Nonlinear nonhomogeneous Robin problems with dependence on the gradient

We consider a nonlinear elliptic equation driven by a nonhomogeneous partial differential operator with Robin boundary condition and a convection term. Using a topological approach based on the Leray–Schauder alternative principle, together with truncation and comparison techniques, we show the existence of a smooth positive solution without imposing any global growth condition on the reaction term.

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with a \(C^{2}\)-boundary ∂Ω, and let \(1< p<+\infty\). In this paper we study the following nonlinear nonhomogeneous Robin problem with convection:

In this problem, \(a\colon\mathbb{R}^{N}\longrightarrow \mathbb{R}^{N}\) is a continuous and strictly monotone map which satisfies certain regularity and growth conditions listed in hypotheses \(H(a)\) below. These hypotheses are mild and incorporate in our framework many differential operators of interest such as the p-Laplacian and the \((p,q)\)-Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian with \(1< q< p<\infty\)). The forcing term has the form of a convection term, that is, it depends also on the gradient of the unknown function. This dependence on the gradient prevents the use of variational methods directly on equation (1.1). In the boundary condition, \(\frac{\partial u}{\partial n_{a}}\) denotes the conormal derivative of u and is defined by extension of the map

to all \(u\in W^{1,p}(\Omega)\), with n being the outward unit normal on ∂Ω. This generalized normal derivative is dictated by the nonlinear Green’s identity (see, e.g., Gasiński and Papageorgiou [1, Theorem 2.4.53, p. 210]) and was used also by Lieberman [2, 3].

Problems with convection were studied in the past using a variety of methods. We mention the works of de Figueiredo et al. [4], Girardi and Matzeu [5] for semilinear equations driven by the Dirichlet Laplacian; the works of Faraci et al. [6], Huy et al. [7], Iturriaga et al. [8] and Ruiz [9] for nonlinear equations driven by the Dirichlet p-Laplacian; and the works of Averna et al. [10], Faria et al. [11] and Tanaka [12] for equations driven by the Dirichlet \((p,q)\)-Laplacian. Finally, we mention also the recent work of Gasiński and Papageorgiou [13] for Neumann problems driven by a differential operator of the form \(\operatorname{div}(a(u)Du)\).

In this paper, in contrast to the aforementioned works, we do not impose any global growth condition on the convection term. Instead we assume that \(f(z,\cdot,y)\) admits a positive root (zero) and all the other conditions refer to the behavior of the function \(x\longmapsto f(z,x,y)\) near zero locally in \(y\in\mathbb{R}^{N}\). Our approach is topological based on the Leray–Schauder alternative principle.

In the analysis of problem (1.1) we will use the following spaces:

By \(\Vert \cdot \Vert \) we denote the norm of the Sobolev space \(W^{1,p}(\Omega )\) defined by

The Banach space \(C^{1}(\overline{\Omega})\) is an ordered Banach space with positive (order) cone given by

This cone has a nonempty interior

which contains the set

In fact \(D_{+}\) is the interior of \(C_{+}\) when \(C^{1}(\overline{\Omega})\) is equipped with the relative \(C(\overline{\Omega})\)-norm topology.

On ∂Ω we consider the \((N-1)\)-dimensional Hausdorff (surface) measure \(\sigma(\cdot)\). Using this measure, we can define the boundary Lebesgue spaces \(L^{q}(\partial\Omega )\) (\(1\leqslant q\leqslant\infty\)) in the usual way. We have that there exists a unique continuous linear map \(\gamma _{0}\colon W^{1,p}(\Omega)\longrightarrow L^{p}(\partial\Omega )\) known as the trace map such that

So, the trace map \(\gamma_{0}\) extends the notion of boundary values to any Sobolev function. We have

The trace map \(\gamma_{0}\) is compact into \(L^{q}(\partial\Omega )\) for all \(q\in[1,\frac{(N-1)p}{N-p})\) if \(p< N\) and into \(L^{q}(\partial\Omega )\) for all \(q\in[1,\infty)\) if \(p\geqslant N\). In what follows, for the sake of notational simplicity, we drop the use of the trace map \(\gamma_{0}\). The restrictions of all Sobolev functions on ∂Ω are understood in the sense of traces.

Now we introduce the conditions on the map \(a(y)\). So, let \(\vartheta \in C^{1}(0,\infty)\) and assume that

for some \(1\leqslant\tau< p\), \(c_{1},c_{2}>0\).

The hypotheses on the map \(a(y)\) are the following:

\(H(a)\)::

\(a(y)=a_{0}(|y|)y\) for all \(y\in\mathbb{R}^{N}\) with \(a_{0}(t)>0\) for all \(t>0\) and

1. (i)

   \(a_{0}\in C^{1}(0,\infty)\), \(t\longmapsto a_{0}(t)t\) is strictly increasing on \((0,\infty)\) and

   $$\lim_{t\to0^{+}}a_{0}(t)t=0\quad\mbox{and}\quad \lim _{t\to0^{+}}\frac{a_{0}'(t)t}{a_{0}(t)}=c>-1; $$
2. (ii)

   there exists \(c_{3}>0\) such that

   $$\bigl\vert \nabla a(y) \bigr\vert \leqslant c_{3} \frac{\vartheta ( \vert y \vert )}{ \vert y \vert }\quad\forall y\in\mathbb{R} ^{N}\setminus\{0\}; $$
3. (iii)

   we have

   $$\bigl(\nabla a(y)\xi,\xi\bigr)_{\mathbb{R}^{N}}\geqslant\frac{\vartheta ( \vert y \vert )}{ \vert y \vert } \vert \xi \vert ^{2}\quad\forall y\in\mathbb{R}^{N}\setminus\{0 \}, \xi\in\mathbb{R}^{N}; $$
4. (iv)

   if \(G_{0}(t)=\int_{0}^{t} a_{0}(s)s\,ds\), then there exists \(q\in (1,p)\) such that

   $$t \longmapsto G_{0}\bigl(t^{\frac{1}{q}}\bigr)\quad \textrm{is convex on }\mathbb{R}_{+}=[0,+\infty), $$
   $$\lim_{t\to0^{+}}\frac{q G_{0}(t)}{t^{q}}=c^{*}>0 $$

   and

   $$0 \leqslant p G_{0}(t)-a_{0}(t)t^{2}\quad\forall t>0. $$

Remark 2.1

Hypotheses \(H(a)\)(i), (ii) and (iii) are dictated by the nonlinear regularity theory of Lieberman [3] and the nonlinear strong maximum principle of Pucci and Serrin [14]. Hypothesis \(H(a)\)(iv) serves the needs of our problem. The examples given below show that hypothesis \(H(f)\)(iv) is mild and it is satisfied in all cases of interest. Note that hypotheses \(H(a)\) imply that \(G_{0}\) is strictly increasing and strictly convex. We set

We have

So, \(G(\cdot)\) is the primitive of \(a(\cdot)\) and \(y\longmapsto G(y)\) is convex with \(G(0)=0\). Hence

Such hypotheses were also used in the works of Gasiński et al. [15] and Papageorgiou and Rădulescu [16–18].

The next lemma is an easy consequence of hypotheses \(H(a)\) which summarizes the basic properties of the map a.

Lemma 2.2

If hypotheses \(H(a)\)(i), (ii) and (iii) hold, then

1. (a)

   \(y\longmapsto a(y)\) is continuous and strictly monotone (hence maximal monotone, too);

2. (b)

   \(|a(y)|\leqslant c_{4}(1+|y|^{p-1})\) for all \(y\in\mathbb{R}^{N}\), for some \(c_{4}>0\);

3. (c)

   \((a(y),y)_{\mathbb{R}^{N}}\geqslant\frac{c_{1}}{p-1}|y|^{p}\) for all \(y\in\mathbb{R}^{N}\).

Using this lemma together with (2.1) and (2.2), we have the following bilateral growth restrictions on the primitive G.

Corollary 2.3

If hypotheses \(H(a)\)(i), (ii) and (iii) hold, then

for some \(c_{5}>0\).

Example 2.4

The following maps a satisfy hypotheses \(H(a)\) (see Papageorgiou and Rădulescu [16]).

1. (a)

   \(a(y)=|y|^{p-2}y\) with \(1< p<\infty\);

   The map corresponds to the p-Laplace differential operator

   $$\Delta_{p} u=\operatorname{div}\bigl(|Du|^{p-2}Du\bigr)\quad \forall u\in W^{1,p}(\Omega). $$
2. (b)

   \(a(y)=|y|^{p-2}y+|y|^{q-2}y\) with \(1< q< p<\infty\).

   This map corresponds to the \((p,q)\)-Laplace differential operator

   $$\Delta_{p} u+\Delta_{q} u\quad\forall u\in W^{1,p}( \Omega). $$

   Such operators arise in problems of mathematical physics (see Cherfils and Il’yasov [19]).

3. (c)

   \(a(y)=(1+|y|^{2})^{\frac{p-2}{2}}y\) with \(1< p<\infty\).

   This operator corresponds to the generalized p-mean curvature differential operator

   $$\operatorname{div}\bigl(\bigl(1+|Du|^{2}\bigr)^{\frac{p-2}{2}}Du\bigr) \quad\forall u\in W^{1,p}(\Omega). $$
4. (d)

   \(a(y)=|y|^{p-2}y (1+\frac{1}{1+|y|^{2}} )\) with \(1< p<\infty\).

In what follows, by \(\langle\cdot,\cdot\rangle\) we denote the duality brackets for the dual pair \((W^{1,p}(\Omega)^{*}, W^{1,p}(\Omega))\). Let \(A\colon W^{1,p}(\Omega)\longrightarrow W^{1,p}(\Omega)^{*}\) be the nonlinear map defined by

The next proposition is a special case of a more general result of Gasiński and Papageorgiou [20].

Proposition 2.5

If hypotheses \(H(a)\)(i), (ii) and (iii) hold, then the map \(A\colon W^{1,p}(\Omega)\longrightarrow W^{1,p}(\Omega)^{*}\) is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone, too) and of type \((S)_{+}\), that is,

“if \(u_{n}\stackrel{w}{\longrightarrow}u\) in \(W^{1,p}(\Omega)\) and \(\limsup_{n\rightarrow+\infty}\langle A(u_{n}),u_{n}-u\rangle\leqslant 0\), then \(u_{n}\longrightarrow u\) in \(W^{1,p}(\Omega)\).”

The hypotheses on the boundary coefficient β are the following:

\(H(\beta)\)::

\(\beta\in C^{0,\alpha}(\partial\Omega )\) with \(\alpha\in(0,1)\) and \(\beta(z)\geqslant0\) for all \(z\in\partial\Omega \).

Remark 2.6

When \(\beta\equiv0\), we recover the Neumann problem.

Let \(\vartheta _{q}\colon W^{1,q}(\Omega)\longrightarrow \mathbb{R}\) be the \(C^{1}\)-functional defined by

Also, we consider the following nonlinear eigenvalue problem:

Here \(1< q<+\infty\) is as in hypothesis \(H(a)\)(iv) and \(\frac{\partial u}{\partial n_{q}}=|Du|^{q-2}(Du,n)_{\mathbb{R}^{N}}\). If the above Robin problem admits a nontrivial solution, then we say that λ̂ is an eigenvalue of \(-\Delta_{q}\) with Robin boundary condition and the nontrivial solution û is an eigenfunction corresponding to λ̂. From Papageorgiou and Rădulescu [17], we know that \(\widehat {u}\in L^{\infty}(\Omega)\), and then from Theorem 2 of Lieberman [2] (see also Lieberman [3]) we have that \(\widehat {u}\in C^{1}(\overline{\Omega})\).

From Papageorgiou and Rădulescu [21], we know that there exists a smallest eigenvalue \(\widehat {\lambda}_{1}(q)\) such that:

• \(\widehat {\lambda}_{1}(q)\geqslant0\) and it is isolated in the spectrum \(\widehat {\sigma}(q)\) (that is, we can find \(\varepsilon >0\) such that \((\widehat {\lambda}_{1}(q),\widehat {\lambda}_{1}(q)+\varepsilon )\cap\widehat {\sigma }(q)=\emptyset\)) and if \(\beta\equiv0\) (Neumann problem), then \(\widehat {\lambda}_{1}(q)=0\), while if \(\beta\not\equiv0\), then \(\widehat {\lambda}_{1}(q)>0\).

• \(\widehat {\lambda}_{1}(q)\) is simple (that is, if \(\widehat {u},\widehat {v}\) are eigenfunctions corresponding to \(\widehat {\lambda}_{1}(q)\), then \(\widehat {u}=\xi\widehat {v}\) for some \(\xi\in\mathbb {R}\setminus\{0\}\)).

• we have

  $$ \widehat {\lambda}_{1}(q)=\inf \biggl\{ \frac{\vartheta _{q}(u)}{ \Vert u \Vert _{q}^{q}}: u\in W^{1,q}(\Omega), u\ne0 \biggr\} . $$

  (2.3)

The infimum in (2.3) is realized on the one-dimensional eigenspace corresponding to \(\widehat {\lambda}_{1}(q)\). It follows that the elements of this eigenspace have constant sign. By \(\widehat {u}_{1}(q)\) we denote the \(L^{q}\)-normalized (that is, \(\Vert \widehat {u}_{1}(q) \Vert _{q}=1\)) positive eigenfunction corresponding to \(\widehat {\lambda}_{1}(q)\). We have \(\widehat {u}_{1}(q)\in C_{+}\) and, using the nonlinear strong maximum principle (see, e.g., Gasiński and Papageorgiou [1, p. 738]), we have \(\widehat {u}_{1}(q)\in D_{+}\). An eigenfunction û corresponding to an eigenvalue \(\widehat {\lambda }\ne\widehat {\lambda}_{1}(q)\) is necessarily nodal. Sometimes, in order to emphasize the dependence on β, we write \(\widehat {\lambda}_{1}(q,\beta)\geqslant0\).

Recall that a function \(f\colon\Omega\times\mathbb{R}\times\mathbb {R}^{N}\longrightarrow \mathbb{R} \) is Carathéodory, if

• for all \((x,y)\in\mathbb{R}\times\mathbb{R}^{N}\), \(z\longmapsto f(z,x,y)\) is measurable;

• for a.a. \(z\in\Omega\), \((x,y)\longmapsto f(z,x,y)\) is continuous.

Such a function is automatically jointly measurable (see Hu and Papageorgiou [22, p. 142]).

The hypotheses on the convection term f in problem (1.1) are the following:

\(H(f)\)::

\(f\colon\Omega\times\mathbb{R}\times\mathbb{R}^{N}\longrightarrow \mathbb{R}\) is a Carathéodory function such that \(f(z,0,y)=0\) for a.a. \(z\in\Omega\), all \(y\in\mathbb{R}^{N}\) and

1. (i)

   there exists \(\eta>0\) such that

   $$\begin{aligned} &f(z,\eta,y)=0\quad\textrm{for a.a. } z\in\Omega, \mbox{ all } y\in \mathbb{R}^{N}, \\ &f(z,x,y)\geqslant0\quad\textrm{for a.a. } z\in\Omega, \mbox{ all } 0\leqslant x \leqslant\eta,\mbox{ all } y\in\mathbb{R}^{N}, \\ &f(z,x,y)\leqslant\widetilde {c}_{1}+\widetilde {c}_{2} \vert y \vert ^{p}\quad\mbox{for a.a. } z\in\Omega, \mbox{ all } 0 \leqslant x\leqslant\eta,\mbox{ all } y\in\mathbb{R}^{N}, \end{aligned}$$

   with \(\widetilde {c}_{1}>0\), \(\widetilde {c}_{2}<\frac{c_{1}}{p-1}\);

2. (ii)

   for every \(M>0\), there exists \(\eta_{M}\in L^{\infty }(\Omega)\) such that

   $$\begin{aligned} &\eta_{M}(z)\geqslant c^{*}\widehat {\lambda}_{1}(q)\quad \mbox{for a.a. } z\in\Omega,\eta_{M}\not\equiv c^{*}\widehat { \lambda}_{1}(q), \\ &\liminf_{x\to0^{+}}\frac{f(z,x,y)}{x^{q-1}}\geqslant\eta_{M}(z) \quad \mbox{uniformly for a.a. } z\in\Omega, \mbox{ all } \vert y \vert \leqslant M \end{aligned}$$

   (here \(q\in(1,p)\) is as in hypothesis \(H(a)\)(iv));

3. (iii)

   there exists \(\xi_{\eta}>0\) such that, for a.a. \(z\in \Omega\), all \(y\in\mathbb{R}^{N}\), the function

   $$x\longmapsto f(z,x,y)+\xi_{\eta}x^{p-1} $$

   is nondecreasing on \([0,\eta]\), for a.a. \(z\in\Omega\), all \(y\in \mathbb{R} ^{N}\) and

   $$ \lambda^{p-1}f\biggl(z,\frac{1}{\lambda} x,y\biggr) \leqslant f(z,x,y) $$

   (2.4)

   and

   $$f(z,x,y)\leqslant\lambda^{p} f\biggl(z,x,\frac{1}{\lambda} y\biggr) $$

   for a.a. \(z\in\Omega\), all \(0\leqslant x\leqslant\eta\), all \(y\in \mathbb{R}^{N}\) and all \(\lambda\in(0,1)\).

Remark 2.7

Since we look for positive solutions and all the above hypotheses are for \(x\geqslant0\), without any loss of generality, we assume that

Note that (2.4) is satisfied if, for example, for a.a. \(z\in \Omega\), all \(y\in\mathbb{R}^{N}\), the function \(x\longmapsto \frac{f(z,x,y)}{x^{p-1}}\) is nonincreasing on \((0,+\infty)\).

Example 2.8

The following function satisfies hypotheses \(H(f)\). For the sake of simplicity, we drop the z-dependence:

with \(\eta>c^{*}\widehat {\lambda}_{1}(q)\geqslant0\), \(p<\min\{r,\mu\}\), \(c<\frac{c_{1}}{2(p-1)}\), \(1<\tau<\infty\).

As we have already mentioned, our approach is topological based on the Leray–Schauder alternative principle, which we recall here (see, e.g., Gasiński and Papageorgiou [1, p. 827]).

Theorem 2.9

If X is a Banach space, \(C\subseteq X\) is nonempty convex and \(\vartheta \colon C\longrightarrow C\) is a compact map, then exactly one of the following two statements is true:

1. (a)

   ϑ has a fixed point;

2. (b)

   the set \(S(\vartheta )=\{u\in C: u=\lambda \vartheta (u), \lambda\in(0,1)\}\) is unbounded.

Finally, let us fix our notation. For \(x\in\mathbb{R}\), we set \(x^{\pm}=\max\{\pm x,0\}\). Then, given \(u\in W^{1,p}(\Omega)\), we define \(u^{\pm}(\cdot)=u(\cdot)^{\pm}\). We know that

Also, if \(u\in W^{1,p}(\Omega)\), then

Consider the following truncation-perturbation of the convection term \(f(z,\cdot,y)\):

Evidently f̂ is a Carathéodoty function.

Given \(v\in C^{1}(\overline{\Omega})\), we consider the following auxiliary Robin problem:

Proposition 3.1

If hypotheses \(H(a)\), \(H(\beta)\) and \(H(f)\) hold, then problem (3.2) admits a positive solution \(u_{v}\in[0,\eta]\cap D_{+}\).

Proof

Let

and consider the \(C^{1}\)-functional \(\widehat {\varphi }_{v}\colon W^{1,p}(\Omega)\longrightarrow \mathbb {R}\) defined by

for all \(u\in W^{1,p}(\Omega)\). From (3.1), Corollary 2.3 and hypothesis \(H(\beta)\), we see that \(\widehat {\varphi }_{v}\) is coercive. Also, using the Sobolev embedding theorem, the compactness of the trace map and the convexity of G, we see that \(\widehat {\varphi }_{v}\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(u_{v}\in W^{1,p}(\Omega)\) such that

Let \(M> \Vert v \Vert _{C^{1}(\overline{\Omega})}\). Hypothesis \(H(f)\)(ii) implies that given \(\varepsilon >0\), we can find \(\delta\in(0,\eta]\) such that

so

(see (3.1)) and thus

Hypothesis \(H(a)\)(iv) implies that

Since \(\widehat {u}_{1}(q)\in D_{+}\), we can find \(t\in(0,1)\) small such that

Then we have

(recall that \(\Vert \widehat {u}_{1}(q) \Vert _{q}=1\)). Using hypothesis \(H(f)\)(ii) and the fact that \(\widehat {u}_{1}(q)\in D_{+}\), we have

Then from (3.7) we have

Choosing \(\varepsilon \in(0,\frac{r_{0}}{\widehat {\lambda}_{1}(q)})\), we see that

so

thus

From (3.3) we have

so

In (3.8) we choose \(h=-u_{v}^{-}\in W^{1,p}(\Omega)\). Using Lemma 2.2 and (3.1), we have

so

Next in (3.8) we choose \(h=(u_{v}-\eta)^{+}\in W^{1,p}(\Omega)\). Then

(see (3.1) and hypothesis \(H(f)\)(i)), so

(see hypothesis \(H(\beta)\) and note that \(A(\eta)=0\)), thus

So, we have proved that

Then, from (3.1), (3.8) and (3.9), we have

so

(see Papageorgiou and Rădulescu [21]). From (3.10) and Papageorgiou and Rădulescu [17], we have

Then from Lieberman [3] (see also Fukagai and Narukawa [23]), we have

Hypothesis \(H(f)\)(iii) implies that

Then from (3.10) we have

From (3.11), the strong maximum principle (see Pucci and Serrin [14, p. 111]) and the boundary point lemma (see Pucci and Serrin [14, p. 120]), we have \(u_{v}\in D_{+}\). □

Next we show that problem (3.2) has a smallest positive solution in the order interval \([0,\eta]\). So, let

From Proposition 3.1 we know that

Given \(\varepsilon >0\) and \(r\in(p,p^{*})\), where

(the critical Sobolev exponent corresponding to p), hypotheses \(H(f)\)(i) and (ii) imply that we can find \(c_{6}=c_{6}(\varepsilon ,r,M)>0\) (recall that \(M> \Vert v \Vert _{C^{1}(\overline{\Omega})}\)) such that

This unilateral growth restriction on \(f(z,\cdot,Dv(z))\) leads to the following auxiliary Robin problem:

Proposition 3.2

If hypotheses \(H(a)\) and \(H(\beta)\) hold, then for all \(\varepsilon >0\) small problem (3.13) admits a unique positive solution \(u^{*}\in D_{+}\).

Proof

First we show the existence of a positive solution for problem (3.13). To this end, let \(\psi\colon W^{1,p}(\Omega)\longrightarrow \mathbb{R}\) be the \(C^{1}\)-functional defined by

Using Corollary 2.3, we obtain

so

for some \(c_{7},c_{8}>0\). Since \(q< p\), it follows that ψ is coercive. Also, from the Sobolev embedding theorem, the compactness of the trace map and the convexity of G, we have that ψ is sequentially weakly lower semicontinuous. Invoking the Weierstrass–Tonelli theorem, we can find \(u^{*}\in W^{1,p}(\Omega)\) such that

As in the proof of Proposition 3.1, using the condition on \(\eta_{M}\) (see hypothesis \(H(f)\)(ii)), we show that, for \(t\in(0,1)\) and \(\varepsilon >0\) small, we have

so

(see (3.14)), thus

From (3.14) we have

so, for all \(h\in W^{1,p}(\Omega)\), we have

In (3.15) we choose \(h=-(u^{*})^{-}\in W^{1,p}(\Omega)\). Then

(see Lemma 2.2 and hypothesis \(H(\beta)\)), so

Hence (3.15) becomes

for all \(h\in W^{1,p}(\Omega)\), thus

(see Papageorgiou and Rădulescu [21]). As before, via the nonlinear regularity theory, we have

From (3.16) we have

(recall \(r>p\)), so

(see Pucci and Serrin [14, pp. 111, 120]).

Next we show that this positive solution is unique. For this purpose, we introduce the integral functional \(j\colon L^{1}(\Omega)\longrightarrow \overline {\mathbb{R}}=\mathbb {R}\cup\{+\infty\}\) defined by

Let \(\operatorname{dom}j=\{u\in L^{1}(\Omega): j(u)<+\infty\}\) (the effective domain of the functional j) and consider \(u_{1},u_{2}\in\operatorname{dom}j\). We set \(u=(1-t)u_{1}+tu_{2}\) with \(t\in[0,1]\). Using Lemma 1 of Díaz and Saá [24], we have

Recalling that \(G_{0}\) is increasing, we have

(see hypothesis \(H(a)\)(iv)), so

thus the map \(\operatorname{dom}j\ni u\longmapsto \int_{\Omega}G(Du^{\frac {1}{q}})\,dz\) is convex.

Since \(q< p\) and \(\beta\geqslant0\), it follows that the map \(\operatorname{dom}j\ni u\longmapsto \frac{1}{p}\int_{\partial \Omega }\beta(z)u^{\frac{p}{q}}\, d\sigma\) is convex.

Therefore the integral functional j is convex.

Suppose that \(\widetilde {u}^{*}\) is another positive solution of (3.13). As we did for \(u^{*}\), we can show that

Hence, given \(h\in C^{1}(\overline{\Omega})\) for \(|t|\) small, we have

Using the convexity of j, we can easily see that j is Gâteaux differentiable at \(u^{*}\) and at \(\widetilde {u}^{*}\) in the direction h. Using the chain rule and the nonlinear Green’s identity (see Gasiński and Papageorgiou [1, p. 210]), we have

and

The convexity of j implies the monotonicity of \(j'\). Therefore

(see (3.13)), so

(since \(q< p< r\)). This proves the uniqueness of the positive solution \(u^{*}\in D_{+}\). □

Proposition 3.3

If hypotheses \(H(a)\), \(H(\beta)\), \(H(f)\) hold and \(u\in S_{v}\), then \(u^{*}\leqslant u\).

Proof

We consider the Carathéodory function \(e\colon\Omega\times\mathbb {R}\longrightarrow \mathbb{R}\) defined by

We set

and consider the \(C^{1}\)-functional \(\tau\colon W^{1,p}(\Omega)\longrightarrow \mathbb{R}\) defined by

From (3.17) it is clear that τ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(\widetilde {u}^{*}\in W^{1,p}(\Omega)\) such that

As before, since \(q< p< r\), we have

so

From (3.18) we have

so

In (3.19) first we choose \(h=-(\widetilde {u}^{*})^{-}\in W^{1,p}(\Omega)\). Then

(see (3.17)), so

(see hypothesis \(H(\beta)\)).

Next in (3.19) we choose \(h=(\widetilde {u}^{*}-u)^{+}\in W^{1,p}(\Omega)\). Then

(see (3.17), (3.12) and recall that \(u\in S_{v}\)), so

(see hypothesis \(H(\beta)\)), thus

We have proved that

Then, from (3.17) and (3.20), equation (3.19) becomes

so \(\widetilde {u}^{*}=u^{*}\) (see Proposition 3.2), thus

 □

Using this proposition, we can show that problem (3.2) admits a smallest positive solution \(\widehat {u}_{v}\in D_{+}\) on \([0,\eta]\).

Proposition 3.4

If hypotheses \(H(a)\), \(H(\beta)\), \(H(f)\) hold, then problem (3.2) admits a smallest positive solution \(\widehat {u}_{v}\in D_{+}\).

Proof

Invoking Lemma 3.10 of Hu and Papageorgiou [22, p. 178], we can find a decreasing sequence \(\{u_{n}\}_{n\geqslant1}\subseteq S_{v}\) such that

For all \(n\geqslant1\), we have

so

Then, on account of hypotheses \(H(f)\)(i), \(H(\beta)\) and Lemma 2.2, we have that the sequence \(\{u_{n}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded. Passing to a subsequence, we may assume that

In (3.22) we choose \(h=u_{n}-\widehat {u}_{v}\in W^{1,p}(\Omega )\), pass to the limit as \(n\to\infty\) and use (3.24). Then

so

(see Proposition 2.5). If in (3.22) we pass to the limit as \(n\to+\infty\) and use (3.25), then

so \(u^{*}\leqslant\widehat {u}_{v}\) (see (3.23)).

From the above it follows that

 □

Let

and let \(\vartheta \colon C\longrightarrow C\) be the map defined by

A fixed point of this map is clearly a positive solution of problem (1.1). We will produce a fixed point for ϑ using the Leray–Schauder alternative principle (see Theorem 2.9). To this end, we will need the following lemma.

Lemma 3.5

If hypotheses \(H(a)\), \(H(\beta)\), \(H(f)\) hold, \(\{v_{n}\}_{n\geqslant1}\subseteq C\), \(v_{n}\to v\) in \(C^{1}(\overline{\Omega})\) and \(u\in S_{v}\), then we can find \(u_{n}\in S_{v_{n}}\) for \(n\geqslant1\) such that \(u_{n}\longrightarrow u\) in \(C^{1}(\overline{\Omega})\).

Proof

Consider the following nonlinear Robin problem:

Since \(u\in S_{v}\subseteq[0,\eta]\cap D_{+}\), we see that

(see (3.1)) and

(see hypothesis \(H(f)\)(i)). Therefore problem (3.26) has a unique nontrivial solution \(u_{n}^{0}\in D_{+}\). Also we have

(see (3.1), hypotheses \(H(f)\)(iii) and (i) and recall that \(u\in S_{v}\subseteq[0,\eta]\cap D_{+}\)), so

(see hypothesis \(H(\beta)\) and note that \(A(\eta)=0\)), thus

So, we have that

Moreover, the nonlinear regularity theory (see Lieberman [3]) and the nonlinear maximum principle (see Pucci and Serrin [14]) imply that

We have

Then \(\{u_{n}^{0}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded (see (3.27), (3.28), Lemma 2.2 and hypothesis \(H(f)\)(i)). So, on account of the nonlinear regularity theory of Lieberman [3], we can find \(\mu\in(0,1)\) and \(c_{9}>0\) such that

The compactness of the embedding \(C^{1,\mu}(\overline{\Omega })\subseteq C^{1}(\overline{\Omega})\) implies that we can find a subsequence \(\{u_{n_{k}}^{0}\}_{k\geqslant1}\) of the sequence \(\{u_{n}^{0}\}_{n\geqslant 1}\) such that

Note that

Since \(u\in S_{v}\) solves (3.29) which has a unique solution, we infer that

Hence, for the original sequence \(\{u_{n}^{0}\}_{n\geqslant1}\), we have

Next consider the following nonlinear Robin problem:

As above, we establish that this problem has a unique solution

Again we have

Continuing this way, we generate a sequence \(\{u_{n}^{k}\}_{k,n\geqslant1}\) such that

and

Fix \(n\geqslant1\). As before we have that the sequence \(\{u_{n}^{k}\}_{k\geqslant1}\subseteq C^{1}(\overline{\Omega })\) is relatively compact. So, we can find a subsequence \(\{u_{n}^{k_{m}}\}_{m\geqslant1}\) of the sequence \(\{u_{n}^{k}\}_{k\geqslant1}\) such that

so

(see (3.30)). Using the nonlinear regularity theory of Lieberman [3], (3.32) and the double limit lemma (see Aubin and Ekeland [25] and Gasiński and Papageorgiou [26, p. 61]), we have

so

and thus

 □

Using this lemma, we can show that the map \(\vartheta \colon C\longrightarrow C\) defined earlier is compact.

Proposition 3.6

If hypotheses \(H(a)\), \(H(\beta)\), \(H(f)\) hold, then the map \(\vartheta \colon C\longrightarrow C\) is compact.

Proof

First we show that ϑ is continuous.

So, suppose that \(v_{n}\longrightarrow v\) in \(C^{1}(\overline{\Omega})\), \(\{v_{n}\}_{n\geqslant 1}\subseteq C\), \(v\in C\), and let \(\widehat {u}_{n}=\vartheta (v_{n})\) for \(n\geqslant1\). We have

From (3.34) we see that \(\{\widehat {u}_{n}\}_{n\geqslant1}\subseteq W^{1,p}(\Omega)\) is bounded and so, according to Lieberman [3], we can find \(\tau\in(0,1)\) and \(c_{10}>0\) such that

So, we may assume that

In (3.34) we pass to the limit as \(n\to\infty\) and use (3.35). Then

From Proposition 3.3 we have

(in this case \(M>\sup_{n\geqslant1} \Vert v_{n} \Vert _{C^{1}(\overline {\Omega})}\)), so

(see (3.35)), thus

We claim that \(\widehat {u}=\vartheta (v)\). According to Lemma 3.5, we can find \(u_{n}\in S_{v_{n}}\), \(n\geqslant1\), such that

We have

so

(see (3.35) and (3.38)), thus

(see (3.37)), and hence ϑ is continuous.

Next we show that ϑ maps bounded sets in C to relatively compact subsets of C. So, let \(B\subseteq C\) be bounded in \(C^{1}(\overline{\Omega})\). As above, we have that the set \(\vartheta (B)\subseteq W^{1,p}(\Omega)\) is bounded. But then the nonlinear regularity theory of Lieberman [3] and the compactness of the embedding \(C^{1,s}(\overline{\Omega})\subseteq C^{1}(\overline{\Omega})\) (with \(0< s<1\)) imply that the set \(\vartheta (B)\subseteq C^{1}(\overline{\Omega})\) is relatively compact, thus ϑ is compact. □

Now we are ready for the existence theorem.

Theorem 3.7

If hypotheses \(H(a)\), \(H(\beta)\), \(H(f)\) hold, then problem (1.1) admits a solution \(\widehat {u}\in[0,\eta ]\cap D_{+}\).

Proof

We consider the set

If \(u\in S(\vartheta )\), then

so

In (3.39) we choose \(h=\frac{u}{\lambda}\in W^{1,p}(\Omega)\). Using Lemma 2.2 and hypothesis \(H(\beta)\), we have

(see (2.4), hypotheses \(H(f)\)(iii) and (i)). Recalling that \(\widetilde {c}_{2}<\frac{\widetilde {c}_{1}}{p-1}\) (see hypothesis \(H(f)\)(i)), we have

for some \(c_{11}>0\), thus

As above, from (3.39) with \(h=\frac{u}{\lambda} \in W^{1,p}(\Omega)\), using hypotheses \(H(f)\)(i), (iii) and (3.40), we obtain

for some \(c_{12}>0\), so

where \(\widehat {\beta}=\frac{p-1}{c_{1}}\beta\) (see (2.3)), thus

hence

(see (3.40)). From (3.39) we have

Hypothesis \(H(f)\)(iii) implies that

Then, from (3.41), (3.42), (3.43) and the nonlinear regularity theory of Lieberman [3], we have

for some \(c_{13}>0\), thus \(S(\vartheta )\subseteq C^{1}(\overline{\Omega})\) is bounded.

Since ϑ is compact (see Proposition 3.6), we can use the Leray–Schauder alternative theorem (see Theorem 2.9) and find \(\widehat {u}\in C\) such that

so \(\widehat {u}\in[0,\eta]\cap D_{+}\) is a solution of (1.1). □

This is the first work producing positive smooth solutions for problems driven by a nonhomogeneous differential operator with Robin boundary condition where the forcing term has the form of a convection term, that is, it depends also on the gradient of the unknown function. In addition, in contrast to the previous works in the field, we do not impose any global growth condition on the convection term. Our formulation incorporates \((p,q)\)-equations which are important in physical applications.

Acknowledgements

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Funding

Leszek Gasiński was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169.

Affiliations

1. Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

   • Yunru Bai
   •  & Leszek Gasiński

2. Department of Mathematics, National Technical University, Athens, Greece

   • Nikolaos S. Papageorgiou

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Leszek Gasiński.

Competing interests

The authors declare that they have no competing interests.

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