactness and noncompactness of the embedding from to for variable exponent which also includes the excluded case in [27]. Based on these results, as the second study we obtain a nontrivial solution of (1) under the compactness conditions with .

Before introducing main results, we fix several notations. denote an open ball centered with radius . is an area of the unit sphere in . Throughout this paper we assume that and for a.e. . A letter denotes various positive constant. If is a radial function in , then we can write as by some function in . For simplicity we write with admitting some ambiguity.

Theorem 1 (noncompactness). If there exist positive constants and and an open set in such thatthen the embedding from to is not compact.

Theorem 2 (compactness). If there exist positive constants , and such thatthen the embedding from to is compact.

Remark 3. In Theorem 2, we do not need the constraint . holds whenever satisfies in and in .

Theorem 4. Assume that satisfies the hypotheses (7) and (8) in Theorem 2 and . Then there exists a nontrivial weak solution of (1) in the sense offor any .

Remark 5. If is a radially symmetric function satisfying the hypotheses of Theorem 4, then we can show that the weak solution obtained in Theorem 4 satisfies and for all Indeed, since and are radially symmetric, it follows that, for all , where . If for any we consider the radial function , then we have Therefore we see that satisfies (9) even for nonradial functions . Finally, by Corollary of Theorem 2 in [28] we have . And also, by Theorem 2.5.1 in [29] we have for all .

We note the difficulties to obtain Theorem 4 caused by the condition . Ambrosetti-Rabinowitz condition (AR) is well known in order to obtain a nontrivial weak solution to the following problem by mountain pass method: where . Especially, condition (AR) has been used to establish not only the mountain pass structure of the energy functional but also the Palais-Smale condition. A weaker condition has also been considered, for instance, Liu-Wang [30] studied (SQ) which is called superquadratic condition. However, assuming that the nonlinear term in (1) is a special case of the general nonlinear term , this does not satisfy even condition (SQ) when . From these facts, it seems to be difficult to confirm whether the energy functional (see Section 4) corresponding to (1) satisfies the Palais-Smale condition or not. In more detail, while the fact that bounded Palais-Smale sequence has a convergent subsequence is straightforward from Theorem 2, boundedness of all Palais-Smale sequence is nontrivial. Besides that, satisfying the mountain pass structure for is not trivial since we cannot apply the fibering map method directly.

To overcome these difficulties, in Section 3, we construct a solution of (1) as a limit of mountain pass solutions of some elliptic equations approaching (1) in the sense of energy functional. In Section 4, we show another proof by using the variant of the mountain pass theorem. More precisely, by introducing the condition (C) (see Section 4) defined in [31] or [32] instead of the Palais-Smale condition, we obtain a solution of (1) in a different way from Section 3.

2. Compactness and Noncompactness of the Embedding

We prove Theorems 1 and 2. Before beginning the proof we recall the point-wise estimate and the compactness theorem introduced in [23, 24] (). For the reader’s convenience, the proofs are in Appendix.

Proposition 6. For any we have

Proposition 7. The embedding from to is compact for .

Proof of Theorem 1. We shall show Theorem 1 in the same way as [17]. Set for . Let be a radial function satisfying on and supp. For , we define Then for any we obtain Since is a bounded sequence in and is reflexive (see, e.g., Proposition 3.20. in [33]), there exist a weakly convergent subsequence and such that in as . By compactness of the embedding from to for , we have in and a.e. in which yields that . On the other hand, we have Since is open in , there exists a smooth subset such that . By using the polar coordinates as we obtain By the assumption (6), we obtain for large , , and . Moreover for and large , it holds that which yields Therefore we obtain for large , where is the dimensional Hausdorff measure. Thus, if we assume that the embedding from to is compact, then we have which contradicts . Hence the embedding from to is not compact.

Proof of Theorem 2. We assume that without loss of generality. Let be a bounded sequence in . We shall show the existence of a strongly convergence subsequence of in . By the reflexivity of , there exist a subsequence and such that in as . Especially it also holds that in as . And also, by Proposition 7 we have in for any andFurthermore, is a bounded sequence and the embedding from to is compact by the assumption (7) (see Remark 2 in [17]). Thus there exist a subsequence of (we use again for simplicity) and such that the following hold true: By (19) and (20), we can check that a.e. in which yields In the similar way as above, we also obtain the following: for any and any as since the embedding from to is compact for any .
Set . In order to make good use of (22) and (23) we divide into three terms as follows:where is sufficiently large.
Firstly, by (22) we haveNext, for we have Thus, by (23) we obtainFinally we shall estimate . Since by Proposition 6 and the boundedness of , we can assume for with large . Therefore by assumption (8) we obtain where as . Since for each , we have Hence we haveWe go back to (24) and by (25), (27), and (31) we have As a consequence we obtain in .

3. Approximation Method: Proof of Theorem 4

In this section, we show Theorem 4 by using Theorem 2. First, we prepare the mountain pass theorem (Theorem 8) introduced in [34, 35] and so on which are based on [36]. Let be a Banach space and . We define a Palais-Smale sequence for as satisfying uniformly in , and in , where is Fréchet derivative and is the dual space of . We say that satisfies (P.-S.) condition if any Palais-Smale sequence has a strongly convergent subsequence.

Theorem 8 (see [34, 35]). Suppose satisfies (P.-S.) condition. Assume that (i)E(0)=0(ii)There exist such that for any with .(iii)There exists such that and . Define Then is a critical value.

Proof of Theorem 4. ⁡
Step 1. We may assume that in the hypotheses of Theorem 2 is sufficiently large such that . Because, if not, we take a sufficiently large which satisfies it instead of . For let be a sequence such that and as . Then we set functions as Define a functional from to by We can check that . Indeed, for fixed and any Then we see that is Gâteaux differentiable in . By the Vitali convergence theorem, we see that is continuous from to its dual space . Hence . Moreover, for each , satisfies as follows: (i) satisfies (P.-S.) condition.(ii),(iii)There exist positive constants such that for any with ,(iv)There exists such that , . By Theorem 8 there exists a critical point of such that where is defined in the same way as in Theorem 8. Thus is a nontrivial weak solution ofWe can also see that by multiplying both sides of (39) by .
Proposition 9. is bounded in .
We will prove this proposition at the end of this section.
Step 2. Since is a bounded sequence, there exists such that weakly in . Put Then we have where as . Moreover, from (56) and (57) in the proof of Proposition 9 it follows that by the generalized Hölder inequality (see, e.g., [9] Theorem 2.1). By the boundedness of in and Theorem 2 we have as . Henceas . Recall that, for , , we have From this inequality and (43) it follows that which is equivalent to strongly in . Thus satisfies Step 3. Finally, we have to show . From the boundedness of and Proposition 6, we see that in for large . Therefore we haveBy the Sobolev inequality it follows thatMoreover, we havePut . From (47), (48), and (49), we obtain where we used that . By Theorem 2 we have Consequently we have .

Proof of Proposition 9. We take a smooth radial function on . Since as , there exists independent of such that . If we set for , then we see that Moreover, we haveOn the other hand, since is a critical point of at we haveand, for any ,In particular,From (54), (55), and (57), it follows that Furthermore, by we haveThus for any there exists a positive constant such thatHere, we take a constant sufficiently large (this will be chosen again later) and we haveby (57) and (60). Set and . Then we obtain where the third inequality comes from the assumption (8). We shall estimate , , and . For , by (59) we haveIn order to estimate and , we prepare an estimate of . For each we haveby the generalized Hölder inequality, where . Now we assume and (if not, the proof is much simpler). By Proposition 2.2. in [27] we have Since we obtainIn the same way as above, we havewhere the second inequality comes from From (67) and (68) we obtainFor , by using (70) and Proposition 6, we have Since there exists a positive constant which is independent of and such thatOn the other hand, for large we obtain which yields From (73) and (75) we have in the same way as the proof of Theorem 2. Thus for sufficiently large we haveIn the same way as , we obtain the estimate of for large as follows: where the last inequality comes from Therefore for sufficiently large we haveFrom (61), (63), (77), and (80) we have As a consequence is bounded.

4. Mountain Pass Theorem under the Condition (C): Proof of Theorem 4

In this section, we show Theorem 4 by a different method from Section 3.

Cerami [31] and Bartolo-Benci-Fortunato [32] have proposed a variant of (P.-S.) condition. In this paper, we use the condition (C) introduced by [31, 32] and the mountain pass theorem under the condition (C) (Theorem 11). Let be a real Banach space and . First, we define the condition (C) based on [31, 32].

Definition 10 ([31, 32] Definition 1.1.). We say that satisfies the condition (C) in , if (i)every bounded sequence , for which is bounded and , possesses a convergent subsequence, and(ii)for any there exist such that and for any with

Theorem 11 (mountain pass theorem under the condition (C)). Let satisfy the condition (C) in . Assume that (i)E(0)=0(ii)There exist such that for any with .(iii)There exists such that and . Define Then is a critical value.

For , we set Note that Theorem 11 can be shown in the same way as the proof of Theorem 6.1 in p.109 in [35] by substituting the following deformation theorem under the condition (C) for Theorem 3.4 in p.83 in [35].

Theorem 12 ([32] Theorem 1.3.). Let satisfy the condition (C) in . If and is any neighborhood of , there exist a bounded homeomorphism of onto and constants such that , satisfying the following properties:(I)(II) if (III) if

We set an energy functional from to as We can check that in the same way as the proof of Theorem 4.

Proposition 13. Assume that satisfies the hypotheses (7) and (8) in Theorem 2 and . Then satisfies the condition (C) on .

Proof. We take with arbitrary. First, we shall show that satisfies (i) in Definition 10. Let be a bounded sequence satisfying and as . Then the following holds true for any :In particular, since is bounded it follows thatLikewise since is bounded, there exists a subsequence written as for simplicity and such that weakly in . Put as in Section 3. In the same way as Step 2 in the proof of Theorem 4 in Section 3 by substituting (86) and (87) for (56) and (57), respectively, we have as by Theorem 2. Recalling that consequently we have This implies that strongly in .
Next, we shall show (ii). For any , we take some with . We will choose suitable again later. By deriving a contradiction, we show that there exists such that for any with We assume that there exists such that with and as . Since as , we have which yields Moreover, in the same way as the proof of Proposition 9, for large we havewhere for and is the same as the proof of Proposition 9. By substituting (93) and (94) for (59) and (70), we obtain the following estimates: where is a positive constant independent of . Therefore we havefor large . If we choose sufficiently large satisfying , then we see that (96) contradicts .
The proof of Proposition 13 is now complete.

Proposition 14. Assume that satisfies the hypotheses (7) and (8) in Theorem 2 and . Then has the mountain pass geometry, that is, satisfies (i), (ii), and (iii) in Theorem 11.

Proof. (i) is obvious. We prove (ii). Let be the best constant of the Sobolev inequality: for . Set . Note that . For with , it follows that where and the second inequality comes from Proposition 6. From this if is sufficiently small, we haveFor and satisfying and (98), we assume that and derive a contradiction. From (98) it follows that . In addition, for sufficiently large we have By using the estimates in the calculation of to show (98) we have and as . For we have We can show that is bounded uniformly for and as in the same way as the estimate of in the proof of Theorem 2. Therefore as , which implies that since as . This contradicts .
Finally, we prove (iii). We take a smooth radial function such that , in , where is in the hypothesis (8). Recall that . By taking sufficiently large we have Since we prove (iii).

Proof of Theorem 4. From Propositions 13 and 14 and Theorem 11, we can show the existence of a nontrivial critical point which is a weak solution to in . Then we also see that in .

Appendix

In this section we show Propositions 6 and 7.

Proof of Proposition 6. It is sufficient to show that (13) holds for with radially symmetric. We have By direct calculation we have Thus it follows that Consequently (13) follows immediately.

Proof of Proposition 7. By (13) we have where . When , that is, , we have Let be a sequence such that weakly in . Firstly we show that the case of . In this case we have Since is bounded from above uniformly, letting and we have strongly in .
Next, for using interpolation of space, we have where . Since and is bounded we have .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

An earlier abstract of this manuscript was presented in Osaka University Differential Equation Seminar in 2017.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are grateful to Professor Michinori Ishiwata (Osaka University) for helpful advice on this subject. Part of this work was supported by JSPS Grant-in-Aid for Fellows (DC2), no. 16J08945 (Masato Hashizume) and no. 16J07472 (Megumi Sano).

References

C. O. Alves and J. L. Barreiro, “Multiple solutions for a class of quasilinear problems involving variable exponents,” Asymptotic Analysis, vol. 96, no. 2, pp. 161–184, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Fu and X. Zhang, “A multiplicity result for p(x)-Laplacian problem in ,” Nonlinear Analysis, vol. 70, no. 6, pp. 2261–2269, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
C. O. Alves and M. C. Ferreira, “Existence of solutions for a class of p(x)-Laplacian equations involving a concave-convex nonlinearity with critical growth in ,” Topological Methods in Nonlinear Analysis, vol. 45, no. 2, pp. 399–422, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
X. L. Fan and X. Han, “Existence and multiplicity of solutions for p(x)-Laplacian equations in ,” Nonlinear Analysis. Theory, Methods & Applications, vol. 59, no. 1-2, pp. 173–188, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
X. Fan, “p(x) -Laplacian equations in with periodic data and nonperiodic perturbations,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 103–119, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhang, “A minimization problem with variable growth on Nehari manifold,” Monatshefte für Mathematik, vol. 181, no. 2, pp. 485–500, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
C. O. Alves and M. A. Souto, “Existence of solutions for a class of problems in involving the p(x)-Laplacian,” in Contributions to nonlinear analysis, vol. 66 of Progress in Nonlinear Differential Equations and Their Applications, pp. 17–32, Birkhäuser, Basel, Switzerland, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
N. Saintier and A. Silva, “Local existence conditions for an equations involving the p(x)-Laplacian with critical exponent in ,” Nonlinear Differential Equations and Applications NoDEA, vol. 24, no. 2, Art. 19, 36 pages, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
O. Kovácik and J. Rákosnk, “On spaces and ,” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991.
View at: Google Scholar | MathSciNet
B. Cekic, R. Mashiyev, and G. T. Alisoy, “On the Sobolev-type inequality for Lebesgue spaces with a variable exponent,” International Mathematical Forum. Journal for Theory and Applications, vol. 1, no. 25-28, pp. 1313–1323, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
R. Mashiyev and B. Cekic, “Sobolev-type inequality for spaces ,” International Journal of Contemporary Mathematical Sciences, vol. 2, no. 9-12, pp. 423–429, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
D. E. Edmunds and J. Rakosnik, “Sobolev embeddings with variable exponent,” Studia Mathematica, vol. 143, no. 3, pp. 267–293, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
D. E. Edmunds and J. Rakosnik, “Sobolev embeddings with variable exponent, II,” Mathematische Nachrichten, vol. 246/247, pp. 53–67, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
L. Diening, “Riesz potential and SOBolev embeddings on generalized Lebesgue and SOBolev spaces and ,” Mathematische Nachrichten, vol. 268, pp. 31–43, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
C. Capone, D. Cruz-Uribe, and A. Fiorenza, “The fractional maximal operator and fractional integrals on variable L^p spaces,” Revista Matemática Iberoamericana, vol. 23, no. 3, pp. 743–770, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
P. Harjulehto and P. Hästö, “Sobolev inequalities with variable exponent attaining the values 1 and n,” Publicacions Matemátiques, vol. 52, no. 2, pp. 347–363, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
K. Kurata and N. Shioji, “Compact embedding from to and its application to nonlinear elliptic boundary value problem with variable critical exponent,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1386–1394, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
X. L. Fan, J. S. Shen, and D. Zhao, “Sobolev embedding theorems for spaces ,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001.
View at: Publisher Site | Google Scholar
Y. Mizuta, T. Ohno, T. Shimomura, and N. Shioji, “Compact embeddings for Sobolev spaces of variable exponents and existence of solutions for nonlinear elliptic problems involving the p(x)-Laplacian and its critical exponent,” Annales Academiæ Scientiarum Fennicæ, vol. 35, no. 1, pp. 115–130, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Basel, Switzerland, 2013.
View at: Publisher Site | MathSciNet
J. Fernández Bonder, N. Saintier, and A. Silva, “On the Sobolev embedding theorem for variable exponent spaces in the critical range,” Journal of Differential Equations, vol. 253, no. 5, pp. 1604–1620, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
J. Fernández Bonder, N. Saintier, and A. Silva, “On the Sobolev trace theorem for variable exponent spaces in the critical range,” Annali di Matematica Pura ed Applicata. Series IV, vol. 193, no. 6, pp. 1607–1628, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
W. A. Strauss, “Existence of solitary waves in higher dimensions,” Communications in Mathematical Physics, vol. 55, no. 2, pp. 149–162, 1977.
View at: Publisher Site | Google Scholar | MathSciNet
P.-L. Lions, “Symétrie et compacité dans les espaces de Sobolev,” Journal of Functional Analysis, vol. 49, no. 3, pp. 315–334, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
J. Chabrowski, “On compact embeddings of radial Sobolev spaces and their applications,” Communications in Partial Differential Equations, vol. 17, no. 5-6, pp. 941–952, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Ebihara and T. P. Schonbek, “On the (non)compactness of the radial Sobolev spaces,” Hiroshima Mathematical Journal, vol. 16, no. 3, pp. 665–669, 1986.
View at: Google Scholar | MathSciNet
X. Fan, Y. Zhao, and D. Zhao, “Compact imbedding theorems with symmetry of Strauss-Lions type for the space ,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 333–348, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
E. DiBenedetto, “ local regularity of weak solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 7, no. 8, pp. 827–850, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
P. Pucci and J. Serrin, The Maximum Principle, vol. 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhauser Verlag, Basel, Switzerland, 2007.
View at: MathSciNet
Z. Liu and Z.-Q. Wang, “On the Ambrosetti-Rabinowitz superlinear condition,” Advanced Nonlinear Studies, vol. 4, no. 4, pp. 563–574, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
G. Cerami, “An existence criterion for the critical points on unbounded manifolds,” Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti. Scienze Matematiche, Fisiche, Chimiche e Geologiche. A, vol. 112, no. 2, pp. 332–336 (1979), 1978.
View at: Google Scholar | MathSciNet
P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 7, no. 9, pp. 981–1012, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, NY, USA, 2011.
View at: MathSciNet
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
View at: Publisher Site | MathSciNet
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34 of Modern Surveys in Mathematics, Springer, Berlin, Germany, 4th edition, 2008.
View at: MathSciNet
A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2018 Masato Hashizume and Megumi Sano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

909

Downloads

877

Citations