Abstract
In this paper, we consider the following coupled gradient-type quasilinear elliptic system
where \(\Omega\) is an open bounded domain in \({\mathbb {R}}^N\), \(N\ge 2\). We suppose that some \(\mathcal {C}^{1}\)–Carathéodory functions \(A, B:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) exist such that \(a(x,t,\xi ) = \nabla _{\xi } A(x,t,\xi )\), \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t} (x,t,\xi )\), \(b(x,t,\xi ) = \nabla _{\xi } B(x,t,\xi )\), \(B_t(x,t,\xi ) =\frac{\partial B}{\partial t}(x,t,\xi )\), and that \(G_u(x, u, v)\), \(G_v(x, u, v)\) are the partial derivatives of a \(\mathcal {C}^{1}\)–Carathéodory nonlinearity \(G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\). Roughly speaking, we assume that \(A(x,t,\xi )\) grows at least as \((1+|t|^{s_1p_1})|\xi |^{p_1}\), \(p_1 > 1\), \(s_1 \ge 0\), while \(B(x,t,\xi )\) grows as \((1+|t|^{s_2p_2})|\xi |^{p_2}\), \(p_2 > 1\), \(s_2 \ge 0\), and that G(x, u, v) can also have a supercritical growth related to \(s_1\) and \(s_2\). Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.
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1 Introduction
The study of partial differential equations involving nonlinearities with critical or supercritical growths is a very complex matter, and for many critical and supercritical problems, some basic issues are mostly unknown or undiscovered. For example, let us consider the quasilinear elliptic problem
where \(\Omega\) is an open bounded domain in \({\mathbb {R}}^N, N\ge 2\), and \(1<p<N\). In spite of the simple looking structure of the problem, if we ask q to be critical or supercritical from the viewpoint of the Sobolev embedding theorem, namely \(q\ge p^*=\frac{Np}{N-p}\), some significant difficulties arise. Among other problems, in general, the lack of compactness which occurs does not guarantee even the existence of solutions, which has been derived only in few cases and frequently under assumptions on the shape of the domain \(\Omega\) (for the classical nonexistence result due to the Pohožaev’s identity see [36], or also [37, Theorem III.1.3]).
The existence of positive solutions of (1.1) has been successfully addressed either by adding some lower-order term to the critical nonlinearity (see [9]) or by considering domains which are not starshaped (see, e.g., [6, 22]) if \(p=2\), \(q=2^*\) and \(\lambda =0\), while the existence of sign-changing solutions of (1.1) has been obtained if \(p=2\), \(q=2^*\) but \(\lambda \ne 0\) (see, e.g., [3, 20]). To our knowledge, all the results carried over so far are built on the key assertion that the functional associated with the critical problem (1.1) satisfies the Palais–Smale condition even if only in certain ranges of energy.
On the other hand, taking \(p\ne 2\), due to the hardship in handling a quasilinear operator, very few results of existence have been derived so far, not even under assumptions of symmetry on the domain (we refer to [31] for a wider discussion). We limit ourselves to point out that as derived in [34], a Pohožaev-type nonexistence result is not yet available for sign-changing solutions of (1.1), as the unique continuation principle for the p-Laplacian is not known, while it has been proved for nonnegative solutions (see [23]). However, the existence of a positive solution in a domain with a sufficiently small hole has been shown for \(\frac{2N}{N + 2}\le p \le 2\), as well as an existence and multiplicity result have been proved under further assumptions of symmetry (see [21, 31, 32, 34, 35] and references therein).
In spite of the mentioned difficulties, in recent years, there has been a marked increase in research in critical and supercritical problems. The interest in these problems is related to their similarity to some variational problems which arise in Geometry and Physics where the lack of compactness also occurs. In this sense, one of the best known challenges is the so-called Yamabe’s problem, but also some examples related to the existence of extremal functions for isoperimetric inequalities, Hardy–Littlewood–Sobolev inequalities and trace inequalities can be addressed (see, e.g., [25, 27, 29]). However, in general, also in the “simplest” cases, some problems are still open, and some classical variational tools, largely used in the subcritical case, do not work in the critical and supercritical ones.
Anyway, recently, quasilinear problems which generalize
have been studied and, by means of a suitable variational setting, the existence of infinitely many weak bounded solutions is proved also if the nonlinear term has a supercritical growth such as \(2< 1 + p< p(s+1)< \mu < p^*(s+1)\), when \(A \in L^\infty (\Omega )\) is such that \(A(x) \ge \alpha _0 >0\) for a.e. \(x \in \Omega\) (see [14] and also, for other approaches, [4, 30]). One of the most remarkable feature of this work is that unlike the results mentioned above, both an existence and a multiplicity result have been provided in the supercritical case for a more general problem without taking any symmetry assumption on the domain \(\Omega\).
Here, following the ideas introduced in [14], we look for solutions of the family of coupled gradient-type quasilinear elliptic systems
where \(\Omega\) is an open bounded domain in \({\mathbb {R}}^N\), \(N \ge 2\), and \(A, B :\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N \rightarrow {\mathbb {R}}\) are given functions with partial derivatives
for a.e. \(x \in \Omega\), for all \((t,\xi ) \in {\mathbb {R}}\times {\mathbb {R}}^N\). Moreover, a nonlinear function \(G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) exists so that
Roughly speaking, here we assume that \(A(x,u,\nabla u)\) grows at least as \((1+|u|^{s_1p_1})|\nabla u|^{p_1}\), \(p_1 > 1\), \(s_1 \ge 0\), while \(B(x,v,\nabla v)\) grows at least as \((1+|v|^{s_2p_2})|\nabla v|^{p_2}\), \(p_2 > 1\), \(s_2 \ge 0\) (see Remark 3.2 and assumption \((h_7)\)), and that G(x, u, v) can also have a supercritical growth depending on \(s_1\) and \(s_2\) (see hypothesis \((g_2)\)).
While subcritical quasilinear systems have been handled through several techniques (see, e.g., [5, 8, 10, 16, 18]), as far as we know very few existence results have been determined for supercritical quasilinear elliptic systems (see, for example, [19, 24] and references therein), even though no result occurs for supercritical systems with coefficients depending on the solution and its gradient themselves, as that one in (1.2). Moreover, as in [14], even if a supercritical growth occurs, the domain \(\Omega\) is only open and bounded as we consider homogeneous Dirichlet boundary condition, and the solutions we are looking for are weak.
Thus, following the same approach used in [16] and [18], but carefully adapting the ideas in [14] to our supercritical setting, we give some sufficient conditions for recognizing the variational structure of problem (1.2), so that investigating solutions of (1.2) reduces to find critical points of functional
in the product Banach space \(X=X_1\times X_2\), with \(X_i=W_0^{1, p_i}(\Omega )\cap L^{\infty }(\Omega )\) if \(i\in \{1, 2\}\).
Moreover, since in the Banach space X our functional \(\mathcal{J}\) does not satisfy the Palais–Smale condition, or one of its standard variants, we are not allowed to use directly existence and multiplicity results as the classical Ambrosetti–Rabinowitz theorems stated in [2] or in [7]. Hence, we have to submit a weaker definition of the Cerami’s variant of Palais–Smale condition, the so-called weak Cerami–Palais–Smale condition (see Definition 2.1). We believe that the use of this definition, introduced in the pioneering paper [11] and employed in the framework of a quasilinear supercritical system, represents another major improvement of the work in this field. In fact, here Definition 2.1 is used for stating an extended Mountain Pass theorem and also its symmetric version of which we avail to gain our existence and multiplicity results (see Theorems 2.2 and 2.3), but we do not exclude the chance that this feature may be also employed to recover other kind of problems (see, e.g., [33]). In fact, we highlight that this technique has been adapted to address problems placed over unbounded domains both in radial and in non-radial setting (see [15], respectively [17]) but so far only in subcritical growth assumptions (in [1], the existence of solutions for some critical and supercritical problems has been proved by using a different (radial) approach, which is not applicable for non-autonomous equations).
On the other hand, this enhancement imposes to pay the price that some technical assumptions on the involved functions are needed. Namely, if we just assume the Carathéodory functions \(A(x, t, \xi )\), \(B(x, t, \xi )\), G(x, u, v) and their partial derivatives fit some proper polynomial growths to show the \(\mathcal {C}^1\) regularity of the functional \(\mathcal{J}\) in (1.6), on the other hand, the proof of the weak Cerami–Palais–Smale condition passes through some fine requirements on the involved functions (see Sect. 3) and a very remarkable result (see Lemma 3.7) which has interest own self and can be employed regardless of this scenario to fix a problem of common trouble in this field.
Now, in order to draw the attention to the enhancement of our main results, we state them here in a “streamlined” version but we refer the reader to Sect. 4 for all the needed hypotheses on the involved functions and the precise statement of the results (see Theorems 4.1 and 4.2).
Theorem 1.1
Suppose that \(A(x,t,\xi )\) grows at least as \((1+|t|^{s_1p_1})|\xi |^{p_1}\), with \(p_1 > 1\), \(s_1 \ge 0\), while \(B(x,t,\xi )\) grows at least as \((1+|t|^{s_2p_2})|\xi |^{p_2}\), with \(p_2 > 1\), \(s_2 \ge 0\). Moreover, assume that the \(\mathcal {C}^1\)–Carathéodory function \(A(x,t,\xi )\), respectively \(B(x,t,\xi )\), and its partial derivatives fits some suitable interaction properties among themselves, while the \(\mathcal {C}^1\)–Carathéodory nonlinear term G(x, u, v) satisfies the Ambrosetti–Rabinowitz condition for systems with coefficients \(\theta _1\), \(\theta _2 > 0\) such that \(\theta _i <\frac{1}{p_i}\), \(i \in \{1,2\}\), and has a proper polynomial growth which can also be supercritical depending on \(s_1\) and \(s_2\). If
with \(\lambda _{i,1}\) first eigenvalue of \(-\Delta _{p_i}\) in \(W_0^{p_i}(\Omega )\), \(i\in \{1, 2\}\), then problem (1.2) admits a nontrivial weak bounded solution.
Theorem 1.2
In the same hypotheses of Theorem 1.1, assume that \(A(x,\cdot ,\cdot )\) and \(B(x,\cdot ,\cdot )\) are even in \({\mathbb {R}}\times {\mathbb {R}}^N\) while \(G(x,\cdot , \cdot )\) is even in \({\mathbb {R}}^2\) for a.e. \(x\in \Omega\). Then, if
problem (1.2) admits infinitely many distinct weak bounded solutions.
Finally, in order to better explain the required hypotheses, we consider the particular setting
and
with \(c_*\ge 0\) and some positive exponents \(p_i\), \(s_i\), \(q_i\), \(\gamma _i\) for each \(i\in \{1, 2\}\). So, \(\mathcal{J}\) in (1.6) reduces to the functional \(\mathcal{J}_0 : X \rightarrow {\mathbb {R}}\) such that
and, in a suitable set of assumptions, system (1.2) turns into the model problem
Hence, the previous results can be reworded in this way.
Theorem 1.3
Let \(A(x,t,\xi )\), \(B(x,t,\xi )\) and G(x, u, v) be as in (1.7) and (1.8) with \(p_i>1\), \(i \in \{1,2\}\), \(c_* \ge 0\) and either \(p_1< N\) or \(p_2< N\). Assume that \(\theta _1, \theta _2\) exist such that
where \(p^*_1\), \(p_2^*\) are the critical Sobolev exponents, and also
Then, if
problem (1.9) admits infinitely many weak bounded distinct solutions.
Our paper is organized as follows. In Sect. 2, we introduce the abstract setting needed to recognize the variational structure of our problem (1.2), as well as some extended versions of the Mountain Pass theorems are shown up. Furthermore, a regularity result for the functional \(\mathcal{J}\) in (1.6) is provided, too. Then, in Sect. 3, some further assumptions on \(A(x, t, \xi )\), \(B(x, t, \xi )\) and G(x, u, v) are addressed in order to show that the functional \(\mathcal{J}\) verifies the weak Cerami–Palais–Smale condition. Lastly, in Sect. 4, our main results are stated and proved.
2 Abstract tools and variational setting
We denote \({\mathbb {N}}= \{1, 2, \dots \}\) and, as long as we introduce our abstract setting, we employ the following notations:
-
\((X, \Vert \cdot \Vert _X)\) is a Banach space with dual \((X',\Vert \cdot \Vert _{X'})\),
-
\((W,\Vert \cdot \Vert _W)\) is a Banach space such that \(X \hookrightarrow W\) continuously, i.e., \(X \subset W\) and a constant \(\sigma _0 > 0\) exists such that
$$\begin{aligned}\Vert y\Vert _W \ \le \ \sigma _0\ \Vert y\Vert _X\qquad \hbox {for all }y \in X, \end{aligned}$$ -
\(J : \mathcal{D} \subset W \rightarrow {\mathbb {R}}\) and \(J \in \mathcal {C}^1(X,{\mathbb {R}})\) with \(X \subset \mathcal{D}\).
In order to avoid any ambiguity and simplify, when possible, the notation, from now on by X we denote the space equipped with its given norm \(\Vert \cdot \Vert _X\) while if the norm \(\Vert \cdot \Vert _{W}\) is involved, we write it explicitly.
Now, taking \(\beta \in {\mathbb {R}}\), we say that a sequence \((y_n)_n\subset X\) is a Cerami–Palais–Smale sequence at level \(\beta\), briefly \((CPS)_\beta\)-sequence, if
As pointed out in [13, Example 4.3], a \((CPS)_{\beta }\) sequence can be constructed so that it is unbounded in \(\Vert \cdot \Vert _X\) but converges with respect to \(\Vert \cdot \Vert _W\). Thus, as in [14], we introduce the following definition.
Definition 2.1
The functional J satisfies the weak Cerami-Palais-Smale condition at level \(\beta\) (\(\beta \in {\mathbb {R}}\)), briefly \((wCPS)_\beta\) condition, if for every \((CPS)_\beta\)-sequence \((y_n)_n\), a point \(y \in X\) exists, such that
- (i):
-
\(\displaystyle \lim _{n \rightarrow +\infty } \Vert y_n - y\Vert _W = 0\quad\) (up to subsequences),
- (ii):
-
\(J(y) = \beta\), \(\; dJ(y) = 0\).
We say that J satisfies the (wCPS) condition in I, I real interval, if J satisfies the \((wCPS)_{\beta }\) condition in X at each level \(\beta \in I\).
Anyway, even if we deal with a weaker version of the Cerami’s variant of the Palais–Smale condition, some classical abstract results can be extended so to fit to our purposes. Actually, as in [14, Lemma 2.2] (see also [12, Lemma 2.3]), a Deformation Lemma can be stated which provides the following extended version of the Mountain Pass theorem given in [2] (see [14, Theorem 2.3] for a detailed proof).
Theorem 2.2
Let \(J\in \mathcal {C}^1(X,{\mathbb {R}})\) be such that \(J(0) = 0\) and the (wCPS) condition holds in \({\mathbb {R}}_+\). Moreover, assume that there exist a continuous map \(\ell : X \rightarrow {\mathbb {R}}\), some constants \(r_0\), \(\varrho _0 > 0\), and \(e \in X\) such that
- (i):
-
\(\; \ell (0) =0 \qquad \hbox {\it{and}}\qquad \ell (y) \ge \Vert y\Vert _W \quad \hbox {\it{for all }}y \in X\);
- (ii):
-
\(\; y \in X, \quad \ell (y) = r_0\qquad \Longrightarrow \qquad J(y) \ge \varrho _0\);
- (iii):
-
\(\; \Vert e\Vert _W > r_0\qquad \hbox {\it{and}}\qquad J(e) < \varrho _0\).
Then, J has a Mountain Pass critical point \(y^* \in X\) such that \(J(y^*) \ge \varrho _0\).
If, in addition, we require that J is symmetric, then a more general version of the Symmetric Mountain Pass Theorem in [2] can be stated, too (for the proof, see [14, Theorem 2.4]).
Theorem 2.3
Let \(J\in \mathcal {C}^1(X,{\mathbb {R}})\) be an even functional such that \(J(0) = 0\) and the (wCPS) condition holds in \({\mathbb {R}}_+\). Moreover, assume that \(\varrho > 0\) exists so that:
-
\((\mathcal{H}_{\varrho })\) three closed subsets \(V_\varrho\), \(Z_\varrho\) and \(\mathcal{M}_\varrho\) of X and a constant \(R_\varrho > 0\) exist which satisfy the following conditions:
-
(i) \(V_\varrho\) and \(Z_\varrho\) are subspaces of X such that
$$\begin{aligned}V_\varrho + Z_\varrho = X,\qquad {{\,\text{codim}\,}}Z_\varrho<\ \dim V_\varrho <\ +\infty ; \end{aligned}$$ -
(ii) \(\mathcal{M}_\varrho = \partial \mathcal{N}\), where \(\mathcal{N}\subset X\) is a neighborhood of the origin which is symmetric and bounded with respect to \(\Vert \cdot \Vert _W\);
-
(iii) \(\ y \in \mathcal{M}_\varrho \cap Z_\varrho \qquad \Longrightarrow \qquad J(y) \ge \varrho\);
-
(iv) \(\ y \in V_\varrho , \quad \Vert y\Vert _X \ge R_\varrho \qquad \Longrightarrow \qquad J(y) \le 0\).
Then, if we put
with
the functional J possesses at least a pair of symmetric critical points in X with corresponding critical level \(\beta _\varrho\) which belongs to \([\varrho ,\varrho _1]\), where \(\varrho _1 \ge \displaystyle \sup _{y \in V_\varrho }J(y) > \varrho\).
Remark 2.4
Since the vector space \(V_{\varrho }\) in Theorem 2.3 has finite dimension, then condition \(({\mathcal {H}}_{\varrho })(iv)\) implies that \(\displaystyle \sup _{y\in V_{\varrho }} \mathcal{J}(y) <+\infty\). Moreover, such hypothesis still holds if we replace \(\Vert \cdot \Vert _{X}\) with \(\Vert \cdot \Vert _{W}\).
Finally, if we can apply Theorem 2.3 infinitely many times, then the following multiplicity abstract result can be stated, too.
Corollary 2.5
Let \(J \in \mathcal {C}^1(X,{\mathbb {R}})\) be an even functional such that \(J(0) = 0\), the (wCPS) condition holds in \({\mathbb {R}}_+\) and assumption \((\mathcal{H}_{\varrho })\) holds for all \(\varrho > 0\). Then, the functional J possesses a sequence of critical points \((y_n)_n \subset X\) such that \(J(y_n) \nearrow +\infty\) as \(n \nearrow +\infty\).
Now, we proceed introducing the notations related to our specific setting. If \(\Omega \subset {\mathbb {R}}^N\) is an open bounded domain, \(N\ge 2\), we denote by:
-
\(L^q(\Omega )\) the Lebesgue space with norm \(|y|_q = \left( \int _\Omega |y|^q dx\right) ^{1/q}\) if \(1 \le q < +\infty\);
-
\(L^\infty (\Omega )\) the space of Lebesgue-measurable and essentially bounded functions \(y :\Omega \rightarrow {\mathbb {R}}\) with norm \(\displaystyle |y|_{\infty } = {{\,\text{{ess sup}}\,}}_{\Omega } |y|\);
-
\(W^{1,p}_0(\Omega )\) the Sobolev space equipped with the norm \(\Vert y\Vert _{W_0^{1, p}} = |\nabla y|_{p}\) if \(1 \le p < +\infty\);
-
\(\rm{ meas }(D)\) the usual Lebesgue measure of a measurable set D in \({\mathbb {R}}^N\);
-
\(|\cdot |\) the standard norm on any Euclidean space, as the dimension of the vector taken into account is clear and no ambiguity occurs.
Moreover, for any \(m\in {\mathbb {N}}\), we say that \(h:\Omega \times {\mathbb {R}}^m\rightarrow {\mathbb {R}}\) is a \(\mathcal {C}^{k}\)–Carathéodory function, \(k\in {\mathbb {N}}\cup \lbrace 0\rbrace\), if \(\ h(\cdot , \nu )\) is measurable in \(\Omega\) for all \(\nu \in {\mathbb {R}}^m\) while \(h(x,\cdot )\) is \(\mathcal {C}^k\) in \({\mathbb {R}}^m\) for a.e. \(x \in \Omega\).
Let \(A,\ B :\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) be such that the following conditions hold:
- \((h_0)\):
-
\(A(x,t,\xi )\) and \(B(x,t,\xi )\) are \(\mathcal {C}^1\)–Carathéodory functions with partial derivatives as in (1.3), respectively (1.4);
- \((h_1)\):
-
two exponents \(p_1>1\), \(p_2 > 1\), and some positive functions \(\Phi _i, \phi _i, \Psi _i, \psi _i \in \mathcal {C}^0({\mathbb {R}},{\mathbb {R}})\), if \(i \in \{0,1,2\}\), exist such that
$$|A(x,t,\xi )|\,\,\, \le \,\,\,\Phi _{0} (t) + \phi _{0} (t)|\xi |^{{p_{1} }} \qquad \mbox{ a.e. in }\Omega ,\mbox{ for all }(t,\xi ) \in \mathbb{R} \times \mathbb{R}^{N} ,$$(2.1)$$\begin{aligned} |A_{t} (x,t,\xi )|{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} & \le {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\Phi _{1} (t) + \phi _{1} (t)|\xi |^{{p_{1} }} \qquad \mbox{ a.e. in }\,\,\Omega ,\,\,\,{\text{for}}\,\,\,{\text{all}}\,\,\,(t,\xi ) \in \mathbb{R} \times \mathbb{R}^{N} , \\ |a(x,t,\xi )|{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\, & \le \,\,\,\Phi _{2} (t) + \phi _{2} (t)|\xi |^{{p_{1} - 1}} \quad\; \mbox{ a.e. in }\,\,\,\Omega ,\,\,{\text{for all}}\,\,(t,\xi ) \in \mathbb{R} \times \mathbb{R}^{N} , \end{aligned}$$(2.2)and
$$\begin{aligned} \vert B(x, t, \xi )\vert\,\,\le\,\, & {} \Psi _0(t) +\psi _0(t)\vert \xi \vert ^{p_2} \qquad \mbox{ a.e. in } \Omega , \mbox{ for all } (t,\xi )\in {\mathbb {R}}\times {\mathbb {R}}^N,\end{aligned}$$(2.3)$$ |B_{t} (x,t,\xi )|\,\,\,\,\, \le \,\, {\rm{ }}\Psi _{1} (t) + \psi _{1} (t)|\xi |^{{p_{2} }} \qquad {\rm{ a}}{\rm{.e}}.\;{\rm{ in }}\;\Omega ,{\rm{ for \; all }}\;(t,\xi ) \in \mathbb{R} \times \mathbb{R}^{N} ,{\rm{ }}|b(x,t,\xi )|\,\,\,\, \le \,\,\,{\rm{ }}\Psi _{2} (t) + \psi _{2} (t)|\xi |^{{p_{2} - 1}} \quad {\rm{ a}}{\rm{.e}}.\;{\rm{ in }}\;\Omega ,{\rm{ for \; all }}\;(t,\xi ) \in \mathbb{R} \times \mathbb{R}^{N} . $$(2.4)
Furthermore, let \(G :\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a map which satisfies the following hypotheses:
- \((g_0)\):
-
G(x, u, v) is a \(\mathcal {C}^1\)–Caratheodory function with partial derivatives as in (1.5), such that
$$\begin{aligned}G(\cdot , 0, 0)\in L^{\infty }(\Omega ) \quad \hbox {and}\quad G_u(x, 0, 0) = G_v(x, 0, 0) = 0 \quad \mbox{ for a.e. } x\in \Omega ; \end{aligned}$$ - \((g_1)\):
-
a constant \(\sigma >0\) and some exponents \(q_i \ge 1\), \(t_i \ge 0\), if \(i\in \{1, 2\}\), exist such that
$$\begin{aligned} \vert G_u(x,u,v)\vert&\le \sigma (1 +\vert u\vert ^{q_1 -1} +\vert v\vert ^{t_1}) \quad \mbox{ for a.e. } x\in \Omega ,\hbox { for all }(u, v)\in {\mathbb {R}}^2,\\ \vert G_v(x,u,v)\vert&\le \sigma (1 +\vert u\vert ^{t_2} +\vert v\vert ^{q_2 -1}) \quad \mbox{ for a.e. } x\in \Omega ,\hbox { for all }(u, v)\in {\mathbb {R}}^2. \end{aligned}$$(2.5)
Remark 2.6
Hypotheses \((g_0)\)–\((g_1)\), the mean value theorem and direct computations ensure the existence of a positive constant \(\sigma _1 > 0\) such that
Now, taking any couple of real numbers \(t_3\), \(t_5 >1\), from Young inequality, we obtain
where for simplicity, we set
Thus, from (2.6) and (2.7), we infer that
with
for a suitable constant \(\sigma _2 > 0\).
In order to recall some features shared by the subcritical systems in [16] and [18], if needed, here we introduce similar notations.
For each \(i\in \lbrace 1, 2\rbrace\) let \(p_i > 1\) be as in assumption \((h_1)\) and let us consider the related Sobolev space
From the Sobolev embedding theorem, for any \(r\in [1, p_i^{*}]\) with \(p_i^{*} = \frac{Np_i}{N-p_i}\) if \(N>p_i\), or any \(r\in [1, +\infty [\) if \(p_i \ge N\), \(W_i\) is continuously embedded in \(L^{r}(\Omega )\), i.e., \(\tau _{i,r} > 0\) exists such that
Furthermore, if \(p_i\ge N\), we place
Here, the notation \((W, \Vert \cdot \Vert _W)\), introduced for the abstract setting at the beginning of this section, is referred to our problem with
Since \((W_i, \Vert \cdot \Vert _{W_i})\) is a reflexive Banach space for both \(i\in \{1, 2\}\), so is \(\left( W, \Vert \cdot \Vert _W\right)\) in (2.12).
Moreover, we consider the Banach space \((X,\Vert \cdot \Vert _X)\) defined as
where
are endowed with the norms
Setting
we have that X in (2.13) can also be written as
and its norm is such that
Clearly, from (2.14), for both \(i\in \{1, 2\}\) we have that the continuous embeddings \(X_i \hookrightarrow W_i\) and \(X_i \hookrightarrow L^\infty (\Omega )\) hold.
Remark 2.7
If \(p_i > N\) for both \(i\in \{1, 2\}\), then the embedding \(W_i \hookrightarrow L^\infty (\Omega )\) means that \(X_i=W_i\). Thus, \(X = W\) and the classical Mountain Pass theorems in [2] may be applied.
Firstly, we note that if conditions \((h_0)\)–\((h_1)\), \((g_0)\)–\((g_1)\) hold, then direct computations imply that \(\mathcal{J}(u,v)\) in (1.6) is well-defined for all \((u, v)\in X\). Moreover, taking any (u, v), \((w, z)\in X\), the Gâteaux differential of functional \(\mathcal{J}\) in (u, v) along the direction (w, z) is given by
For simplicity, we set
hence, from (2.16), it follows that
and
Taking \((u, v)\in X\), since \(d\mathcal{J}(u, v)\in X^{\prime }\), then
and
Furthermore, above remarks and direct computations give not only the estimates
but also
At last, from (2.19), we infer that
Finally, we can state the regularity of functional \(\mathcal{J}\) defined in (1.6) (for the proof, see [18, Proposition 3.5]).
Proposition 2.8
Assume that conditions \((h_0)\)–\((h_1)\), \((g_0)\)–\((g_1)\) hold. Let \(((u_n, v_n))_n \subset X\) and \((u, v) \in X\) be such that
If \(M > 0\) exists such that
then
Hence, \(\mathcal{J}\) is a \(\mathcal {C}^1\) functional on X with Fréchet differential defined as in (2.16).
3 The set up for the weak Cerami–Palais–Smale condition
In order to prove some more properties of functional \(\mathcal{J}\) in (1.6), let \(p_1>1\) and \(p_2>1\) as in the earlier hypothesis \((h_1)\). Then, assume that \(R \ge 1\) exists such that the following conditions hold:
- \((h_2)\):
-
some constants \(\eta _1, \eta _2> 0\) exist such that
$$\begin{aligned}&A(x, t, \xi )\le \eta _1 a(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \hbox { if } \vert (t, \xi )\vert \ge R, \end{aligned}$$(3.1)$$\begin{aligned}&B(x, t, \xi )\le \eta _1 b(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \hbox { if } \vert (t, \xi )\vert \ge R, \end{aligned}$$(3.2)and
$$\begin{aligned} \sup _{\vert (t, \xi )\vert \le R} \vert A(x, t, \xi )\vert \le \eta _2, \quad \sup _{\vert (t, \xi )\vert \le R}\vert B(x, t, \xi )\vert \le \eta _2 \quad \mbox{ a.e. in } \Omega ; \end{aligned}$$(3.3) - \((h_3)\):
-
some exponents \(s_1, s_2\ge 0\) and a constant \(\mu _0 > 0\) exist so that
$$\begin{aligned}\begin{aligned}&a(x, t, \xi )\cdot \xi \ge \mu _0 (1 +\vert t\vert ^{s_1p_1})\vert \xi \vert ^{p_1} \quad \mbox{ a.e. in } \Omega , \mbox{ for all } (t, \xi )\in {\mathbb {R}}\times {\mathbb {R}}^N,\\&b(x, t, \xi )\cdot \xi \ge \mu _0 (1+\vert t\vert ^{s_2p_2})\vert \xi \vert ^{p_2} \quad \mbox{ a.e. in } \Omega , \mbox{ for all } (t, \xi )\in {\mathbb {R}}\times {\mathbb {R}}^N; \end{aligned} \end{aligned}$$ - \((h_4)\):
-
a constant \(\mu _1 >0\) exists such that
$$\begin{aligned}\begin{aligned}&a(x, t, \xi )\cdot \xi + A_t(x, t, \xi ) t \ge \mu _1 a(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \mbox{ if } \vert (t, \xi )\vert \ge R,\\&b(x, t, \xi )\cdot \xi + B_t(x, t, \xi ) t \ge \mu _1 b(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \mbox{ if } \vert (t, \xi )\vert \ge R; \end{aligned} \end{aligned}$$ - \((h_5)\):
-
some constants \(\theta _1, \theta _2, \mu _2 >0\) exist such that
$$\begin{aligned} \theta _1<\frac{1}{p_1},\qquad \theta _2 <\frac{1}{p_2}, \end{aligned}$$(3.4)and
$$\begin{aligned}\begin{aligned} A(x, t, \xi ) - \theta _1 a(x, t,\xi )\cdot \xi -\theta _1 A_t(x, t, \xi ) t&\ge \mu _2 a(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \mbox{ if } \vert (t, \xi )\vert \ge R,\\ B(x, t, \xi ) - \theta _2 b(x, t,\xi )\cdot \xi - \theta _2 B_t(x, t, \xi ) t&\ge \mu _2 b(x, t, \xi )\cdot \xi \quad \mbox{ a.e. in } \Omega \mbox{ if } \vert (t, \xi )\vert \ge R;\\ \end{aligned} \end{aligned}$$ - \((h_6)\):
-
for all \(\xi , \xi ^{\prime }\in {\mathbb {R}}^N\), with \(\xi \ne \xi ^{\prime }\), it is
$$\begin{aligned}\begin{aligned}{}[a(x, t, \xi ) -a(x, t, \xi ^{\prime })]&\cdot [\xi -\xi ^{\prime }]>0 \;\quad \rm{ a.e. in } \ \Omega , \ \mbox{ for all } t\in {\mathbb {R}},\\ [b(x, t, \xi ) -b(x, t, \xi ^{\prime })]&\cdot [\xi -\xi ^{\prime }] >0 \quad \mbox{ a.e. in } \Omega , \ \mbox{ for all } t\in {\mathbb {R}}; \end{aligned} \end{aligned}$$ - \((g_2)\):
-
for \(i\in \{1, 2\}\), taking \(p_i\) as in hypothesis \((h_1)\), \(q_i, t_i\) as in assumption \((g_1)\) and \(s_i\) as in \((h_3)\), we assume that
$$\begin{aligned} 1 \le q_1< p_1^{*}(s_1 +1),\qquad 1 \le q_2 < p_2^{*}(s_2+1), \end{aligned}$$(3.5)and
$$\begin{aligned} 0 \le t_1< \frac{p_1}{N} \left( 1 - \frac{1}{p^*_1(s_1+1)}\right) p_2^*(s_2+1), \; 0 \le t_2 < \frac{p_2}{N} \left( 1 - \frac{1}{p^*_2(s_2+1)}\right) p_1^*(s_1+1); \end{aligned}$$(3.6) - \((g_3)\):
-
taking \(\theta _1, \theta _2\) as in \((h_5)\), we assume that
$$\begin{aligned}0 < G(x, u, v)\le \theta _1 G_u(x, u, v) u +\theta _2 G_v(x, u, v) v \quad \mbox{ a.e. in } \Omega , \ \mbox{ if } \vert (u, v)\vert \ge R. \end{aligned}$$
Remark 3.1
Assumption (3.5) shows up the supercritical nature of our problem which vanishes if \(s_1 = s_2 =0\) as it reduces exactly to the subcritical condition \((g_1)\) in [16, 18].
However, in general, if one or both \(s_1 >0\), \(s_2>0\) hold, then a supercritical growth on the nonlinear term G(x, u, v) is allowed. Moreover, we emphasize that the growth hypothesis \((g_2)\) is needed to prove that the functional \(\mathcal{J}\) satisfies the (wCPS) condition, but has not been required for the variational principle stated in Proposition 2.8.
Remark 3.2
If we consider hypothesis \((h_4)\) with \(t = 0\) and \(\vert \xi \vert \ge R\), then assumption \((h_3)\) gives \(\mu _1\le 1\). Moreover, we note that \((h_4)\) and \((h_5)\) yield
which, together with condition \((h_3)\), imply that
Hence, from (3.3) and (3.9), respectively (3.10), and direct computations, we have that \(\begin{aligned}\mbox{ a.e. in } \Omega \ \mbox{ for all } \ (t, \xi )\in {\mathbb {R}}\times {\mathbb {R}}^N, \end{aligned}\)
for a suitable constant \(\eta _3>0\). On the other hand, from \((h_2)\) and (2.2), respectively (2.4), direct computations imply that
for a suitable constant \(\eta _4>0\). Then, if hypotheses \((h_2)\)–\((h_5)\) hold, the growth conditions on \(A(x, t, \xi )\) and \(B(x, t, \xi )\) stated in (2.1) and (2.3) are a direct consequence of (2.2), respectively (2.4), as (2.1) follows from (3.11) and (3.13), while (2.3) follows from (3.12) and (3.14). Hence, even if (2.1) and (2.3) are part of assumption \((h_1)\), they can be ruled-out from the hypotheses if \((h_2)\)–\((h_5)\) hold, too.
Remark 3.3
In the set of hypotheses \((h_2)\) and \((h_5)\) a more precise growth condition on both the functions \(A(x, t, \xi )\) and \(B(x, t, \xi )\) can be pointed out. In fact, (3.1), respectively (3.2), \((h_5)\) and direct calculations imply that
Now, taking \(t=0\) and \(|\xi |\ge R\) in both \((h_2)\) and \((h_5)\), without loss of generality we can choose \(\mu _2\) small enough so that
Thus, from (2.1), (3.9), (3.15), respectively (2.3), (3.10), (3.16), and direct computations, we obtain that
for a suitable \(\eta _5>0\), and then from (3.7), respectively (3.8), we have that
Finally, from (3.17), (3.18) and assumption \((h_3)\), we infer that
We note that if
then we can always choose \(\eta _1\) in \((h_2)\) large enough so that (3.19) is satisfied.
Remark 3.4
Conditions in (3.20) not only relate the exponents \(s_1, s_2\) provided in assumption \((h_3)\) with the powers \(p_1, p_2 >1\) used in the growth conditions \((h_1)\) and \(\theta _1, \theta _2\) claimed in (3.4), but also they tell us how far we can take it. In particular, it implies that in our set of hypotheses, a supercritical growth is allowed as long as \(s_1, s_2\) cover the whole range stated in (3.20).
Remark 3.5
Assumptions \((g_0)\)–\((g_1)\), \((g_3)\) and direct calculations imply that for each \(i\in \{1, 2\}\) a function \(h_i\in L^{\infty }(\Omega )\), \(h_i(x) >0\) for a.e. \(x\in \Omega\), exists such that
Thus, from (2.6), we obtain that
for a positive constant \(\sigma _3 >0\). Then, (3.20) and (3.21) imply that
while from (3.4), it is
So, if condition (3.5) holds, without loss of generality, we can take \(q_1\), \(q_2\) in \((g_1)\) large enough so that
In order to show that the (wCPS) condition holds also in our supercritical setting, we need some preliminary results.
Firstly, we note that taking \(p > 1\) and \(s \ge 0\), then straightforward computations give
Such an equality allows us to prove the following Rellich-type embedding theorem (for the proof, see [14, Lemma 3.8]).
Lemma 3.6
Taking \(1<p < N\) and \(s >0\), let \((y_n)_n\subset W^{1,p}_0(\Omega ) \cap L^\infty (\Omega )\) be a sequence such that
Then, \(y\in W^{1,p}_0(\Omega )\) exists such that \(\vert y\vert ^{s} y\in W^{1,p}_0(\Omega )\), too, and, up to subsequences, we have that
Furthermore, we state the following boundedness result (for the proof, see [26, Theorem II.5.1]).
Lemma 3.7
Let \(\Omega\) be an open bounded subset of \({\mathbb {R}}^N\) and consider \(y \in W_0^{1, p}(\Omega )\) with \(p \le N\). Suppose that \(\gamma > 0\) and \(k_0\in {\mathbb {N}}\) exist such that
with \(\Omega ^{+}_{k} =\lbrace x\in \Omega : y(x) > k\rbrace\) and r, m, \(\alpha _j\), \(\varepsilon _j\) positive constants such that
Then, \(\displaystyle {{\,\text{ {ess sup}}\,}}_{\Omega } y\) is bounded from above by a positive constant which can be chosen so that it depends only on \(\text{meas}(\Omega )\), N, p, \(\gamma\), \(k_0\), r, m, \(\varepsilon _j\), \(\alpha _j\), \(\vert y\vert _{p^{*}}\) (eventually, \(\vert y\vert _{l}\) for some \(l > r\) if \(p^{*} = +\infty\)).
As pointed out in Remark 3.1, the upper bounds in \((g_2)\) were not required so far, but will be essential in the incoming results. Therefore, some consequences of the estimates in (3.5) and (3.6) are needed.
Remark 3.8
Suppose \(1< p_1 < N\), \(1< p_2 < N\) and take \(t_1\), \(t_2\) as in (3.6). Following the ideas in Remark 2.6, we can choose \(t_3\) and \(t_5\) in (2.7) so that
as (3.6) implies that
and also
Then, \(t_4\) and \(t_6\) in (2.8) are such that
Clearly, (3.24) and (3.25) still hold if \(p_1 =N\) and/or \(p_2 =N\).
Remark 3.9
In the set of hypotheses \((g_0)\)–\((g_3)\), by reasoning as in Remark 2.6, we can consider estimate (2.9) with \({\overline{q}}_1\) and \({\overline{q}}_2\) as in (2.10) but taking \(t_j\), \(j \in \{3,4,5,6\}\), as in Remark 3.8. Hence, from (3.5), (3.22), (3.24) and (3.25), we infer that
Finally, we are able to prove that the weak Cerami–Palais–Smale condition holds.
Proposition 3.10
Under assumptions \((h_0)\)–\((h_6)\) and \((g_0)\)–\((g_3)\) functional \(\mathcal{J}: X \rightarrow {\mathbb {R}}\), defined as in (1.6), satisfies (wCPS) condition in \({\mathbb {R}}\).
Proof
Taking any \(\beta \in {\mathbb {R}}\), let \(((u_n, v_n))_n\subset X\) be a sequence such that
We want to prove that a couple \((u, v)\in X\) exists such that
- (i):
-
\((u_n, v_n) \rightarrow (u, v)\) in W (up to subsequences),
- (ii):
-
\(\mathcal{J}(u, v) = \beta\), \(\ d\mathcal{J}(u, v) = 0\).
To this aim, for simplicity, we organize our proof in the following steps:
-
1.
Both the sequences
$$\begin{aligned} \left( \int _{\Omega }(1 +\vert u_n\vert ^{s_1p_1}) \vert \nabla u_n\vert ^{p_1}dx\right) _n\; \hbox {and}\; \left( \int _{\Omega } (1 +\vert v_n\vert ^{s_2p_2}) \vert v_n\vert ^{p_2} dx\right) _n \mbox{ are bounded, } \end{aligned}$$(3.28)so, by applying Lemma 3.6, a couple \((u, v)\in W\) exists such that also \((\vert u\vert ^{s_1} u, \vert v\vert ^{s_2} v)\in W\) and, up to subsequences, we have:
$$\begin{aligned} (u_n, v_n)\ \rightharpoonup \ (u, v) \quad \mbox{ weakly in } W, \end{aligned}$$(3.29)$$\begin{aligned}(\vert u_n\vert ^{s_1} u_n, \vert v_n\vert ^{s_2} v_n)\ \rightharpoonup \ (\vert u\vert ^{s_1} u, \vert v\vert ^{s_2} v)\quad \mbox{ weakly in } W, \end{aligned}$$(3.30)$$\begin{aligned}(u_n, v_n)\ \rightarrow \ (u, v) \; \mbox{ in } L^{r_1}(\Omega )\times L^{r_2}(\Omega ) \; \mbox{ if } 1 \le r_i < p_i^{*}(s_i +1),\, i \in \{1,2\}, \end{aligned}$$(3.31)$$\begin{aligned}(u_n, v_n)\ \rightarrow \ (u, v) \quad \mbox{ a.e. in } \Omega ; \end{aligned}$$(3.32) -
2.
\((u, v)\in L^{\infty }(\Omega )\times L^{\infty }(\Omega )\);
-
3.
for any \(k>0\), define \(T_k:{\mathbb {R}}\longrightarrow {\mathbb {R}}\) such that
$$\begin{aligned}T_k t:= {\left\{ \begin{array}{*{20}l} t &{}\hbox { if } \vert t\vert \le k\\ k \frac{t}{\vert t\vert } &{}\hbox { if } \vert t\vert > k \end{array}\right. } \end{aligned}$$and
$$\begin{aligned}\mathbb {T}_k:(y_1,y_2)\in {\mathbb {R}}^2\mapsto \mathbb {T}_k(y_1,y_2) = (T_ky_1, T_ky_2)\in {\mathbb {R}}^2, \end{aligned}$$then, if \(k\ge \max \lbrace \Vert (u, v)\Vert _{L}, R\rbrace + 1\) (with \(R\ge 1\) as in our set of hypotheses), it is
$$\begin{aligned}\Vert d\mathcal{J}(\mathbb {T}_k (u_n, v_n))\Vert _{X^{\prime }}\ \rightarrow \ 0 \quad \hbox {and}\quad \mathcal{J}(\mathbb {T}_k(u_n, v_n))\ \rightarrow \ \beta ; \end{aligned}$$ -
4.
\(\Vert \mathbb {T}_k (u_n, v_n) -(u, v) \Vert _{W}\ \rightarrow \ 0\) and then (i) holds;
-
5.
(ii) is satisfied.
For simplicity, here and in the following, we will use the notation \((\varepsilon _n)_n\) for any infinitesimal sequence depending only on \(((u_n, v_n))_n\). Moreover, we denote by \(c_i\) every positive constant which arises during our computations.
Step 1. Firstly, we note that (2.20) and (3.27) imply
Thus, if we take \(\theta _1\), \(\theta _2\) as in \((h_5)\), \((g_3)\), and \(s_1\), \(s_2\) as in \((h_3)\), by reasoning as in [18Step 1 in Proposition 4.8], from (1.6), (2.17), (2.18), (3.27), (3.33), assumptions \((h_1)\), \((h_3)\), \((h_5)\), \((g_3)\) together with estimate (2.9), and direct computations it follows that
which implies that (3.28) is satisfied and, up to subsequences, \((u,v) \in W\) exists such that (3.29)–(3.32) hold.
Step 2. Due to the Sobolev Embedding Theorem, this step requires a proof only if either \(p_1 \le N\) or \(p_2 \le N\). So, if \(p_1 < N\) (when \(p_1 = N\) the arguments can be simplified), we want to prove that \(u \in L^{\infty }(\Omega )\). Arguing by contradiction, we assume that \(u\notin L^{\infty }(\Omega )\) as either
or
If (3.34) holds, then for any \(k\in {\mathbb {N}}\) , we have that
and for an integer \({\tilde{k}}>0\) , we consider the function \(R^{+}_{{\tilde{k}}}: t\in {\mathbb {R}}\mapsto R^{+}_{{\tilde{k}}} t\in {\mathbb {R}}\) defined as
Now, we consider condition (3.36) for a fixed integer \(k > R\) (with \(R\ge 1\) as in our setting of hypotheses) and, taking \({\tilde{k}} = k^{s_1 +1}\), for simplicity, we put
and, as \(|t|^{s_1} t>{\tilde{k}}\iff t>k\), we have that
Thus, from condition (3.30) and the sequentially weakly lower semicontinuity of \(\Vert \cdot \Vert _{W_1}\) , we have that
i.e.,
with
On the other hand, by definition (3.37) with \({\tilde{k}}\) replaced with k, it is \(\Vert R^{+}_k u_n\Vert _{X_1} \le \Vert u_n\Vert _{X_1}\), so from (2.20), (3.27) and (3.36), an integer \(n_k\in {\mathbb {N}}\) exists such that
Then, by reasoning as in [18, Step 2 in Proposition 4.8], from (2.17), hypotheses \((h_3)\), \((h_4)\) with \(\mu _1 \le 1\) (see Remark 3.2), equality (3.23) and estimate (3.41) we obtain that
We claim that
In fact, from (3.32) and \((g_0)\) , we have that
while thanks to assumption \((g_2)\), formulae (2.5), (2.7), (3.24), (3.25) and (3.31) ensure the existence of \(h\in L^1(\Omega )\) such that
so the dominated convergence theorem implies (3.43). Thus, summing up, via (3.40), (3.42), (3.43) and again (2.5), from definition (3.37) (\({\tilde{k}}\) replaced with k), we infer that
or better, from (2.7) but according to the choises in Remark 3.8, by taking \({\overline{q}}_1 > 1\) as in (2.10) and being \(u >1\) in \(\Omega _{u,k}^{+}\), definition (3.38) implies that
We claim that
In fact, if \(t_4= 0\) then (3.45) reduces to
while if \(t_4 > 0\) from Step 1 we have that \(z = |v|^{s_2}v \in W_2\), so (3.25) gives \(\frac{p^*_2(s_2+1)}{t_4} >1\) and the Hölder inequality with such an exponent, together with (2.11), implies that
On the other hand, if, for simplicity, we put \(r = \frac{{\overline{q}}_1}{s_1+1}\), from (3.26), direct computations and, again, (2.11) we have that
which, together with (3.45), allows us to reduce (3.44) to the estimate
with \(c_4 = c_4(\Vert w\Vert _{W_1}, \Vert z\Vert _{W_2}) > 0\). At last, as \(p_1 < N\), from (3.25), we have that
so, from (3.26) and (3.39), since (3.46) holds for all \({\tilde{k}}\) large enough, we have that Lemma 3.7 applies and \(\displaystyle {{\,\text{ {ess sup}}\,}}_{\Omega } w < +\infty\) in contradiction with (3.34). Similar arguments, but modified in a suitable way, ensures that even (3.35) cannot occur, then it has to be \(u \in L^\infty (\Omega )\), and also that it has to be \(v \in L^\infty (\Omega )\).
Step 3 The proof can be obtained by reasoning as in the proof of [18, Step 3 in Proposition 4.8] but with \(m=2\) and by replacing the estimates in [18, Remark 4.5] with those ones in Remark 3.8 together with (3.26), and also by using (3.31) at the place of [18, (4.19)].
Steps 4 and 5. The proofs are as in the corresponding steps of [18, Proposition 4.8] (see also [11, Proposition 4.6]). \(\square\)
4 Existence and multiplicity results
Now, we can state our leading results. To this aim, we refer to the decomposition of X already introduced in [16, Section 5]. For the sake of convenience, here we recall the main issues. For \(i\in \lbrace 1,2\rbrace\), the first eigenvalue of \(- \Delta _{p_i}\) in \(W_i\) is given by
Such an eigenvalue is simple, positive, isolated and has a unique eigenfunction \(\varphi _{i,1}\) such that
(see, e.g., [28]). Furthermore, a sequence of positive real numbers exists such that
with corresponding pseudo-eigenfunctions \((\psi _{i, m})_m\) which not only generate the whole space \(W_i\), but are in \(L^{\infty }(\Omega )\), too. Thus, \((\psi _{i, m})_m\subset X_i\), and, for any fixed \(m\in {\mathbb {N}}\), we consider
and denote \(Y_{i, m}\) its topological complement in \(W_i\) so that \(W_i = V_{i,m}\oplus Y_{i,m}\) and the inequality
is satisfied (cf. [11, Proposition 5.4]).
Thus, for any \(m\in {\mathbb {N}}\) definition (2.12) implies that
while from (2.15), it follows that
where, for \(i\in \lbrace 1, 2\rbrace\), it is \(Y_{m}^{X_i} = Y_{i, m}\cap L^{\infty }(\Omega )\subset X_i\) and \(X_i= V_{i, m}\oplus Y_m^{X_i}\), with
Now, we are ready to provide our existence and multiplicity results.
Theorem 4.1
Suppose that \(A(x, t, \xi )\), \(B(x, t, \xi )\) comply with assumptions \((h_0)\)–\((h_6)\) and that a given function G(x, u, v) satisfies hypotheses \((g_0)\)–\((g_3)\). Furthermore, assume that a constant \(\alpha _2>0\) exists such that the following conditions hold:
- \((h_7)\):
-
taking \(p_1\), \(p_2\) as in hypothesis \((h_1)\) and \(s_1\), \(s_2\ge 0\) as in assumption \((h_3)\), we have that
$$\begin{aligned}\begin{aligned}&A(x, t, \xi )\ge \alpha _2(1+|t|^{s_1p_1})\vert \xi \vert ^{p_1}\quad { a.e.\; in } \;\Omega , { for\; all } (t, \xi )\in {\mathbb {R}}\times {\mathbb {R}}^N,\\&B(x, t, \xi )\ge \alpha _2(1+|t|^{s_2p_2})\vert \xi \vert ^{p_2} \quad { a.e. \;in }\; \Omega , { for \;all } (t, \xi )\in {\mathbb {R}}\times {\mathbb {R}}^N; \end{aligned} \end{aligned}$$ - \((g_4)\):
-
taking \(\lambda _{1,1}\) and \(\lambda _{2,1}\) as in (4.1), we have that
$$\begin{aligned}\limsup _{(u,v)\rightarrow (0,0)} \frac{G(x, u, v)}{\vert u\vert ^{p_1}+|v|^{p_2}} <\ \alpha _2\min \{\lambda _{1,1}, \lambda _{2,1}\}\quad {uniformly\; a.e. \;in }\Omega . \end{aligned}$$
Thus, functional \(\mathcal{J}\) in (1.6) possesses at least one nontrivial critical point in X; hence, problem (1.2) admits a nontrivial weak bounded solution.
Theorem 4.2
Suppose that \(A(x,t,\xi )\), \(B(x,t,\xi )\) and G(x, u, v) satisfy hypotheses \((h_0)\)–\((h_6)\), \((g_0)\)–\((g_3)\). Moreover, if we assume also that:
- \((h_8)\):
-
\(A(x,\cdot ,\cdot )\) and \(B(x,\cdot ,\cdot )\) are even in \({\mathbb {R}}\times {\mathbb {R}}^N\) for a.e. \(x\in \Omega\);
- \((g_5)\):
-
taking \(\theta _1\), \(\theta _2\) as in hypotheses \((h_5)\) and \((g_3)\), we have that
$$\begin{aligned}\liminf _{|(u, v)|\rightarrow +\infty }\frac{G(x, u, v)}{\vert u\vert ^{\frac{1}{\theta _1}} +\vert v\vert ^{\frac{1}{\theta _2}}}\ >\ 0 \quad \hbox {uniformly a.e. in } \Omega ; \end{aligned}$$ - \((g_6)\):
-
\(G(x,\cdot , \cdot )\) is even in \({\mathbb {R}}^2\) for a.e. \(x\in \Omega\);
then functional \(\mathcal{J}\) in (1.6) possesses an unbounded sequence of critical points \(((u_m, v_m))_m\subset X\) such that \(\mathcal{J}(u_m, v_m)\nearrow +\infty\); hence, problem (1.2) admits infinitely many distinct weak bounded solutions.
Finally, by reasoning as in [16, Corollary 5.4], we can state this further multiplicity result since the supercritical growth in (3.26) does not affect its proof.
Corollary 4.3
Let \(p_1, p_2>1\) and suppose that the functions \(A(x, t, \xi )\), \(B(x, t, \xi )\) and G(x, u, v) satisfy assumptions \((h_0)\)–\((h_6)\), \((h_8)\), \((g_0)\)–\((g_3)\) and \((g_6)\). Furthermore, if
- \((g_7)\):
-
\(\; \inf \{G(x,w,z):\ x \in \Omega ,\ (w,z) \in {\mathbb {R}}^2 \ \hbox {such that |(w,z)|=R}\} > 0\), with R as in \((g_2)\);
- \((g_8)\):
-
\(\; \theta _1=\theta _2\), with \(\theta _1\), \(\theta _2\) as in \((h_5)\) and \((g_3)\);
are satisfied too, the even functional \(\mathcal{J}\) in (1.6) possesses a sequence of critical points \(((u_m,v_m))_m\) in X such that \(\mathcal{J}(u_m,v_m)\nearrow +\infty\); hence, problem (1.2) admits infinitely many distinct weak bounded solutions.
Before turning to the proof of our main results, we observe that if assumption \((h_3)\), and then \((h_7)\), holds with \(s_1 = s_2 =0\), then Theorem 4.1 reduces to [18, Theorem 5.1] while Theorem 4.2 reduces to [18, Theorem 5.2] but with \(m=2\). Actually, the same holds true if both \(p_1\ge N\) and \(p_2\ge N\). Thus, in order to improve such previous results, here we assume that either \(s_1 >0\) or \(s_2 >0\) and we define
and then
From definitions (4.5) and (4.6), we have that
and
Moreover, taking \({\bar{p}}=\min \{p_1, p_2\}\), direct computations imply that
Remark 4.4
For both \(i \in \{1,2\}\) definition (2.14) and identity (3.23) imply that the function \(y\mapsto \Vert |y|^{s_i} y\Vert _{W_i}\) is continuous and well-defined in \((X_i, \Vert \cdot \Vert _{X_i})\) and so \(\ell _i:X_i\rightarrow {\mathbb {R}}\) is continuous, too. Thus, from (2.15), we have that \((u, v)\mapsto \Vert (|u|^{s_1} u, |v|^{s_2} v)\Vert _{W}\) is continuous and well-defined in \((X, \Vert \cdot \Vert _X)\), then also \(\ell :X\rightarrow {\mathbb {R}}\) is continuous with respect \(\Vert \cdot \Vert _{X}\) and definition (4.6) implies that \(\ell (u, v) \ge \Vert (u, v)\Vert _{W}\) for all \(\ (u, v)\in X\) with \(\ell (0,0) = 0\).
Throughout the remaining part of this section, for simplicity, we assume that
with \(\mathbf{0}_N =(0,\dots , 0)\in {\mathbb {R}}^N\), and
Differently, one can always replace \(\mathcal{J}(u, v)\) in (1.6) with the new functional
since they share the same differential on X and so the same critical points.
For simplicity, we denote by \(c_i\) every positive constant which arises during computations.
Now, we can prove our existence result.
Proof of Theorem 4.1
Firstly, hypothesis \((g_4)\) allows us to take \({\bar{\lambda }} > 0\) such that
Thus, from (4.12) and direct computations, estimate (2.9) ensures the existence of a constant \(\sigma ^*>0\) such that
with \({\overline{q}}_1, {\overline{q}}_2\) as in (2.10) so that (3.26) holds. Moreover, taking \(s_1\), \(s_2\) as in our setting of hypotheses and fixing any couple \((u,v) \in X\), from definition (1.6), condition \((h_7)\), estimate (4.13) together with (3.23) and (4.1), it follows that
where from (3.26) and the Sobolev inequality (2.11), we have that
for a suitable \(c_1 > 0\) independent of i. Then, by using (4.15) in (4.14), from (4.12), a positive constant \(c_2 > 0\) exists such that definition (4.5), estimate (4.7) and direct computations imply that
for a suitable \(c_3>0\); hence, from (3.26) and (4.8), we obtain that
We note that again from (3.26), a radius \(r_0 > 0\) and a constant \(\varrho _1\) can be found so that
thus, from (4.8) and (4.16), we infer that a constant \(\varrho _0 > 0\) exists such that
On the other hand, from \((h_0)\)–\((h_2)\) and \((h_5)\) we have that [11, Proposition 6.5] implies the existence of some constants \(b^*_1\), \(b_2^* >0\) such that
a.e. in \(\Omega\) and for all \((t,\xi ) \in {\mathbb {R}}\times {\mathbb {R}}^N\), with \(\eta _1\), \(\mu _2\) as in \((h_2)\), respectively \((h_5)\), and, without loss of generality, we can assume \(\frac{1}{\theta _1}\left( 1-\frac{\mu _2}{\eta _1}\right) - p_1 >0\) (a priori, we can take either \(\mu _2\) small enough or \(\eta _1\) large enough). Thus, taking \(\varphi _{1,1}\in X_1\) as in (4.2), from (1.6), (3.21), (4.10) and direct computations, we obtain that
for a suitable \(c_4 > 0\), which implies, from (3.4) that
as (4.2) and Remark 3.5 ensure that \(\int _{\Omega } h_1(x)|\varphi _{1,1}|^{\frac{1}{\theta _1}} dx>0\). Hence, considering \(r_0\), \(\varrho _0\) so that (4.17) holds, a point \(e_1\in X_1\) can be found so that
Finally, from (1.6), (4.10) and (4.11), it is \(\mathcal{J}(0, 0)=0\), which, together with Remark 4.4, (4.17), (4.18) and Propositions 2.8 and 3.10, ensures that Theorem 2.2 applies and a critical point (u, v) exists in X such that \(\mathcal{J}(u, v)\ge \varrho _0 >0\). \(\square\)
In order to prove our multiplicity theorem, some geometric conditions are needed. In particular, if assumptions \((h_0)\)–\((h_6)\) and \((g_0)\)–\((g_3)\) hold, we are able to state the following results.
Proposition 4.5
For any fixed \(\varrho \in {\mathbb {R}}\), an integer \(m=m(\varrho )\ge 1\) and a radius \(R_m >0\) exist such that
Proof
Firstly, we note that (3.23) and (4.7) imply that
Then, taking \((u,v)\in X\), from (1.6), (3.11), (3.12), (4.19), together with (2.9) where \({\bar{q}}_1\), \({\bar{q}}_2\) satisfy (3.26), we obtain that
for some \(c_1>0\). We note that for each \(i \in \{1,2\}\) condition (3.26) allows us to take \(r_i >0\) so that
then, reasoning as in [14, Proposition 4.5], from classical interpolation arguments, (2.11) and (4.5), we obtain that
for a suitable constant \(c_2 > 0\) independent of i.
Thus, fixing any \(m\in {\mathbb {N}}\), from (4.4) and, again, (4.5), it follows that
where from (3.26), it is
Hence, taking any couple \((u, v)\in Y_m^{X_1}\times Y_m^{X_2}\), by using estimate (4.21) in (4.20), we obtain that
or better, from (4.8) and (4.22), we have that
Now, for each \(i \in \{1,2\}\), from (4.22), we can define \(R_{i,m} > 0\) so that
and, since from (4.3), it follows that \(R_{i,m} \nearrow +\infty\) as \(m \rightarrow +\infty\), we have that
which implies \(R_m \ge 2\) for all \(m \ge m_0\) if \(m_0\in {\mathbb {N}}\) is large enough. So, for any \(m \ge m_0\), taking \((u,v)\in Y_m^{X_1}\times Y_m^{X_2}\) such that \(\ell (u,v) = R_m\), from (4.9), we have that
while from (4.23), by using (4.22), (4.24) and the definition in (4.25), we obtain
Thus, for any \(m \ge m_0\) estimate (4.26) implies that
Finally, we note that the proof follows from (4.25) and (4.27). \(\square\)
At last, by reasoning as in the first part of the proof of [18, Theorem 5.2] (we note that the computations do not involve the supercritical growth of G(x, u, v) but only its lower bound coming from assumption \((g_5)\)), the following result can be stated, too.
Proposition 4.6
If also hypothesis \((g_5)\) holds, then for any finite-dimensional subspace V of X a suitable radius \(R_V>0\) exists such that
In particular, the functional \(\mathcal{J}\) is bounded form above in V.
Now, we can prove our multiplicity results.
Proof of Theorem 4.2
Firstly, we observe that (1.6), (4.10) and (4.11) give \(\mathcal{J}(0, 0)=0\), while assumptions \((h_8)\) and \((g_6)\) imply that the functional \(\mathcal{J}\) is even in X. Furthermore, taking any \(r>0\), we set
By definition, \(\mathcal{M}_r\) is the boundary of a symmetric neighborhood of the origin which is bounded with respect to \(\Vert \cdot \Vert _W\). Now, fixing any \(\varrho >0\), from Proposition 4.5, it follows that an integer \(m_{\varrho }\ge 1\) and a radius \(r_\varrho =r_\varrho (m_{\varrho }) >0\) exist so that
while, by choosing \(m>m_{\varrho }\), the m-dimensional space \(V_m\) is such that \(\rm{ codim } Y_{m_{\varrho }} < \rm{ dim } V_m\), and from Proposition 4.6 a radius \(R_{V_m} >0\) exists so that
Hence, assumption \(({\mathcal {H}}_{\varrho })\) in Theorem 2.3 is verified. Then, the arbitrariness of \(\varrho >0\) so that \(({\mathcal {H}}_{\varrho })\) holds, together with Propositions 2.8 and 3.10, allows us to apply Corollary 2.5 and the existence of a sequence of diverging critical levels for the functional \(\mathcal{J}\) in X is provided. \(\square\)
Proof of Theorem 1.3
Taking \(A(x,t,\xi )\) and \(B(x,t,\xi )\) as in (1.7), from (1.10), it follows that conditions \((h_0)\)–\((h_4)\) and \((h_6)\) hold. Moreover, if G(x, u, v) is as in (1.8), assumptions (1.10)–(1.12) and Young inequality imply that \((g_0)\)–\((g_2)\) are satisfied with
On the other hand, again from (1.10), direct computations allow us to prove that hypotheses \((h_5)\) and \((g_3)\) are verified, too. At last, also condition \((g_5)\) holds as (1.10) and direct computations allow us to prove that for any \(R \ge 2\) it is
Then, since the symmetric assumptions \((h_8)\) and \((g_6)\) are trivially satisfied, the thesis follows from Theorem 4.2. \(\square\)
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Candela, A.M., Sportelli, C. Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth. Annali di Matematica 201, 2341–2369 (2022). https://doi.org/10.1007/s10231-022-01202-0
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DOI: https://doi.org/10.1007/s10231-022-01202-0
Keywords
- Coupled gradient-type quasilinear elliptic system
- p-Laplacian-type operator
- Supercritical growth
- Weak Cerami–Palais–Smale condition
- Ambrosetti–Rabinowitz condition
- Mountain Pass theorem
- Critical Sobolev exponent
- Nontrivial weak bounded solution
- Pseudo-eigenvalue