 Research
 Open access
 Published:
Existence and multiplicity results for a degenerate quasilinear elliptic system near resonance
Boundary Value Problems volume 2014, Article number: 184 (2014)
Abstract
We establish existence and multiplicity results for weak solutions of a degenerate quasilinear elliptic system. By using Ekeland’s variational principle, the mountain pass theorem and the saddle point theorem in critical point theory, we obtain the existence of one or three solutions for an elliptic system with Dirichlet boundary conditions under some restriction on \lambda. This kind of results was firstly obtained by Mawhin and Schmitt (Ann. Pol. Math. 51:241248, 1990) for a semilinear twopoint boundary value problem.
1 Introduction and main results
In this paper, we consider the following degenerate quasilinear elliptic systems:
where \mathrm{\Omega} is a bounded and connected subset of {R}^{N} (N\ge 2), \lambda is a nonnegative parameter, F\in {C}^{1}(\overline{\mathrm{\Omega}}\times {R}^{2},R), \mathrm{\nabla}F=({F}_{u},{F}_{v}) denotes the gradient of F with respect to (u,v)\in {R}^{2}.
(f1) There exists a positive constant {C}_{1} such that, for some \theta \in (1,p), \tau \in (1,q),
for all (u,v)\in {R}^{2} and x\in \mathrm{\Omega}.

(H)
p>1, q>1, \alpha \ge 0, \beta \ge 0, such that
\frac{\alpha +1}{p}+\frac{\beta +1}{q}=1.
The degeneracy of this system is considered in the sense that the measurable, nonnegative diffusion coefficients {h}_{1}, {h}_{2} are allowed to vanish in \mathrm{\Omega} (as well as at the boundary \partial \mathrm{\Omega}) and/or to blow up in \overline{\mathrm{\Omega}}. The consideration of suitable assumptions on the diffusion coefficients will be based on [1], where the degenerate scalar equation was studied. We introduce the function space (N)_{ p } which consists of functions h:\mathrm{\Omega}\subset {R}^{N}\to R, such that h\in {L}^{1}(\mathrm{\Omega}), {h}^{\frac{1}{p1}}\in {L}^{1}(\mathrm{\Omega}) and {h}^{s}\in {L}^{1}(\mathrm{\Omega}), for some p>1, s>max\{\frac{N}{p},\frac{1}{p1}\} satisfying ps\le N(s+1).
Then for the weight functions {h}_{1}, {h}_{2} we assume the following hypothesis:
(N) There exist functions {m}_{1} satisfying condition (N)_{ p }, for some {s}_{p}, and {m}_{2} satisfying condition (N)_{ q }, for some {s}_{q}, such that
a.e. in \mathrm{\Omega}, for some constants {k}_{1}>1 and {k}_{2}>1.
The mathematical modeling of various physical processes, ranging from physics to biology, where spatial heterogeneity has a primary role, is reduced to nonlinear evolution equations with variable diffusion or dispersion. Note that problem (1) is closely related (see [1]) to the following system:
This system has successfully dealt with a variety of physical phenomena, such as the heat propagation in heterogeneous materials, the transport of electron temperature in a confined plasma, the propagation of varying amplitude waves in a nonlinear medium, the electromagnetic phenomena in nonhomogeneous superconductors, the dynamics of Josephson functions, and the spread of microorganisms. For more details as regards this type of system, see [2]–[4] and the references therein.
An example of the physical motivation of the assumptions (N), (N)_{ p } may be found in [3]. These assumptions are related to the modeling of reaction diffusion processes in composite materials occupying a bounded domain \mathrm{\Omega}, which at some points behave as perfect insulators. When at some points the medium is perfectly insulating, it is natural to assume that {h}_{1}(x) and/or {h}_{2}(x) vanish in \mathrm{\Omega}. For more information, we refer the interesting reader to [4] and the references therein.
For the perturbed problem, Mawhin and Schmitt [5] first considered the twopoint boundary value problem
Under the assumption that f is bounded and satisfies a sign condition, if the parameter \lambda is sufficiently close to {\lambda}_{1} from left, problem (3) has at least three solutions, if {\lambda}_{1}\le \lambda <{\lambda}_{2}, problem (3) has at least one solution, where {\lambda}_{1}, {\lambda}_{2} are the first, second eigenvalues of the corresponding linear problem. Ma et al.[6] considered the boundary value problem \mathrm{\Delta}u+\lambda u+f(x,u)=h(x) defined on a bounded open set \mathrm{\Omega}\subset {R}^{N}, no matter whether the boundary conditions are Dirichlet or Neumann conditions, as the parameter \lambda approaches {\lambda}_{1} from left, there exist three solutions. Moreover, the existence of three solutions was obtained for the quasilinear problem in bounded domains as the parameter \lambda approaches {\lambda}_{1} from left in [7], [8], these results were extended to the perturbed pLaplacian equation in {R}^{N}. Especially, the existence of three solutions has been extended to cooperative elliptic systems in [9]. Motivated by the above idea, we have the goal in this paper of extending these results to some degenerate quasilinear elliptic systems with the Dirichlet boundary conditions.
Next, we recall the basic results of the following eigenvalue problem (for details see [2]):
where {h}_{1}, {h}_{2} satisfy (N), and the coefficient functions a, d, and b satisfy conditions (A), (D), and (B), respectively (the conditions (A), (D), (B) see Section 2). Zographopoulos proved the theorem below.
Theorem 1.1
Let\mathrm{\Omega}be a bounded domain of{R}^{N} (N\ge 2). Assume that hypotheses (H) and (N) are satisfied and the coefficient functionsa, dandbsatisfy conditions (A), (D) and (B), respectively. Denote byLthe set
whereZ={D}_{0}^{1,p}(\mathrm{\Omega},{h}_{1})\times {D}_{0}^{1,q}(\mathrm{\Omega},{h}_{2}) (we will determine in Section 2whatZmeans). Then the system (4) admits a positive principal eigenvalue{\lambda}_{1}, satisfying
The associated normalized eigenfunction{\varphi}_{1}=({u}_{1},{v}_{1})belongs toZand each component is nonnegative. In addition,

(i)
the set of all eigenfunctions corresponding to the principal eigenvalue {\lambda}_{1} forms a onedimensional manifold {E}_{1}\subset Z, which is defined by
{E}_{1}=\{({c}_{1}{u}_{1},{c}_{1}^{p/q}{v}_{1}):{c}_{1}\in R\}.

(ii)
{\lambda}_{1} is the only eigenvalue of (4) to which corresponds a componentwise nonnegative eigenfunction.

(iii)
{\lambda}_{1} is isolated in the following sense: there exists \eta >0, such that the interval (0,{\lambda}_{1}+\eta ) does not contain any other eigenvalue than {\lambda}_{1}.
Now we are ready to state our main results. For the related definitions we refer to Section 2.
Theorem 1.2
Let the hypotheses of Theorem 1.1and (f1) be satisfied. In addition,
uniformly inx\in \mathrm{\Omega}. Then, for\lambda <{\lambda}_{1}sufficiently close to{\lambda}_{1}, the system (1) has at least three solutions, where{\lambda}_{1}>0is the first eigenvalue of the system (4).
Theorem 1.3
Let the hypotheses of Theorem 1.1and (f1) be satisfied. In addition,
uniformly inx\in \mathrm{\Omega}. Then for\lambda \in ({\lambda}_{1},{\lambda}_{2}), the system (1) has at least one solution (we will determine in Section 3what{\lambda}_{2}mean).
2 Space and operator setting
Let h(x) be a nonnegative weight function in \mathrm{\Omega} which satisfies condition (N)_{ p }. We consider the weighted Sobolev space {D}_{0}^{1,p}(\mathrm{\Omega},h) to be defined as the closure of {C}_{0}^{\mathrm{\infty}}(\mathrm{\Omega}) with respect to the norm
The space {D}_{0}^{1,p}(\mathrm{\Omega},h) is a reflexive Banach space. For a discussion of the space setting we refer to [1] and the references therein. Let
Lemma 2.1
[1]
Assume that\mathrm{\Omega}is a bounded domain in{R}^{N}and the weighthsatisfies (N)_{ p }. Then the following embeddings hold:

(I)
{D}_{0}^{1,p}(\mathrm{\Omega},h)\hookrightarrow {L}^{{p}_{s}^{\ast}}(\mathrm{\Omega}) continuously for 1<{p}_{s}^{\ast}<N,

(II)
{D}_{0}^{1,p}(\mathrm{\Omega},h)\hookrightarrow {L}^{r}(\mathrm{\Omega}) compactly for any r\in [1,{p}_{s}^{\ast}).
In the sequel we denote by {p}^{\ast} and {q}^{\ast} the quantities {p}_{{s}_{p}}^{\ast} and {q}_{{s}_{q}}^{\ast}, respectively, where {s}_{p} and {s}_{q} are induced by condition (N), recall that {h}_{1}, {h}_{2} satisfy (N). The assumptions concerning the coefficient functions of systems (1) and systems (4) are the following.

(A)
a\in {L}^{\frac{{p}^{\ast}}{{p}^{\ast}p}}(\mathrm{\Omega}) and either there exists {\mathrm{\Omega}}_{a}^{+}\subset \mathrm{\Omega} of positive Lebesgue measure, i.e., {\mathrm{\Omega}}_{a}^{+}>0, such that a(x)>0, for all x\in {\mathrm{\Omega}}_{a}^{+}, neither a(x)\equiv 0, in \mathrm{\Omega}.

(D)
d\in {L}^{\frac{{q}^{\ast}}{{q}^{\ast}p}}(\mathrm{\Omega}) and either there exists {\mathrm{\Omega}}_{d}^{+}\subset \mathrm{\Omega} of positive Lebesgue measure, i.e., {\mathrm{\Omega}}_{d}^{+}>0, such that d(x)>0, for all x\in {\mathrm{\Omega}}_{d}^{+}, neither d(x)\equiv 0, in \mathrm{\Omega}.

(B)
b(x)\ge 0, a.e. in \mathrm{\Omega}, b\not\equiv 0 and b\in {L}^{\omega}(\mathrm{\Omega}), where \omega ={(1\frac{\alpha +1}{{p}^{\ast}}\frac{\beta +1}{{q}^{\ast}})}^{1}.
The space setting for our problem is the product space Z={D}_{0}^{1,p}(\mathrm{\Omega},{h}_{1})\times {D}_{0}^{1,q}(\mathrm{\Omega},{h}_{2}) equipped with the norm
Observe that inequalities (2) in condition (N) imply that the functional spaces {D}_{0}^{1,p}(\mathrm{\Omega},{h}_{1})\times {D}_{0}^{1,q}(\mathrm{\Omega},{h}_{2}) and {D}_{0}^{1,p}(\mathrm{\Omega},{m}_{1})\times {D}_{0}^{1,q}(\mathrm{\Omega},{m}_{2}) are equivalent. Next, let us introduce the functionals I,J:Z\to R in the following way:
It is a standard procedure (see [10], Lemma 2.1) to prove the following properties of the functionals.
Lemma 2.2
The functionalsI, Jare well defined. Moreover, Iis continuous andJis compact.
Notation
For simplicity we use the symbol {\parallel \cdot \parallel}_{{h}_{1}} for the norm {\parallel \cdot \parallel}_{{D}_{0}^{1,p}(\mathrm{\Omega},{h}_{1})} and {\parallel \cdot \parallel}_{{h}_{2}} for the norm {\parallel \cdot \parallel}_{{D}_{0}^{1,q}(\mathrm{\Omega},{h}_{2})}.
3 Proof of theorems
Let {\mathrm{\Phi}}_{\lambda}:Z\to R be the functional defined by
where
Since the potential F has sublinear growth, by a standard argument, it follows that {\mathrm{\Phi}}_{\lambda}\in {C}^{1}(Z,R). In addition, (u,v)\in Z is a weak solution of systems (1) if and only if (u,v) is a critical point of {\mathrm{\Phi}}_{\lambda}.
By (5) we can deduce that
Let V=span\{{\varphi}_{1}\} and W=\{z\in Z:\u3008{J}^{\prime}({\varphi}_{1}),z\u3009=0\}, we have from the simplicity of {\lambda}_{1}, Z=V\oplus W. Then, since {\lambda}_{1} is also isolated, we have
which satisfies {\lambda}_{1}<{\lambda}_{2}. In addition,
On the other hand, from the condition (f1) and by Young’s inequality, for every fixed ({u}_{0},{v}_{0})\in {R}^{2}, it follows that
where {C}_{2} is a positive constant independent to ({u}_{0},{v}_{0}), therefore, from Lemma 2.1, we have
for all (u,v)\in Z, where {C}_{4}={C}_{2}{\int}_{\mathrm{\Omega}}F(x,0,0)\phantom{\rule{0.2em}{0ex}}dx and {C}_{3} is a positive constant.
Next, we will prove Theorem 1.2 by using Ekeland’s variational principle and the mountain pass theorem and Theorem 1.3 by using the saddle point theorem.
Proof of Theorem 1.2
The proof will be divided into four steps.
Step 1. The functional {\mathrm{\Phi}}_{\lambda} is coercive in Z, {\mathrm{\Phi}}_{\lambda} is bounded from below on W and there is a constant M, independent of \lambda, such that {inf}_{z\in W}{\mathrm{\Phi}}_{\lambda}(z)\ge M.
For \lambda <{\lambda}_{1}, from (8) and (11), we get
for all (u,v)\in Z, where {a}_{1}=min\{(1+\alpha )/p,(1+\beta )/q\}>0, which shows that {\mathrm{\Phi}}_{\lambda} is coercive in Z.
Similarly, from (9) and (11), we obtain
for all (u,v)\in W. Hence {\mathrm{\Phi}}_{\lambda} is coercive in W and {\mathrm{\Phi}}_{\lambda} is bounded from below on W. Moreover, there is a constant M, independent of \lambda, such that {inf}_{z\in W}{\mathrm{\Phi}}_{\lambda}(z)\ge M.
Step 2. If the parameter \lambda is sufficiently close to {\lambda}_{1} from the left, we have {t}^{}<0<{t}^{+} such that {\mathrm{\Phi}}_{\lambda}({t}^{\pm}{\varphi}_{1})<M.
We choose {t}^{+}>0 sufficiently large, we get from (6) that N({t}^{+}{\varphi}_{1})>M+1, so that
where {a}_{2}=max\{(1+\alpha )/p,(1+\beta )/q\}. Then for \lambda sufficiently close to {\lambda}_{1} from the left, {\mathrm{\Phi}}_{\lambda}({t}^{+}{\varphi}_{1})<M. The same conclusion holds for a {t}^{}<0.
Step 3. If \lambda <{\lambda}_{1}, the functional {\mathrm{\Phi}}_{\lambda} satisfies the (P.S.) condition. In addition, let
{\mathrm{\Phi}}_{\lambda} satisfies {(P.S.)}_{c,{\mathrm{\Theta}}^{+}} and {(P.S.)}_{c,{\mathrm{\Theta}}^{}} for all c<M.
On one hand, if ({u}_{n},{v}_{n}) be a (P.S.) sequence of {\mathrm{\Phi}}_{\lambda}, that is,
as n\to \mathrm{\infty}. From Step 1 and (15), ({u}_{n},{v}_{n}) must be bounded in Z, that is, there is K>0 such that
for all n\in N. Thus, there is a subsequence of ({u}_{n},{v}_{n}), without any loss of generality still denoted by ({u}_{n},{v}_{n}), and (u,v)\in Z such that ({u}_{n},{v}_{n})\rightharpoonup (u,v) weakly in Z, ({u}_{n},{v}_{n})\to (u,v) strongly in {L}^{p}(\mathrm{\Omega})\times {L}^{q}(\mathrm{\Omega}). Consequently, from (15) and (16), one has
From the condition (f1), Hölder’s inequality, Lemma 2.1, and (16), it follows that
as n\to \mathrm{\infty}, where S is the embedding constant. Combining (17), (18), and Lemma 2.2, we get
Similarly, we also obtain
hence, we conclude that
Therefore, using a wellknown lemma, see for instance [11], Lemma 3.1] or [12], Lemma 4.1], we get {u}_{n}\to u in {D}_{0}^{1,p}(\mathrm{\Omega},{h}_{1}) as n\to \mathrm{\infty}. Similarly, we get {v}_{n}\to v in {D}_{0}^{1,q}(\mathrm{\Omega},{h}_{2}) as n\to \mathrm{\infty}, that is, ({u}_{n},{v}_{n}) has a convergent subsequence.
On the other hand, let \{{z}_{n}\}\subset {\mathrm{\Theta}}^{+} satisfy {\mathrm{\Phi}}_{\lambda}({z}_{n})\to c<M and {\mathrm{\Phi}}_{\lambda}^{\prime}({z}_{n})\to 0 as n\to \mathrm{\infty}. Since {\mathrm{\Phi}}_{\lambda} is coercive and the potential F satisfies the condition (f1), there is z\in Z such that {z}_{n}\to z strongly in Z. If z\in \partial {\mathrm{\Theta}}^{+}=W, from the second conclusion of Step 1, we get {\mathrm{\Phi}}_{\lambda}({z}_{n})\to c\ge M, which is impossible. Hence z\in {\mathrm{\Theta}}^{+} and {\mathrm{\Phi}}_{\lambda} satisfies the {(P.S.)}_{c,{\mathrm{\Theta}}^{+}} condition. Similarly we see that {(P.S.)}_{c,{\mathrm{\Theta}}^{}} holds for all c<M.
Step 4. Three solutions are obtained.
If \lambda <{\lambda}_{1} is sufficiently close {\lambda}_{1}, from Step 1 and Step 2, we get \mathrm{\infty}<{inf}_{{\mathrm{\Theta}}^{\pm}}{\mathrm{\Phi}}_{\lambda}<M, which implies that {\mathrm{\Phi}}_{\lambda} is bounded below in {\mathrm{\Theta}}^{+}. Consequently, from Ekeland’s variational principle, there exists \{{z}_{n}\}\subset {\mathrm{\Theta}}^{+} such that {\mathrm{\Phi}}_{\lambda}({z}_{n})\to {inf}_{{\mathrm{\Theta}}^{+}}{\mathrm{\Phi}}_{\lambda} and {\mathrm{\Phi}}_{\lambda}^{\prime}({z}_{n})\to 0 as n\to \mathrm{\infty}. Since {\mathrm{\Phi}}_{\lambda} satisfies {(P.S.)}_{c,{\mathrm{\Theta}}^{+}} for all c<M, there is {z}^{+}\in {\mathrm{\Theta}}^{+} such that {\mathrm{\Phi}}_{\lambda}(z)={inf}_{{\mathrm{\Theta}}^{+}}{\mathrm{\Phi}}_{\lambda}, that is, the infimum is attained in {\mathrm{\Theta}}^{+}. A similar conclusion holds in {\mathrm{\Theta}}^{}. So {\mathrm{\Phi}}_{\lambda} has two distinct critical points, denoted by {z}^{+} and {z}^{}.
To fix ideas, suppose that {\mathrm{\Phi}}_{\lambda}({z}^{})\le {\mathrm{\Phi}}_{\lambda}({z}^{+}), if {z}^{+} is not an isolated critical point, then {\mathrm{\Phi}}_{\lambda} has at least three solutions. Otherwise, putting
we have \mathrm{\Psi}(0)=0, \mathrm{\Psi}(e)\le 0, and there exist r>0, \rho >0 such that \mathrm{\Psi}(z)\ge \rho if \parallel z\parallel =r. Then, since {\mathrm{\Phi}}_{\lambda}^{\prime}={\mathrm{\Psi}}^{\prime} and \mathrm{\Psi} also satisfies the (P.S.) condition, from the mountain pass theorem, the number
where
is a critical value of {\mathrm{\Phi}}_{\lambda}. Noting that all paths joining {z}^{+} to {z}^{} pass through W, we have c\ge M. Therefore we have obtained a third critical point of {\mathrm{\Phi}}_{\lambda}. The proof is now complete. □
Proof of Theorem 1.3
The proof will be divided into two steps.
Step 1 (the {(P.S.)}_{c} condition). Let \{{z}_{n}=({u}_{n},{v}_{n})\}\subset Z be such that there exists c>0 such that
and there exists a strictly decreasing sequence {\{{\epsilon}_{n}\}}_{n=1}^{\mathrm{\infty}} with {lim}_{n\to \mathrm{\infty}}{\epsilon}_{n}=0 such that
We first prove that \{{z}_{n}\} is bounded in Z, then by a standard argument, \{{z}_{n}\} has a convergent subsequence. Suppose by contradiction that \parallel {z}_{n}\parallel \to \mathrm{\infty}, we have
Letting n tend to infinity, we get
where C is a constant, which contradicts (7). Thus, \{{z}_{n}\} is bounded. Since the nonlinearity F satisfies the conditions (f1), by a standard argument, \{{z}_{n}\} has a convergent subsequence.
Step 2 (the saddle point theorem). It is well known that the (P.S.) condition can be replaced by the {(P.S.)}_{c} condition in the saddle point theorem of Rabinowitz (see [13], [14]). Then to conclude that {\mathrm{\Phi}}_{\lambda} has a critical point it suffices to show that
where z\in V=span\{{\varphi}_{1}\} and w\in W.
For {\lambda}_{1}<\lambda <{\lambda}_{2}, from (11), we get
Since \theta \in (1,p), \tau \in (1,q), the first part of (21) holds.
For {\lambda}_{1}<\lambda <{\lambda}_{2} and for any w\in W, from (13), we obtain the second statement of (21). The proof is now complete. □
References
Drábek P, Kufner A, Nicolosi F: Quasilinear Elliptic Equations with Degenerations and Singularities. de Gruyter, Berlin; 1997.
Zographopoulos NB: On the principal eigenvalue of degenerate quasilinear elliptic systems. Math. Nachr. 2008, 281: 13511365. 10.1002/mana.200510683
Dautray R, Lions JL: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin; 1990.
Karachalios NI, Zographopoulos NB: On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence. Calc. Var. Partial Differ. Equ. 2006, 25: 361393. 10.1007/s0052600503474
Mawhin J, Schmitt K: Nonlinear eigenvalue problems with the parameter near resonance. Ann. Pol. Math. 1990, 51: 241248.
Ma TF, Ramost M, Sanchez L: Multiple solutions for a class of nonlinear boundary value problem near resonance: a variational approach. Nonlinear Anal. 1997, 30: 33013311. 10.1016/S0362546X(96)00380X
Ma TF, Pelicer ML:Perturbations near resonance for the pLaplacian in {R}^{N}. Abstr. Appl. Anal. 2002, 7: 323334. 10.1155/S1085337502203073
Ma TF, Ramost M: Three solutions of a quasilinear elliptic problem near resonance. Math. Slovaca 1997, 47: 451457.
Ou ZQ, Tang CL: Existence and multiplicity results for some elliptic systems at resonance. Nonlinear Anal. 2009, 71: 26602666. 10.1016/j.na.2009.01.106
Drábek P, Stavrakakis NM, Zographopoulos NB: Multiple nonsemitrivial solutions for quasilinear elliptic systems. Differ. Integral Equ. 2003, 16: 15191532.
Brock F, Iturriaga J, Sánchez J, Ubilla P:Existence of positive solution for pLaplacian problems with weight. Commun. Pure Appl. Anal. 2006, 5: 941952. 10.3934/cpaa.2006.5.941
Ghoussoub N, Yuan C: Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 2000, 352: 57035743. 10.1090/S0002994700025605
Bartolo P, Benci V, Fortunato D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 1983, 7: 9811012. 10.1016/0362546X(83)901153
Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS 1986.
Acknowledgements
This work was supported by the Science and Technology Foundation of Guizhou Province (No. LKB[2012]19; No. [2013]2141).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
An, YC., Lu, X. & Suo, HM. Existence and multiplicity results for a degenerate quasilinear elliptic system near resonance. Bound Value Probl 2014, 184 (2014). https://doi.org/10.1186/s1366101401845
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366101401845