 Research
 Open access
 Published:
Competing anisotropic and Finsler \((p,q)\)Laplacian problems
Boundary Value Problems volumeÂ 2024, ArticleÂ number:Â 39 (2024)
Abstract
The aim of this paper is to prove the existence of generalized variational solutions for nonlinear Dirichlet problems driven by anisotropic and Finsler Laplacian competing operators. The main difficulty consists in the lack of ellipticity and monotonicity in the principal part of the equations. This difficulty is overcome by developing a Galerkintype procedure.
1 Introduction
Recently, Galewski and Motreanu [5] studied the coercive competitive equation
driven by a competing \((p,q)\)Laplaciantype operator with weight depending on the gradient. Problem (1.1) is variational, but the driving operator is competing which means that the ellipticity condition is not satisfied. For this reason, one cannot establish the existence of a weak solution, but one can find a socalled generalized variational solution. Moreover, an abstract setting related to a Galerkintype scheme is built in [5] that we briefly recall here for treating new problems with competing operators.
Let E be a separable and reflexive Banach space. We recall that E is separable if there exists a countable dense subset \(\{h_{i}\}_{i\geq 1}\) of E. It is said that a sequence of finitedimensional subspaces \((E_{n})_{n=1}^{\infty}\subset E\) has the approximation property if
There always exists such a sequence \((E_{n})_{n=1}^{\infty}\) by defining \(E_{n}\) for \(n\in \mathbb{N}\) as the linear hull of \(\{h_{1}, \dots, h_{n}\}\).
Let \(A: E\to E^{*}\) be a potential operator, which means that \(A =J'\) (the differential of J) for a GÃ¢teaux differentiable function \(J: E \to \mathbb{R}\) called the potential of A. Note that the critical points of J coincide with the solutions to the equation
or equivalently (in the weak sense),
Taking advantage of the variational structure of problem (1.2) involving the functional J, the following definition sets forth a new type of solution.
Definition 1.1
An element \(u\in E\) is said to be a generalized variational solution to problem (1.2) if there exists a sequence of finitedimensional subspaces \((E_{n})_{n=1}^{\infty}\subset E\) with the approximation property and a sequence of elements \((u_{n})_{n=1}^{\infty}\) with \(u_{n}\in E_{n}\) such that

(a)
\(u_{n} \rightharpoonup u\) in E as \(n\to \infty \);

(b)
\(\inf_{v\in E_{n}} J (v) = J (u_{n})\);

(c)
\(A(u_{n})\rightharpoonup 0\) in \(E^{*}\) and \(\langle A(u_{n}), u_{n}u\rangle \to 0\).
We quote the following abstract result from [5, Theorem 4] (stated here in the particular case \(k=0\)).
Theorem 1.1
Assume that the operator \(A: E\to E^{*}\) is bounded (i.e., A maps bounded sets into bounded sets) and potential with a coercive potential \(J: E\to \mathbb{R}\) (i.e., \(\lim_{\u\\to \infty}J(u)=+\infty \)). Then problem (1.2) has at least one generalized variational solution in the sense of Definition 1.1.
Remark 1.1
Without additional assumptions, there is no relation between the notion of (weak) solution in (1.3) and the notion of generalized variational solution given in Definition 1.1. Indeed, the fact that \(u\in E\) fulfills (1.3) amounts to saying that \(J'(u)=0\), but then one cannot generally expect that the minimization in Definition 1.1(b) is verified. Conversely, if \(u\in E\) satisfies Definition 1.1, one cannot generally have the strong convergence \(u_{n}\to u\) in E which would result in (1.3) unless the operator A fulfills the Sproperty (meaning that \(u_{n}\rightharpoonup u\) and \(\langle A(u_{n}),u_{n}u\rangle \to 0\) imply \(u_{n}\to u\)).
The first aim of this paper is to investigate the anisotropic counterpart of problem (1.1), namely the nonlinear elliptic problem
on a bounded domain Î© in \(\mathbb{R}^{N}\) (\(N\geq 2\)) with a Lipschitz boundary âˆ‚Î©. In the statement of problem (1.4), \(g_{i}:\mathbb{R}\to \mathbb{R}\) are continuous functions for which there are constants \(0< a_{g_{i}}\leq g_{i}(t)\leq b_{g_{i}}\) for all \(t\geq 0\) and \(i=1,2,\dots,N\), and \(h:\Omega \times \mathbb{R}\to \mathbb{R}\) is a CarathÃ©odory function (i.e., \(h(x,t)\) is measurable in x and continuous in t).
A prototype of the driving operator in the lefthand side of equation (1.4) is the competing anisotropic operator \(\Delta _{\overrightarrow{p}}+\Delta _{\overrightarrow{q}}\), where
is the anisotropic pLaplacian with \(\overrightarrow{p}=(p_{1},\dots,p_{n})\), and
is the anisotropic qLaplacian with \(\overrightarrow{q}=(q_{1},\dots,q_{n})\). In (1.4) we have an extension of \(\Delta _{\overrightarrow{p}}\) constructed by means of the weights \((g_{1},\dots,g_{N})\), specifically,
We assume that \(1< p_{1},\dots,p_{N}<\infty \), \(1< q_{1},\dots,q_{N}<\infty \), \(q_{i}< p_{i}\) for all \(i=1,\dots,N\), and
Let us introduce
and assume that
The anisotropic Sobolev space \(W_{0}^{1,\overrightarrow{p}}(\Omega )\) is defined as the completion of the set of smooth functions with compact support \(C^{\infty}_{c}(\Omega )\) with respect to the norm
This space is separable and uniformly convex, thus a reflexive Banach space. The dual of the space \(W_{0}^{1,\overrightarrow{p}}(\Omega )\) is denoted \(W^{1,\overrightarrow{p}'}(\Omega )\).
We also introduce
The quantity Î¼ in (1.5) is finite due to the compact embedding \(W_{0}^{1,\overrightarrow{p}}(\Omega )\subset L^{p^{}}(\Omega )\) (see [4, Theorem 1]). For various results regarding anisotropic Sobolev spaces, we refer to [1, 3, 4, 6â€“12].
In order to simplify the notation, for any real number \(r>1\) we denote \(r':=r/(r1)\) (the HÃ¶lder conjugate of r).
The following condition is assumed to hold:
 \((H1)\):

There exist a nonnegative function \(\sigma \in L^{(p^{*})'}(\Omega )\) and a constant \(b\geq 0\) such that
$$\begin{aligned} \bigl\vert h(x,t) \bigr\vert \leq \sigma (x)+b \vert t \vert ^{p^{*}1} \end{aligned}$$for a.e. \(x\in \Omega \) and all \(t\in \mathbb{R}\).
In addition, we formulate the condition:
 \((H2)_{\xi,\alpha}\):

Given positive constants Î¾ and Î±, it holds
$$\begin{aligned} H(x,t):= \int _{0}^{t} h(x,s)\,ds\leq c_{1}\bigl( \vert t \vert ^{\alpha}+1\bigr) \end{aligned}$$for a.e. \(x\in \Omega \) and all \(t\in \mathbb{R}\), with a positive constant \(c_{1}<\xi \).
The usual arguments fail to apply for obtaining a weak solution to problem (1.4), which means an element \(u\in W_{0}^{1,\overrightarrow{p}}(\Omega )\) satisfying
for all \(v\in W_{0}^{1,\overrightarrow{p}}(\Omega )\). The reason is the lack of ellipticity condition for equation (1.4). Notice also that the driving operator in (1.4) is not monotone. The idea is to weaken the notion of solution, still keeping the main characteristics of problem (1.4) as, for instance, its variational structure. Hence the Euler functional \(J:W_{0}^{1,\overrightarrow{p}}(\Omega )\to \mathbb{R}\) associated to problem (1.4) is well defined and given by
for all \(u\in W_{0}^{1,\overrightarrow{p}}(\Omega )\), where
for a.e. \(x\in \Omega \), all \(t\in \mathbb{R}\), and \(1\leq i\leq N\).
Now we state the existence result.
Theorem 1.2
Assume that the conditions \((H1)\) and \((H2)_{\xi,\alpha}\) hold with \(\xi =\mu a^{*}\) for \(a^{*}=\min \{a_{g_{i}}:1\leq i\leq N\}\) and \(\alpha =p^{}\). Then there exists a generalized variational solution to problem (1.4) in the sense of DefinitionÂ 1.1.
The second aim of the paper is to study the Dirichlet problem
on a bounded domain Î© in \(\mathbb{R}^{N}\) (\(N\geq 2\)) with a Lipschitz boundary âˆ‚Î©. Problem (1.7) is driven by the competing Finsler \((p,q)\)Laplaciantype operator \(\mathcal{Q}^{g}_{p}+\mathcal{Q}_{q}\), with \(1< q< p<+\infty \), that we now describe.
Let \(F:\mathbb{R}^{N}\to [0,+\infty )\) be a convex function of class \(C^{2}(\mathbb{R}^{N}\backslash \{0\})\), which is even and satisfies \(F (\xi )>0\) for each \(\xi \neq 0\), and \(F (t\xi ) = tF(\xi )\) for all \(t\in \mathbb{R}\), \(\xi \in \mathbb{R}^{N}\). Given \(p\in (1,+\infty )\), we assume that there exists a constant \(\gamma >0\) such that
with some positive constant Î³, for all \(\eta \in \mathbb{R}^{N}\backslash \{0\}\) and \(\zeta \in \mathbb{R}^{N}\).
The Finsler pLaplacian operator \(\mathcal{Q}_{p}:W^{1,p}_{0}(\Omega )\to W^{1,p'}_{0}(\Omega )\) is defined as
If \(F(\xi ) = \xi \) (the Euclidean norm) and \(p=2\), then it becomes the ordinary Laplacian.
We denote by \(\lambda _{1}\) the first eigenvalue of \(\mathcal{Q}_{p}\), that is,
For more details on the operator \(\mathcal{Q}_{p}\), we refer to [2, 13]. As a real life application, we mention Wulffâ€™s work [14] on crystal shapes.
In (1.7) we have a weighted version of the Finsler pLaplacian \(\mathcal{Q}_{p}\) extending (1.8). Specifically, corresponding to a continuous function \(g:\mathbb{R}\to \mathbb{R}\) for which there exist constants \(a_{g} > 0\) and \(b_{g} > 0\) such that \(a_{g}\leq g(t)\leq b_{g}\) for all \(t\geq 0\), one sets
The underlying space for problem (1.7) is \(W^{1,p}_{0}(\Omega )\). Denote by \(p^{*}\) the critical Sobolev exponent, that is, \(p^{*} = \frac{Np}{Np}\) if \(N> p\) and \(p^{*} =+\infty \) if \(N\leq p\).
We are in a position to state our result regarding problem (1.7).
Theorem 1.3
Assume that \(h:\Omega \times \mathbb{R}\to \mathbb{R}\) is a CarathÃ©odory function for which the conditions \((H1)\) and \((H2)_{\xi,\alpha}\) with \(\xi =\frac{\lambda _{1} a_{g}}{p}\) and \(\alpha =p\) hold. Then there exists a generalized variational solution to problem (1.7) in the sense of Definition 1.1.
Remark 1.2
Problems (1.4) and (1.7) cannot be reduced one to another due to the different structure of the leading operators. For example, the driving operator in (1.4) is orthotropic whose properties depend on directions, whereas the driving operator in (1.7) is homogeneous. Such features are reflected in the distinct choices for the constants Î¾ and Î± in hypothesis \((H2)_{\xi,\alpha}\), as well as in the different proofs of Theorems 1.2 and 1.3.
Remark 1.3
The limit cases when \(q_{i}=p_{i}\) for all \(i=1,\dots,N\) in problem (1.4) and \(q=p\) in problem (1.7) generally do not give rise to competing operators, which is the object of our work. For instance, taking \(g_{i}\equiv 2\) for all \(i=1,\dots,N\), we obtain an equation driven by the (negative) pseudopLaplacian which is an elliptic operator.
In the rest of the paper, we prove the existence of generalized variational solutions for problems (1.4) and (1.7). SectionsÂ 2 and 3 contain the proofs of Theorems 1.2 and 1.3, respectively.
2 Generalized variational solutions for competing anisotropic Laplacian
In this section, we prove the existence of generalized variational solutions for problem (1.4) by Theorem 1.2, i.e., we present the proof of Theorem 1.2.
We show that we can fit in the setting of Theorem 1.1. Problem (1.4) can be regarded as an operator equation (1.2) with \(E =W_{0}^{1,\overrightarrow{p}}(\Omega )\) and
Consider the Nemytskij operator \(N_{h}:W_{0}^{1,\overrightarrow{p}}(\Omega )\to W^{1, \overrightarrow{p}'}(\Omega )\) induced by the CarathÃ©odory function \(h:\Omega \times \mathbb{R}\to \mathbb{R}\) as
Assumption \((H1)\), HÃ¶lderâ€™s inequality, and the continuous embedding \(W_{0}^{1,\overrightarrow{p}}(\Omega )\subset L^{p^{*}}(\Omega )\) (see [4, Theorem 1]) imply that there is a constant \(C > 0\) such that
for all \(v,w\in W^{1,\overrightarrow{p}}(\Omega )\). It turns out that
By HÃ¶lderâ€™s inequality, we see that
where \(\Omega \) denotes the Lebesgue measure of Î©. This ensures the continuous embedding \(W_{0}^{1,\overrightarrow{p}}(\Omega )\subset W_{0}^{1, \overrightarrow{q}}(\Omega )\), guaranteeing that the sum in (2.1) is well defined on \(W_{0}^{1,\overrightarrow{p}}(\Omega )\). It follows that the operator \(A:W_{0}^{1,\overrightarrow{p}}(\Omega )\to W^{1,\overrightarrow{p}'}( \Omega )\) expressed by (2.1) is well defined and bounded.
Standard arguments relying on assumption \((H1)\) and Lebesgueâ€™s dominated convergence theorem ensure that the functional \(J:W_{0}^{1,\overrightarrow{p}}(\Omega )\to \mathbb{R}\) in (1.6) is GÃ¢teaux differentiable and its differential \(J'\) satisfies \(J'=A\). Therefore the operator \(A:W_{0}^{1,\overrightarrow{p}}(\Omega )\to W^{1,\overrightarrow{p}'}( \Omega )\) introduced in (2.1) is a potential operator with the potential given by the functional \(J:W_{0}^{1,\overrightarrow{p}}(\Omega )\to \mathbb{R}\) in (1.6).
We claim that the functional \(J:W_{0}^{1,\overrightarrow{p}}(\Omega )\to \mathbb{R}\) in (1.6) is coercive. Towards this, we note that assumption \((H2)_{\xi,\alpha}\), with \(\xi =\mu a^{*}\), \(a^{*}=\min \{a_{g_{i}}:1\leq i\leq N\}\), \(\alpha =p^{}\), and (1.5) imply
Then, in view of (1.6), (2.2), and \(G_{i}(t)\geq a_{g_{i}}t\) for all \(t\geq 0\) and \(i=1,\dots, N\), we are led to
Since \(q_{i} < p_{i}\) for all \(i=1,\dots, N\) and \(c_{1}<\mu a^{*}\), we obtain that J coercive.
We have checked all the hypotheses required to apply Theorem 1.1 to the functional J in (1.6). As a consequence, according to Definition 1.1, the existence of a generalized variational solution to problem (1.2) with A given in (2.1) is established. This is just the stated result for the original problem (1.4), thus completing the proof of Theorem 1.2.
3 Generalized variational solutions for competing Finsler operator
In this section, we prove the existence of generalized variational solutions for problem (1.7) by Theorem 1.3, i.e., we present the proof of Theorem 1.3.
We apply Theorem 1.1 taking \(E =W_{0}^{1,p}(\Omega )\) and \(A:W_{0}^{1,p}(\Omega )\to W^{1,p'}(\Omega )\) given by
Observe that problem (1.7) can be written as the operator equation (1.2) with A in (3.1).
By HÃ¶lderâ€™s inequality, we see that
Since for the function F there exist two constants \(0< a< b<+\infty \) such that \(a\xi \leq F(\xi )\leq b\xi \) for all \(\xi \in \mathbb{R}^{N}\), the operator \(\mathcal{Q}^{g}_{p}+\mathcal{Q}_{q}\) is well defined, continuous, and bounded on \(W^{1,p}_{0}(\Omega )\).
The CarathÃ©odory function \(h(x,t)\) entering equation (1.7) determines the Nemytskij operator \(N_{h}:W_{0}^{1,p}(\Omega )\to W^{1,p'}(\Omega )\) by
Assumption \((H1)\), HÃ¶lderâ€™s inequality, and Sobolev embedding theorem imply that there is a constant \(C > 0\) such that
for all \(v,w\in W^{1,p}(\Omega )\). Hence we find the estimate
We infer from (3.2) that the operator \(N_{h}:W_{0}^{1,p}(\Omega )\to W^{1,p'}(\Omega )\) is well defined and bounded. Taking into account (3.1), it follows that the operator \(A=\mathcal{Q}^{g}_{p}+\mathcal{Q}_{q}N_{h}\) is well defined and bounded from \(W_{0}^{1,p}(\Omega )\) to \(W^{1,p'}(\Omega )\).
We are going to show that the operator A in (3.1) is potential. To this end, we define the functional \(J:W_{0}^{1,p}(\Omega )\to \mathbb{R}\) by
for all \(u\in W_{0}^{1,p}(\Omega )\), where
for a.e. \(x\in \Omega \) and all \(t\in \mathbb{R}\).
The boundedness of g implies the existence of a constant \(c > 0\) with \(G(t)\leq c(t+ 1)\) for all \(t\in \mathbb{R}\). Then, arguing on the basis of assumption \((H1)\), we can prove through Lebesgueâ€™s dominated convergence theorem that the functional J in (3.3) is GÃ¢teaux differentiable with the differential
By (3.1) and (3.4), we note that \(Au = J'(u)\) for all \(u\in W_{0}^{1,p}(\Omega )\). As a consequence, we can infer that A in (3.1) is a potential operator with the potential J given by (3.3).
Now we focus on the coerciveness of the functional J in (3.3). Assumption \((H2)_{\xi,\alpha}\), with \(\xi =\frac{\lambda _{1} a_{g}}{p}\) and \(\alpha =p\), and (1.9) imply
Then, in view of (3.3) and \(G(t)\geq a_{g}t\) for all \(t\geq 0\), we are led to
for all \(u\in W_{0}^{1,p}(\Omega )\). Since \(q < p\) and \(c_{1}<\frac{a_{g}\lambda _{1}}{p}\), we obtain that J is coercive.
All the hypotheses required to apply Theorem 1.1 to the functional J in (3.3) are fulfilled. Then the existence of a generalized variational solution to problem \(Au=0\) with A given in (3.1) is established. This completes the proof concerning the original problem (1.7).
Data Availability
No datasets were generated or analysed during the current study.
References
Bonanno, G., Dâ€™AguÃ¬, G., Sciammetta, A.: Multiple solutions for a class of anisotropic \(\overrightarrow{p}\)Laplacian problems. Bound. Value Probl. 2023, Article IDÂ 89 (2023)
Della Pietra, F., Gavitone, N., Piscitelli, G.: On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators. Bull. Sci. Math. 155, 10â€“32 (2019)
Fan, X.: Anisotropic variable exponent Sobolev spaces and \(\overrightarrow{p}(x)\)Laplacian equations. Complex Var. Elliptic Equ. 56, 623â€“642 (2011)
Fragala, I., Gazzola, F., Kawohl, B.: Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. Henri PoincarÃ©, Anal. Non LinÃ©aire 21, 715â€“734 (2004)
Galewski, M., Motreanu, D.: On variational competing \((p,q)\)Laplacian Dirichlet problem with gradient depending weight. Appl. Math. Lett. 148, Article IDÂ 108881 (2024)
MihÄƒilescu, M., Pucci, P., RÄƒdulescu, V.D.: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J. Math. Anal. Appl. 340, 687â€“698 (2008)
Motreanu, D., Tornatore, E.: Dirichlet problems with anisotropic principal part involving unbounded coefficients. Electron. J. Differ. Equ. 2024, Article IDÂ 11 (2024)
RÃ¡kosnÃk, J.: Some remarks to anisotropic Sobolev spaces I. Beitr. Anal. 13, 55â€“68 (1979)
Razani, A.: Entire weak solutions for an anisotropic equation in the Heisenberg group. Proc. Am. Math. Soc. 151(11), 4771â€“4779 (2023). https://doi.org/10.1090/proc/16488
Razani, A., Figueiredo, G.M.: Existence of infinitely many solutions for an anisotropic equation using genus theory. Math. Methods Appl. Sci. 45, 7591â€“7606 (2022)
Razani, A., Figueiredo, G.M.: Degenerated and competing anisotropic \((p,q)\)Laplacians with weights. Appl. Anal. 102, 4471â€“4488 (2023)
Razani, A., Figueiredo, G.M.: A positive solution for an anisotropic \((p,q)\)Laplacian. Discrete Contin. Dyn. Syst., Ser. S 16(6), 1629â€“1643 (2023)
Wang, G., Xia, C.: Blowup analysis of a Finslerâ€“Liouville equation in two dimensions. J. Differ. Equ. 252, 1668â€“1700 (2012)
Wulff, G.: Zur Frage der Geschwindigkeit des Wachstums und der AuflÃ¶sung der KristallflÃ¤chen. Z. Kristallogr. 34, 449â€“530 (1901)
Acknowledgements
We thank the reviewers for careful reading and helpful comments.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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 articleâ€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleâ€™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 http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Motreanu, D., Razani, A. Competing anisotropic and Finsler \((p,q)\)Laplacian problems. Bound Value Probl 2024, 39 (2024). https://doi.org/10.1186/s13661024018471
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661024018471