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Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms

Abstract

We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation \(f(z,x,y)\) which is only locally defined in \(x \in {\mathbb {R}} \). Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.

Introduction

Let \(\Omega \subseteq \mathbb {R}^{N}\) be a bounded domain with \(C^{2}\)-boundary Ω. In this paper we study the following nonhomogeneous parametric singular Dirichlet problem with gradient dependence (convection):

figurea

In this problem, the map \(a : \mathbb {R}^{N} \mapsto \mathbb {R}^{N}\) in the differential operator is continuous and strictly monotone (thus, maximal monotone too) and satisfies certain other growth and regularity conditions listed in hypotheses \({\mathrm{H}_{0}}\) below (see Sect. 2). These conditions are general and provide a broad framework in which we can fit many differential operators of interest, such as the p-Laplacian and the \((p,q)\)-Laplacian (that is, the sum of a p-Laplacian and of q-Laplacian). In the reaction (right-hand side) of (\(P_{\lambda }\)), we have the competing effects of a parametric singular term \(u \mapsto \lambda u^{-\eta }\) with \(\lambda >0\) being the parameter and of a perturbation \(u \mapsto f(z,u,Du)\) which is a Carathéodory function (that is, for all \(x \in \mathbb {R}\), \(y \in \mathbb {R}^{N}\) the function \(z \mapsto f(z,x,y)\) is measurable and for a.a. \(z \in \Omega \), \((x,y) \mapsto f(z,x,y)\) is continuous). So, this perturbation is gradient dependent and on \(f(z,\cdot ,y)\) we do not impose any growth condition. Instead we assume that near zero the function \(x \mapsto f(z,x,y)\) exhibits a kind of oscillatory behavior.

The gradient dependence of the reaction means that problem (\(P_{\lambda }\)) is not variational and so eventually our method of proof is going to be topological. We use the so-called “frozen variable method”. According to this approach, in the perturbation \(f(z,x,y)\), we fix (“freeze”) the y-variable. This leads to a variational problem, which a priori can be solved using tools from the critical point theory. However, the presence of the singular term leads to an energy functional which is not \(C^{1}\) and so we have difficulty in applying the minimax theorems of critical point theory. So, we need to find a way to bypass the singularity and deal with \(C^{1}\)-functionals. This is done by considering the purely singular problem (that is, \(f \equiv 0\)), which we show that for every \(\lambda > 0\) it has a unique positive smooth solution which converges to zero in \(C^{1}_{0}(\overline{\Omega })\) as \(\lambda \rightarrow 0^{+}\). We use this solution and its properties and truncation techniques, to show that, for all small values of the parameter \(\lambda > 0\), the “frozen problem” has at least one positive smooth solution. In order to use topological tools (fixed point theory), we need to find a canonical way to choose such a positive solution. This is done by showing that the frozen problem has a smallest positive solution (minimal positive solution). Then we show that the minimal solution map satisfies all the requirements of Leray–Schauder alternative principle (see Sect. 2) and so we can produce a positive solution for (\(P_{\lambda }\)) when \(\lambda > 0\) is small.

Recently there have been published some existence results for nonlinear problem with convection. We mention the work of Bai [1], Bai–Gasinski–Papageorgiou [2], Candito–Gasinski–Papageorgiou [4], Faraci–Motreanu–Puglisi [5], Gasinski–Krech–Papageorgiou [6], Gasinski–Papageorgiou [10], Gasinski–Winkert [11], Liu–Papageorgiou [18], Papageorgiou–Rădulescu–Repovš [20], Papageorgiou–Vetro–Vetro [24], Tanaka [31]. However, none of the aforementioned works involves singular terms. The only work examining the combined effects of singular and convection terms, is the recent paper of Papageorgiou–Rădulescu–Repovš [22], which deals with a nonparametric Neumann problem driven by the p-Laplacian. Nonlinear singular Dirichlet problems were also investigated in the paper of Papageorgiou–Winkert [25] for different settings and conditions.

Mathematical background—hypotheses

The main spaces in the analysis of our problem (\(P_{\lambda }\)), are the Sobolev space \(W^{1,p}_{0}(\Omega )\) and the Banach space \(C^{1}_{0}(\overline{\Omega })= \{ u \in C^{1}(\overline{\Omega }) : u\vert _{\partial \Omega }=0 \} \). By \(\Vert \cdot \Vert \) we denote the norm of the Sobolev space \(W^{1,p}_{0}(\Omega )\). On account of the Poincaré inequality, we have

$$ \Vert u \Vert = \Vert Du \Vert _{p} \quad \text{for all } u \in W^{1,p}_{0}(\Omega ). $$

The Banach space \(C^{1}_{0}(\overline{\Omega })\) is an ordered Banach space with positive (order) cone \(C_{+}= \{ u\in C^{1}_{0}(\overline{\Omega }) : u(z) \geqslant 0 \text{ for all } z \in \overline{\Omega } \} \). This cone has a nonempty interior given by

$$ \operatorname{int} C_{+}= \biggl\{ u \in C_{+} : u(z) > 0 \text{ for all } z \in \Omega , \frac{\partial u}{\partial n}\bigg| _{\partial \Omega } < 0 \biggr\} , $$

with \(n(\cdot )\) being the outward unit normal on Ω.

Let X be a Banach space and \(\xi : X \mapsto X\). We say that \(\xi (\cdot )\) is “compact” if it is continuous and for every \(B \subseteq X\) bounded, the set \(\overline{\xi (B)} \subseteq X\) is compact. The “Leray–Schauder alternative principle” asserts the following.

Theorem 1

If X is a Banach space, \(\xi : X \mapsto X\)is compact and

$$ \mathcal{D}(\xi )= \bigl\{ u \in X : u=t\xi (u) \textit{ for some } 0< t< 1 \bigr\} , $$

then the following alternative holds:

  1. (a)

    \(\mathcal{D}(\xi )\)is unbounded; or

  2. (b)

    \(\xi (\cdot )\)has a fixed point.

Let \(\beta \in C^{1}(0,\infty )\) with \(\beta (t) > 0\) for all \(t > 0\) and assume that

$$\begin{aligned} & 0 < \hat{c} \leqslant \frac{\beta '(t)t}{\beta (t)} \leqslant c_{0}, \\ & c_{1} t^{p-1} \leqslant \beta (t) \leqslant c_{2} \bigl(t^{s-1} +t^{p-1} \bigr), \end{aligned}$$

for all \(t > 0\), some \(1 \leqslant s < p < N\), \(0 < c_{1} < c_{2}\).

Then our hypotheses on the map \(a(\cdot )\) are the following.

\({ \mathrm{H}_{0}}\)::

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

  1. (i)

    \(a_{0} \in C^{1}(0,\infty )\), \(t \mapsto a_{0}(t) t\) is strictly increasing on \((0,\infty )\), \(a_{0}(t) t \rightarrow 0^{+}\) as \(t \rightarrow 0^{+}\) and \(\lim_{t \rightarrow 0^{+}} \frac{a_{0}'(t)t}{a_{0}(t)} > -1\);

  2. (ii)

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

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

    \(\frac{\beta (\vert y\vert )}{\vert y\vert } \vert \xi \vert ^{2} \leqslant (\nabla a(y) \xi , \xi )_{\mathbb {R}^{N}}\) for all \(y \in \mathbb {R}^{N} \setminus \{0\}\), all \(\xi \in \mathbb {R}^{N}\);

  4. (iv)

    if \(G_{0}(t)=\int ^{t}_{0} a_{0}(s)s \,ds\), \(t \geqslant 0\), then there exist \(q \in (1,p)\) and \(c_{*}>0\) such that

    $$ \limsup_{t \rightarrow 0^{+}} \frac{ qG_{0}(t)}{t^{q}} \leqslant c_{*}. $$

Remark 1

Such conditions on the differential operator were used in the context of singular or convection problems, also by Papageorgiou–Rădulescu–Repovš [23] and Candito–Gasinski–Papageorgiou [4]. Hypotheses \({\mathrm{H}_{0}(\mathrm{i}), (\mathrm{ii}), (\mathrm{iii})}\) are motivated by the nonlinear regularity theory of Lieberman [17] and the nonlinear maximum principle of Pucci–Serrin [28] (p. 111). Hypothesis \({\mathrm{H}_{0}(\mathrm{iv})}\) serves the needs of our problem and it is mild. As we will see in the examples listed below, it is satisfied in all cases of interest.

Clearly the above hypotheses imply that the primitive \(G_{0}(\cdot )\) is strictly convex and strictly increasing. Let \(G(y)=G_{0}(\vert y\vert )\) for all \(y \in \mathbb {R}^{N}\). Evidently \(G \in C^{1}(\mathbb {R}^{N},\mathbb {R})\), it is convex and we have

$$\begin{aligned} & \nabla G(y) = G_{0}'\bigl( \vert y \vert \bigr) \frac{y}{ \vert y \vert }=a_{0}\bigl( \vert y \vert \bigr) y=a(y) \quad \text{for all } y \in \mathbb {R}^{N} \setminus \{0\}, \\ & \nabla G(0)=0. \end{aligned}$$

So, \(G(\cdot )\) is the primitive of \(a(\cdot )\) and on account of the convexity of \(G(\cdot )\), we have

$$\begin{aligned} G(y) \leqslant \bigl(a(y),y \bigr)_{\mathbb {R}^{N}} \quad \text{for all } y \in \mathbb {R}^{N}. \end{aligned}$$
(1)

Hypotheses \({\mathrm{H}_{0}}\) lead easily to the following properties of the map \(a(\cdot )\).

Lemma 2

The mapping \(a(\cdot )\)is continuous, strictly monotone (hence maximal monotone too) and

  1. (a)

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

  2. (b)

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

Using this lemma and (1) we are led to the following growth restrictions on \(G(\cdot )\).

Corollary 3

We have \(\frac{c_{1}}{p-1} \vert y\vert ^{p} \leqslant G(y) \leqslant c_{5} ( 1 + \vert y\vert ^{p} )\)for some \(c_{5} > 0\), all \(y \in \mathbb {R}^{N}\).

Examples

The following maps \(a(\cdot )\) satisfy hypotheses \({\mathrm{H}_{0}}\) (see Papageorgiou–Rădulescu [19]):

  1. (i)

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

This map corresponds to the p-Laplace differential operator defined by

$$ \Delta _{p} u = \operatorname{div} \bigl( \vert Du \vert ^{p-2}Du \bigr) \quad \text{for all } u \in W^{1,p}_{0}( \Omega ). $$
  1. (ii)

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

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

$$ \Delta _{p} u+ \Delta _{q} u \quad \text{for all } u \in W^{1,p}_{0}( \Omega ). $$

Such operators arise often in the mathematical models of physical processes and recently there have been published several works dealing with equations driven by such operators. We mention the works of Bobkov–Tanaka [3], Papageorgiou–Zhang [26, 27], Rădulescu [29], Ragusa–Tachikawa [30].

  1. (iii)

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

This map corresponds to the extended capillary differential operator defined by

$$ \operatorname{div} \bigl( \bigl(1 + \vert Du \vert ^{2} \bigr)^{\frac{p-2}{2}} Du \bigr) \quad \text{for all } u \in W^{1,p}_{0}( \Omega ). $$

Let \(A : W^{1,p}_{0}(\Omega )\mapsto W^{-1,p'}(\Omega )={W^{1,p}_{0}( \Omega )}^{*}\) (\(\frac{1}{p} + \frac{1}{p'}=1\)) be the nonlinear operator defined by

$$ \bigl\langle A(u), h\bigr\rangle = \int _{\Omega } \bigl(a(Du),Dh \bigr)_{ \mathbb {R}^{N}} \,dz \quad \text{for all } u, h \in W^{1,p}_{0}(\Omega ). $$

From Gasinski–Papageorgiou [9] (Problem 2.192, p. 279), we have the following properties for this operator.

Proposition 4

The operator \(A(\cdot )\)is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type \((S)_{+}\), that is,

$$\begin{aligned}& \textit{``}u_{n} \xrightarrow{w} u \quad \textit{in } W^{1,p}_{0}(\Omega )\quad \textit{and}\quad \limsup _{n \rightarrow \infty } \bigl\langle A(u_{n}), u_{n}-u \bigr\rangle \leqslant 0 \\& \quad \Rightarrow \quad u_{n} \rightarrow u \quad \textit{in } W^{1,p}_{0}( \Omega ).\textit{''} \end{aligned}$$

In the sequel by \(\hat{\lambda }_{1}(r)\) we denote the first (principal) eigenvalue of \((-\Delta _{r}, W^{1,r}_{0}(\Omega ) )\) (\(1 < r < \infty \)). We know that

$$\begin{aligned} &\bullet \quad \hat{\lambda }_{1}(r) > 0, \quad\quad \text{it is simple and isolated}. \\ &\bullet \quad \hat{\lambda }_{1}(r) = \inf \biggl\{ \frac{ \Vert Du \Vert _{r}^{r}}{ \Vert u \Vert _{r}^{r}} : u \in W^{1,r}_{0}(\Omega ), u \neq 0 \biggr\} . \end{aligned}$$
(2)

In (2) the infimum is realized on the corresponding one-dimensional eigenspace the elements of which do not change sign. By \(\hat{u}_{1}(r)\) we denote the \(L^{r}\)-normalized (that is, \(\Vert \hat{u}_{1}(r)\Vert _{r}=1\)), positive eigenfunction corresponding to \(\hat{\lambda }_{1}(r)\). The nonlinear regularity theory and the nonlinear maximum principle (see Gasinski–Papageorgiou [8], pp. 738–739), imply \(\hat{u}_{1}(r) \in \operatorname{int} C_{+}\). So, we have \(\Vert D{\hat{u}_{1}(r)}\Vert _{r}^{r}=\hat{\lambda }_{1}(r)\). We mention that every eigenvalue λ̂ of \((-\Delta _{r}, W^{1,r}_{0}(\Omega ) )\) distinct from \(\hat{\lambda }_{1}(r)\) has eigenfunctions which are nodal (sign-changing).

Given \(x \in \mathbb {R}\), we set \(x^{\pm }=\max \{ \pm x, 0 \} \). Then for \(u \in W^{1,p}_{0}(\Omega )\) we define \(u^{\pm }(z) = {u(z)}^{\pm }\) for all \(z \in \Omega \). We know that

$$ u^{\pm }\in W^{1,p}_{0}(\Omega ), \quad\quad u=u^{+}-u^{-}, \quad\quad \vert u \vert =u^{+}+u^{-}. $$

If u, \(v : \Omega \mapsto \mathbb {R}\) are measurable functions and \(u \leqslant v\), then we define

$$ [u,v]= \bigl\{ h \in W^{1,p}_{0}(\Omega ): u(z) \leqslant h(z) \leqslant v(z) \text{ for a.a. } z \in \Omega \bigr\} . $$

A set \(S \subseteq W^{1,p}_{0}(\Omega )\) is said to be the downward directed, if for every u, \(v \in S\), we can find \(y \in S\) such that \(y \leqslant u\) and \(y \leqslant v\).

Now we introduce the hypotheses on perturbation \(f(z,x,y)\).

\({\mathrm{H}_{1}}\)::

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

  1. (i)

    there exist \(0 < \delta _{0} <\theta \) and \(c_{6} > 0\) such that

    $$\begin{aligned} & \bigl\vert f(z,x,y) \bigr\vert \leqslant c_{6} \bigl(1 + \vert y \vert ^{p-1} \bigr) \quad \text{for a.a. } z \in \Omega , \text{all } 0 \leqslant x \leqslant \theta , \text{all } y \in \mathbb {R}^{N}, \\ & \theta ^{-\eta } + f(z,\theta ,y) \leqslant 0, \quad \text{for a.a. } z \in \Omega , \text{all } y \in \mathbb {R}^{N}, \\ & f(z,x,y) \geqslant 0 \quad \text{for a.a. } z \in \Omega , \text{all } 0 \leqslant x \leqslant \delta _{0}, \text{all } y \in \mathbb {R}^{N}; \end{aligned}$$
  2. (ii)

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

    $$\begin{aligned} & \eta _{M}(z) \geqslant \frac{c_{*}}{q} \hat{\lambda }_{1}(q)\quad \text{for a.a. } z \in \Omega ,\quad\quad \eta _{M} \not \equiv \frac{c_{*}}{q} \hat{\lambda }_{1}(q), \\ & \liminf_{x \rightarrow 0^{+}} \frac{f(z,x,y)}{x^{q-1}} \geqslant \eta _{M}(z) \quad \text{uniformly for a.a. } z \in \Omega , \text{all } \vert y \vert \leqslant M; \end{aligned}$$
  3. (iii)

    for a.a. \(z \in \Omega \), all \(0 \leqslant x \leqslant \theta \), all \(y \in \mathbb {R}^{N}\) and all \(0 < t < 1\), we have

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

Remark 2

The above hypotheses concern only the behavior of \(f(z,\cdot,y)\) near zero. No global growth condition is imposed on \(f(z,\cdot ,y)\). Hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\) dictates an oscillatory behavior for \(f(z,\cdot ,y)\). It starts positive and by the time we have reached \(x = \theta \), the perturbation \(f(z,\cdot ,y)\) has become negative. In the case of the equation driven by the q-Laplacian, hypothesis \({\mathrm{H}_{1}(\mathrm{ii})}\) is a nonuniform nonresonance condition at zero. Hypothesis \({\mathrm{H}_{1}(\mathrm{iii})}\) is satisfied if for a.a. \(z \in \Omega \) and all \(y \in \mathbb {R}^{N}\) the quotient function \(x \mapsto \frac{f(z,x,y)}{x^{p-1}}\) is nonincreasing on \((0,\theta ]\).

Example

The following function satisfies hypotheses \({\mathrm{H}_{1}}\) above. For the sake of simplicity, we drop the z-dependence. We have

$$ f(x,y ) = \eta \bigl(x^{p-1}- 2x^{r-1} \bigr) \bigl(1+ \vert y \vert ^{p-1} \bigr) \quad \text{for all } x \geqslant 0, \text{all } y \in \mathbb {R}^{N}, $$

with \(\eta > \max \{ \frac{c_{*}}{q}\hat{\lambda }_{1}(q),1 \} \), \(1< q < r\). For \(\theta =1\) hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\) is satisfied.

As we already mentioned in the Introduction, due to the presence of the singular term, we have an energy functional which is not \(C^{1}\) and this prevents us from using the tools of critical point theory. So, we need to find a way to bypass the singularity and deal with \(C^{1}\)-functionals. For this reason in the next section we deal with the purely singular problem \((f \equiv 0)\). A solution of this problem will help us isolate the singularity.

Purely singular problem

In this section we examine the following purely singular problem:

figureb

Proposition 5

If hypotheses \({\mathrm{H}_{0}}\)hold, then for every \(\lambda > 0\)problem (\(Q_{\lambda }\)) admits a unique solution \(\overline{u} \in \operatorname{int} C_{+}\)and \(\overline{u}_{\lambda }\rightarrow 0\)in \(C^{1}_{0}(\overline{\Omega })\)as \(\lambda \rightarrow 0^{+}\).

Proof

Let \(g \in L^{p}(\Omega )\) and \(\varepsilon > 0\). We consider the following auxiliary Dirichlet problem:

figurec

Recall that \(A : W^{1,p}_{0}(\Omega )\mapsto W^{-1,p'}(\Omega )\) is the operator defined by

$$ \bigl\langle A(u), h\bigr\rangle = \int _{\Omega } \bigl(a(Du)\bigr), Dh )_{ \mathbb {R}^{N}} \,dz \quad \text{for all } u, h \in W^{1,p}_{0}(\Omega ). $$

On account of Proposition 4, the operator \(A(\cdot )\) is continuous and maximal monotone. In addition, from Lemma 2(b) we see that \(A(\cdot )\) is coercive. Therefore \(A(\cdot )\) is surjective (see Papageorgiou–Rădulescu–Repovš [21], Corollary 2.8.7, p. 135). So, we can find \(\hat{u}_{\varepsilon }\in W^{1,p}_{0}(\Omega )\) such that

$$ A(\hat{u}_{\varepsilon })= \frac{\lambda }{ ( \vert g(z) \vert + \varepsilon )^{\eta }}. $$

Moreover, the strict monotonicity of \(A(\cdot )\) implies that this solution is unique. Note that \(\frac{\lambda }{ (\vert g(\cdot )\vert + \varepsilon )^{\eta }} \in {L^{\infty }(\Omega )}_{+}\). So, the nonlinear regularity theory of Lieberman [17] and the nonlinear maximum principle of Pucci–Serrin [28] (p. 111, 120) imply that \(\hat{u}_{\varepsilon }\in \operatorname{int} C_{+}\).

We consider the solution map \(\gamma _{\varepsilon }: L^{p}(\Omega ) \mapsto L^{p}(\Omega )\) defined by

$$ \gamma _{\varepsilon }(g) = \hat{u}_{\varepsilon }. $$

Consider the following perturbation of problem (\(Q_{\lambda }\)):

figured

Evidently a fixed point of \(\gamma _{\varepsilon }(\cdot )\) is a solution of (\(Q_{\lambda }^{\varepsilon }\)). Clearly \(\gamma _{\varepsilon }(\cdot )\) is continuous. Moreover, using Lemma 2(b) we have

$$\begin{aligned} & \frac{c_{1}}{p-1} \Vert D{\hat{u}_{\varepsilon }} \Vert _{p}^{p} \leqslant \int _{\Omega }\frac{\lambda }{ ( \vert g(z) \vert + \varepsilon )^{\eta }} \hat{u}_{\varepsilon }\,dz \leqslant \frac{\lambda }{\varepsilon ^{\eta }} c_{7} \Vert \hat{u}_{\varepsilon } \Vert \quad \text{for some } c_{7} > 0 \\ &\quad\Rightarrow\quad \gamma _{\varepsilon }\bigl(L^{p}(\Omega ) \bigr) \subseteq W^{1,p}_{0}( \Omega )\text{ is bounded}. \end{aligned}$$

The fact that \(W^{1,p}_{0}(\Omega )\hookrightarrow L^{p}(\Omega )\) compactly implies that

$$ \gamma _{\varepsilon }\bigl(L^{p}(\Omega )\bigr)\subseteq L^{p}(\Omega ) \text{ is relatively compact}. $$

Then, by the Schauder–Tychonov fixed point theorem (see Papageorgiou–Rădulescu–Repovš [21], Theorem 4.3.21, p. 298), we can find \(\widetilde{u}_{\varepsilon }\in W^{1,p}_{0}(\Omega )\) such that

$$ \gamma _{\varepsilon }(\widetilde{u}_{\varepsilon }) = \widetilde{u}_{\varepsilon }. $$

This is the unique positive solution of (\(Q_{\lambda }^{\varepsilon }\)) and since \(\gamma _{\varepsilon }(\cdot )\) has values in \(\operatorname{int} C_{+}\), we infer that \(\widetilde{u}_{\varepsilon }\in \operatorname{int} C_{+}\).

Claim: \(\{\widetilde{u}_{\varepsilon }\}_{\varepsilon \in (0,1]} \subseteq \operatorname{int} C_{+}\) is nonincreasing.

Let \(0 < \varepsilon ' < \varepsilon \). We have

$$\begin{aligned} -\operatorname{div} a(D{\widetilde{u}_{\varepsilon '}}) = \frac{\lambda }{ (\widetilde{u}_{\varepsilon '} + \varepsilon ' )^{\eta }} \geqslant \frac{\lambda }{ (\widetilde{u}_{\varepsilon '} + \varepsilon )^{\eta }} \quad \text{in } \Omega . \end{aligned}$$
(3)

We introduce the Carathéodory function \(k_{\varepsilon }: \Omega \times \mathbb {R}\mapsto \mathbb {R}\) defined by

$$\begin{aligned} k_{\varepsilon }(z,x) = \textstyle\begin{cases} \frac{\lambda }{(x^{+} + \varepsilon )^{\eta }} &\text{if } x \leqslant \widetilde{u}_{\varepsilon '}(z), \\ \frac{\lambda }{ (\widetilde{u}_{\varepsilon '}(z)+ \varepsilon )^{\eta }} &\text{if } \widetilde{u}_{\varepsilon '}(z) < x. \end{cases}\displaystyle \end{aligned}$$
(4)

We set \(K_{\varepsilon }(z,x)=\int ^{x}_{0} k_{\varepsilon }(z,s) \,ds\) and consider the \(C^{1}\)-functional \(\psi _{\varepsilon }: W^{1,p}_{0}(\Omega )\mapsto \mathbb {R}\) defined by

$$ \psi _{\varepsilon }(u) = \int _{\Omega }G(Du) \,dz - \int _{\Omega }K_{\varepsilon }(z,u) \,dz \quad \text{for all } u \in W^{1,p}_{0}(\Omega ). $$

From Corollary 3 and (4), we see that the functional \(\psi _{\varepsilon }(\cdot )\) is coercive. Also, using the Sobolev embedding theorem, we see that \(\psi _{\varepsilon }(\cdot )\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(\widetilde{u}_{\varepsilon }^{*} \in W^{1,p}_{0}(\Omega )\) such that

$$\begin{aligned} \psi _{\varepsilon }\bigl(\widetilde{u}_{\varepsilon }^{*} \bigr) = \min \bigl\{ \psi _{\varepsilon }(u) : u \in W^{1,p}_{0}( \Omega ) \bigr\} . \end{aligned}$$
(5)

From hypothesis \({\mathrm{H}_{0}(\mathrm{iv})}\) and Corollary 3, we have

$$\begin{aligned} G(y) \leqslant c_{8} \bigl( \vert y \vert ^{q} + \vert y \vert ^{p} \bigr) \quad \text{for some } c_{8} > 0, \text{all } y \in \mathbb {R}^{N}. \end{aligned}$$
(6)

For \(t \in (0,1)\) we have

$$ \psi _{\varepsilon }(t \widetilde{u}_{\varepsilon '}) \leqslant \frac{t^{p}}{p} \Vert D{\widetilde{u}_{\varepsilon '}} \Vert _{p}^{p} + \frac{t^{q}}{q} \Vert D{ \widetilde{u}_{\varepsilon '}} \Vert _{q}^{q}- \frac{t^{1-\eta }}{1-\eta } \int _{\Omega }{ \widetilde{u}_{\varepsilon '}}^{1- \eta } \,dz \quad (\text{see (6)}). $$

Since \(0 < \eta <1 < q <p\), for \(t \in (0,1)\) small, we see that

$$\begin{aligned} & \psi _{\varepsilon }(t \widetilde{u}_{\varepsilon '}) < 0 \\ &\quad\Rightarrow\quad \psi _{\varepsilon }\bigl(\widetilde{u}_{\varepsilon }^{*} \bigr) < 0= \psi _{\varepsilon }(0)\quad (\text{see (5)}) \\ &\quad\Rightarrow\quad \widetilde{u}_{\varepsilon }^{*} \neq 0. \end{aligned}$$

From (5) we have

$$\begin{aligned} & \psi _{\varepsilon }\bigl(\widetilde{u}_{\varepsilon }^{*} \bigr)=0 \\ &\quad\Rightarrow\quad \bigl\langle A\bigl(\widetilde{u}_{\varepsilon }^{*} \bigr), h \bigr\rangle = \int _{\Omega }k_{\varepsilon }\bigl(z,\widetilde{u}_{\varepsilon }^{*} \bigr) h \,dz \quad \text{for all } h \in W^{1,p}_{0}(\Omega ). \end{aligned}$$
(7)

In (7) first we choose \(h=-(\widetilde{u}_{\varepsilon }^{*})^{-} \in W^{1,p}_{0}(\Omega )\). We have

$$\begin{aligned} & \frac{c_{1}}{p-1} \bigl\Vert D\bigl(\widetilde{u}_{\varepsilon }^{*} \bigr)^{-} \bigr\Vert _{p}^{p} \leqslant 0 \quad \bigl(\text{see Lemma 2(b) and (4)}\bigr) \\ &\quad\Rightarrow\quad \widetilde{u}_{\varepsilon }^{*} \geqslant 0, \quad\quad \widetilde{u}_{\varepsilon }^{*} \neq 0. \end{aligned}$$

Next we test (7) with \(h=(\widetilde{u}_{\varepsilon }^{*}-\widetilde{u}_{\varepsilon '})^{+} \in W^{1,p}_{0}(\Omega )\). We have

$$\begin{aligned}& \begin{aligned} \bigl\langle A\bigl(\widetilde{u}_{\varepsilon }^{*}\bigr), \bigl( \widetilde{u}_{\varepsilon }^{*}-\widetilde{u}_{\varepsilon '} \bigr)^{+}\bigr\rangle &= \int _{\Omega }\frac{\lambda }{ (\widetilde{u}_{\varepsilon '}+\varepsilon )^{\eta }}\bigl( \widetilde{u}_{\varepsilon }^{*}- \widetilde{u}_{\varepsilon '}\bigr)^{+} \,dz \quad (\text{see (4)}) \\ & \leqslant \bigl\langle A(\widetilde{u}_{\varepsilon '}), \bigl( \widetilde{u}_{\varepsilon }^{*}-\widetilde{u}_{\varepsilon '} \bigr)^{+}\bigr\rangle \quad (\text{see (3)}) \end{aligned} \\& \quad \Rightarrow \quad \widetilde{u}_{\varepsilon }^{*} \leqslant \widetilde{u}_{ \varepsilon '}. \end{aligned}$$

So, we have proved that

$$\begin{aligned} \widetilde{u}_{\varepsilon }^{*} \in [0, \widetilde{u}_{\varepsilon '}], \quad\quad \widetilde{u}_{\varepsilon }^{*} \neq 0. \end{aligned}$$
(8)

From (8), (4) and (7) it follows that

$$\begin{aligned} & \widetilde{u}_{\varepsilon }^{*}=\widetilde{u}_{\varepsilon } \\ &\quad\Rightarrow\quad \widetilde{u}_{\varepsilon }\leqslant \widetilde{u}_{ \varepsilon '}. \end{aligned}$$

This proves the claim.

Next, let \(\varepsilon _{n}= \frac{1}{n}\) and \(\widetilde{u}_{n}=\widetilde{u}_{\varepsilon _{n}} \in \operatorname{int} C_{+}\) for all \(n \in \mathbb {N}\). We have

$$\begin{aligned} & \bigl\langle A(\widetilde{u}_{n}),h\bigr\rangle = \int _{\Omega }\frac{\lambda }{ (\widetilde{u}_{n} + \frac{1}{n} )^{\eta }} h \,dz\quad \text{for all } h \in W^{1,p}_{0}(\Omega ), \text{all } n \in \mathbb {N}, \end{aligned}$$
(9)
$$\begin{aligned} & \widetilde{u}_{1} \leqslant \widetilde{u}_{n} \quad \text{for all } n \in \mathbb {N}\ (\text{by the claim}). \end{aligned}$$
(10)

Consider the Banach space \(C_{0}(\overline{\Omega })= \{ u \in C(\overline{\Omega }) : u\vert _{ \partial \Omega } =0 \} \) (with the supremum norm). This is an ordered Banach space with positive cone

$$ K_{+}= \bigl\{ u \in C_{0}(\overline{\Omega }) : u(z) \geqslant 0 \text{ for all } z \in \overline{\Omega } \bigr\} . $$

This cone has a nonempty interior given by

$$ \operatorname{int} K_{+} = \{ u \in C_{+} : c_{u} \hat{d} \leqslant u \text{ for some } c_{u} > 0 \} , $$

with \(\hat{d}(z) = d(z,\partial \Omega )\) for all \(z \in \overline{\Omega }\). Since \(\widetilde{u}_{1} \in \operatorname{int} C_{+}\), we can find \(0 < c_{9} < c_{10}\) such that

$$\begin{aligned} c_{9} \hat{d} \leqslant \widetilde{u}_{1} \leqslant c_{10} \hat{d} \quad (\text{see Guo--Webb [13], Lemma 2.3}). \end{aligned}$$
(11)

Let \(s > N\) and note that \({\hat{u}_{1}(p)}^{1/s} \in K_{+}\). Using Proposition 4.1.22, p. 274, of Papageorgiou–Rădulescu–Repovš [21], we can find \(c_{11} > 0\) such that

$$\begin{aligned} & {\hat{u}_{1}(p)}^{1/s} \leqslant c_{11} \hat{d} \\ &\quad\Rightarrow\quad {\hat{d}}^{-\eta } \leqslant c_{12} { \hat{u}_{1}(p)}^{- \eta /s} \quad \text{for some } c_{12} >0. \end{aligned}$$
(12)

Since \(0 < \eta < 1\), the Lemma in Lazer–McKenna [16] implies that \({\hat{u}_{1}(p)}^{-\eta /s} \in L^{s}(\Omega )\). Therefore

$$\begin{aligned} {\hat{d}}^{-\eta } \in L^{s}(\Omega )\quad ( \text{see (12)}). \end{aligned}$$
(13)

From (11) we have

$$\begin{aligned} & {\widetilde{u}_{1}}^{-\eta } \leqslant c_{13}{\hat{d}}^{-\eta } \quad \text{for some } c_{13}>0 \\ &\quad\Rightarrow\quad {\widetilde{u}_{1}}^{-\eta } \in L^{s}(\Omega ) \quad (\text{see (13)}). \end{aligned}$$
(14)

We return to (9) and use \(h=\widetilde{u}_{n} \in W^{1,p}_{0}(\Omega )\). Then from Lemma 2(b) we have

$$\begin{aligned} & \frac{c_{1}}{p-1} \Vert D{\widetilde{u}_{n}} \Vert _{p}^{p} \leqslant \int _{\Omega }\frac{\lambda }{ (\widetilde{u}_{n}+\frac{1}{n} )^{\eta }} \widetilde{u}_{n} \,dz \leqslant \lambda c_{14} \Vert \widetilde{u}_{n} \Vert ^{1- \eta } \quad \text{for some } c_{14} > 0, \text{all } n \in \mathbb {N}\\ &\quad\Rightarrow\quad \{ \widetilde{u}_{n} \} _{n \in \mathbb {N}} \subseteq W^{1,p}_{0}(\Omega )\text{ is bounded}. \end{aligned}$$

So, we may assume that

$$\begin{aligned} \widetilde{u}_{n} \xrightarrow{w} \overline{u}_{\lambda }\quad \text{in } W^{1,p}_{0}( \Omega )\quad \text{and} \quad \widetilde{u}_{n} \rightarrow \overline{u}_{\lambda }\quad \text{in } L^{p}(\Omega ). \end{aligned}$$
(15)

Note that

$$\begin{aligned} 0 \leqslant \frac{\lambda }{ (\widetilde{u}_{n} + \frac{1}{n} )^{\eta }} \leqslant \frac{\lambda }{{\widetilde{u}_{1}}^{\eta }} \in L^{s}(\Omega ) \quad (\text{see (14)}). \end{aligned}$$
(16)

On account of (15), we can also say (at least for a subsequence) that

$$\begin{aligned} \frac{\lambda }{ (\widetilde{u}_{n}(z) + \frac{1}{n} )^{\eta }} \rightarrow \frac{\lambda }{{\overline{u}_{\lambda }(z)}^{\eta }} \quad \text{for a.a. } z \in \Omega . \end{aligned}$$
(17)

From (16) and (17) it follows that

$$\begin{aligned} &\frac{\lambda }{ (\widetilde{u}_{n} + \frac{1}{n} )^{\eta }} \xrightarrow{w} \frac{\lambda }{{\overline{u}_{\lambda }}^{\eta }} \quad \text{in } L^{s}(\Omega ) \end{aligned}$$
(18)

(see Gasinski–Papageorgiou [9], Problem 1.19, p. 38).

Again we return to (9) and choose \(h=\widetilde{u}_{n}-\overline{u}_{\lambda }\in W^{1,p}_{0}(\Omega )\), pass to the limit as \(n \rightarrow \infty \) and use (15) and (18). We obtain

$$\begin{aligned} & \lim_{n \rightarrow \infty } \bigl\langle A(\widetilde{u}_{n}), \widetilde{u}_{n}-\overline{u}_{\lambda }\bigr\rangle =0 \\ &\quad\Rightarrow\quad \widetilde{u}_{n} \rightarrow \overline{u}_{\lambda }\text{ in } W^{1,p}_{0}(\Omega ) \quad (\text{see Proposition 4}). \end{aligned}$$
(19)

Therefore if in (9) we pass to the limit as \(n \rightarrow \infty \) and use (19), then

$$ \left \{ \textstyle\begin{array}{l} \langle A(\overline{u}_{\lambda }),h\rangle = \int _{\Omega }\frac{\lambda }{{\overline{u}_{\lambda }}^{\eta }} h \,dz \quad \text{for all } h \in W^{1,p}_{0}(\Omega ), \\ \widetilde{u}_{1} \leqslant \overline{u}_{\lambda } \quad (\text{see (10)}). \end{array}\displaystyle \right \} $$
(20)

From (20) we infer that

$$ \overline{u}_{\lambda }\text{ is a positive solution of ($Q_{\lambda }$)}. $$

Let \(\theta =\frac{\lambda }{{\overline{u}_{\lambda }}^{\eta }} \in L^{s}( \Omega )\) and consider the linear Dirichlet problem

$$ -\Delta y(z) = \theta (z) \quad \text{in } \Omega , y| _{\partial \Omega }=0. $$

From Theorem 9.15, p. 241, of Gilbarg–Trudinger [12], we know that this problem has a unique solution \(y \in W^{2,s}(\Omega )\), \(y \geqslant 0\), \(y \neq 0\). Since \(s > N\), by the Sobolev embedding theorem, we have \(W^{2,s}(\Omega ) \hookrightarrow C^{1,\alpha }(\overline{\Omega })\) continuously with \(\alpha =1-\frac{N}{s} \in (0,1)\). Let \(w=Dy \in C^{0,\alpha }(\overline{\Omega },\mathbb {R}^{N})\). We have

$$ -\operatorname{div} \bigl(a\bigl(D{\overline{u}_{\lambda }(z)}\bigr)-w(z)\bigr)=0, \quad \text{in } \Omega , \overline{u}_{\lambda }| _{\partial \Omega }=0. $$

Then the nonlinear regularity theory of Lieberman [17] implies that \(\overline{u}_{\lambda }\in C_{+}\setminus \{0\}\). Using the nonlinear maximum principle of Pucci–Serrin [28] (pp. 111, 120), we infer that \(\overline{u}_{\lambda }\in \operatorname{int} C_{+}\).

From (20) and (14) and Theorem 7.1, p. 286, of Ladyzhenskaya–Uraltseva [15] (see also Gasinski–Papageorgiou [8], p. 737), we know that there exists \(c_{15} > 0\) such that

$$ \Vert \overline{u}_{\lambda } \Vert _{\infty }\leqslant c_{15} \quad \text{for all } 0 < \lambda \leqslant 1. $$

Then the nonlinear regularity theory of Lieberman [17], says that we can find \(\alpha \in (0,1)\) and \(c_{16} > 0\) such that

$$\begin{aligned} \overline{u}_{\lambda }\in C^{1,\alpha }_{0}( \overline{\Omega })=C^{1, \alpha }(\overline{\Omega }) \cap C^{1}_{0}( \overline{\Omega }), \quad\quad \Vert \overline{u}_{\lambda } \Vert _{C^{1,\alpha }_{0}(\overline{\Omega })} \leqslant c_{16} \quad \text{for all } \lambda \in (0,1]. \end{aligned}$$
(21)

Since \(C^{1,\alpha }_{0}(\overline{\Omega }) \hookrightarrow C^{1}_{0}( \overline{\Omega })\) compactly, from (14), (20) and (21), we conclude that

$$ \overline{u}_{\lambda }\rightarrow 0 \quad \text{in } C^{1}_{0}( \overline{\Omega }) \text{ as } \lambda \rightarrow 0^{+}. $$

This proof is now complete. □

The frozen variable method

In this section we develop the method described in the Introduction (the frozen variable method).

So, fix \(v \in C^{1}_{0}(\overline{\Omega })\) and consider the following Dirichlet problem (the “frozen problem”):

figuree

Since we have fixed the gradient variable in f, the resulting problem (\(P_{\lambda }^{v}\)) has a variational structure. However, as we already mentioned in the Introduction, the presence of the singular term \(u \mapsto \lambda u^{-\eta }\) leads to an energy functional which is not \(C^{1}\) and so we cannot use the minimax theorems of the critical point theory. To remedy this situation, we use the solution \(\overline{u}_{\lambda }\) of (\(Q_{\lambda }\)) to bypass the singularity and deal with a \(C^{1}\)-functional.

Let \(M \geqslant \Vert Dv\Vert _{\infty }\). On account of hypotheses \({\mathrm{H}_{1}(\mathrm{ii})}\), given \(\varepsilon > 0\) small, we can find \(\delta \in (0,\delta _{0}]\) such that

$$\begin{aligned} f(z,x,y) \geqslant \bigl(\eta _{M}(z)-\varepsilon \bigr)x^{q-1} \quad \text{for a.a } z \in \Omega , \text{all } 0 \leqslant x \leqslant \delta , \text{all } \vert y \vert \leqslant M. \end{aligned}$$
(22)

Proposition 6

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold, then we can find \(\hat{\lambda }_{*} > 0\)such that for all \(\lambda \in (0,\hat{\lambda }_{*}]\)problem (\(P_{\lambda }^{v}\)) admits a positive solution \(\hat{u}_{v} \in \operatorname{int} C_{+}\).

Proof

Proposition 5 implies that we can find \(\hat{\lambda }_{*} \in (0,1]\) such that

$$\begin{aligned} 0 \leqslant \overline{u}_{\lambda }(z) \leqslant \delta \quad \text{for all } z \in \overline{\Omega }, \text{all } 0 < \lambda \leqslant \hat{ \lambda }_{*}. \end{aligned}$$
(23)

We introduce the Carathéodory function \(\tau _{v}^{\lambda }: \Omega \times \mathbb {R}\mapsto \mathbb {R}\) defined by

$$\begin{aligned} \tau _{v}^{\lambda }(z,x)= \textstyle\begin{cases} \lambda {\overline{u}_{\lambda }(z)}^{-\eta } + f(z,\overline{u}_{\lambda }(z),Dv(z)) & \text{if } x < \overline{u}_{\lambda }(z), \\ \lambda x^{-\eta } + f(z,x,D v(z)) & \text{if } \overline{u}_{\lambda }(z) \leqslant x \leqslant \theta , \\ \lambda \theta ^{-\eta } + f(z,\theta ,Dv(z)) & \text{if }\theta \leqslant x; \end{cases}\displaystyle \end{aligned}$$
(24)

recall that \(\delta _{0} < \theta \), see hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\). We set \(T _{v}^{\lambda }(z,x) = \int ^{x}_{0} \tau _{v}^{\lambda }(z,s) \,ds\) and consider the \(C^{1}\)-functional \(\varphi _{v}^{\lambda }: W^{1,p}_{0}(\Omega )\mapsto \mathbb {R}\) defined by

$$ \varphi _{v}^{\lambda }(u) = \int _{\Omega }G(Du) \,dz- \int _{\Omega }T_{v}^{\lambda }(z,u) \,dz \quad \text{for all } u \in W^{1,p}_{0}(\Omega ). $$

Corollary 3 and (24) imply that \(\varphi _{v}^{\lambda }(\cdot )\) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(\hat{u}_{v} \in W^{1,p}_{0}(\Omega )\) such that

$$\begin{aligned} \varphi _{v}^{\lambda }(\hat{u}_{v}) = \min \bigl\{ \varphi _{v}^{\lambda }(u) : u \in W^{1,p}_{0}(\Omega ) \bigr\} . \end{aligned}$$
(25)

Hypothesis \({\mathrm{H}_{0}(\mathrm{iv})}\) implies that, given \(\varepsilon ' \in (0,\varepsilon ]\), we can find \(\delta ' \in (0,\delta ]\) such that

$$\begin{aligned} G(y) \leqslant \frac{c_{*} + \varepsilon '}{q} \vert y \vert ^{q} \quad \text{for all } \vert y \vert \leqslant \delta '. \end{aligned}$$
(26)

Let \(t \in (0,1)\) be small such that

$$ 0 \leqslant t\hat{u}_{1}(q) (z) \leqslant \min \bigl\{ \overline{u}_{\lambda }(z), \delta _{0} \bigr\} \quad \text{for all } z \in \overline{\Omega } $$

(recall that \(\overline{u}_{\lambda }\in \operatorname{int} C_{+}\) and use Proposition 4.1.22, p. 274, of Papageorgiou–Rădulescu–Repovš [21]). We have

$$\begin{aligned} \varphi _{v}^{\lambda }\bigl(t\hat{u}_{1}(q)\bigr) & \leqslant \frac{t^{p}}{p} \bigl\Vert D{ \hat{u}_{1}(q)} \bigr\Vert _{p}^{p} + \frac{t^{q}}{q} \bigl(c_{*} + \varepsilon '\bigr) \hat{\lambda }_{1}(q) - \frac{t^{q}}{q} \int _{\Omega }\bigl(\eta _{M}(z)- \varepsilon \bigr) { \hat{u}_{1}(q)}^{q} \,dz \\ & \quad (\text{see (26), (22), (23), (24)}) \\ & \leqslant \frac{t^{p}}{p} \bigl\Vert D{\hat{u}_{1}(q)} \bigr\Vert _{p}^{p} + \frac{t^{q}}{q} \biggl( \int _{\Omega } \bigl(c_{*} \hat{\lambda }_{1}(q)- \eta _{M}(z) \bigr){\hat{u}_{1}(q)}^{q} \,dz + 2 \varepsilon \biggr) \\ & \quad \bigl(\text{recall } \bigl\Vert \hat{u}_{1}(q) \bigr\Vert _{q}=1\bigr). \end{aligned}$$

Note that

$$ \mu = \int _{\Omega } \bigl(\eta _{M}(z)-c_{*}\hat{ \lambda }_{1}(q) \bigr){\hat{u}_{1}}^{q} \,dz >0 $$

(see hypothesis \({\mathrm{H}_{1}(\mathrm{ii})}\) and recall \(\hat{u}_{1}(q) \in \operatorname{int} C_{+}\)).

We have

$$\begin{aligned} \varphi _{v}^{\lambda }\bigl(t\hat{u}_{1}(q)\bigr) \leqslant c_{17} t^{p}-c_{18}t^{q} \quad \text{for some } c_{17}, c_{18} > 0 \ (\text{with } \varepsilon > 0 \text{ small}). \end{aligned}$$

Since \(q < p\), choosing \(t \in (0,1)\) even smaller, we have

$$\begin{aligned} & \varphi ^{\lambda }_{v}\bigl(t\hat{u}_{1}(q)\bigr) < 0 \\ &\quad\Rightarrow\quad \varphi _{v}^{\lambda }( \hat{u}_{v}) < 0 = \varphi _{v}^{\lambda }(0) \quad ( \text{see (25)}) \\ &\quad\Rightarrow\quad \hat{u}_{v} \neq 0. \end{aligned}$$

From (25) we have

$$\begin{aligned} & \bigl(\varphi _{v}^{\lambda }\bigr)'( \hat{u}_{v})=0 \\ &\quad\Rightarrow\quad \bigl\langle A(\hat{u}_{v}), h\bigr\rangle = \int _{\Omega }\tau _{v}^{\lambda }(z, \hat{u}_{v})h \,dz \quad \text{for all } h \in W^{1,p}_{0}( \Omega ). \end{aligned}$$
(27)

In (27) first we choose \(h = (\overline{u}_{\lambda }-\hat{u}_{v} )^{+} \in W^{1,p}_{0}( \Omega )\). Then

$$\begin{aligned} & \bigl\langle A(\hat{u}_{v}), (\overline{u}_{\lambda }- \hat{u}_{v} )^{+}\bigr\rangle \\ &\quad = \int _{\Omega } \bigl(\lambda {\overline{u}_{\lambda }}^{-\eta }+f(z, \overline{u}_{\lambda },Dv) \bigr) (\overline{u}_{\lambda }- \hat{u}_{v} )^{+} \,dz \quad (\text{see (24)}) \\ &\quad \geqslant \int _{\Omega }\lambda {\overline{u}_{\lambda }}^{-\eta } ( \overline{u}_{\lambda }-\hat{u}_{v} )^{+} \,dz \\ & \quad\quad \bigl(\text{since } f(z,\overline{u}_{\lambda },Dv) \geqslant 0 \quad \text{(see (23)) and hypothesis } {\mathrm{H}_{1}( \mathrm{ii})}\bigr) \\ &\quad = \bigl\langle A(\overline{u}_{\lambda }), (\overline{u}_{\lambda }- \hat{u}_{v} )^{+}\bigr\rangle \quad (\text{see Proposition 5}) \\ &\quad\Rightarrow\quad \overline{u}_{\lambda }\leqslant \hat{u}_{v}. \end{aligned}$$

Next, in (27) we choose \(h = (\hat{u}_{v}-\theta )^{+} \in W^{1,p}_{0}(\Omega )\). Then

$$\begin{aligned} & \bigl\langle A(\hat{u}_{v}), (\hat{u}_{v}-\theta )^{+} \bigr\rangle \\ &\quad = \int _{\Omega } \bigl(\lambda \theta ^{-\eta } + f(z,\theta ,Dv) \bigr) (\hat{u}_{v}-\theta )^{+} \,dz \quad (\text{see (24)}) \\ &\quad \leqslant \int _{\Omega } \bigl( \theta ^{-\eta } + f(z,\theta ,Dv) \bigr) (\hat{u}_{v}-\theta )^{+} \,dz \quad (\text{recall } 0 < \lambda \leqslant \hat{\lambda }_{*} \leqslant 1) \\ &\quad \leqslant 0\quad \bigl(\text{see hypothesis } {\mathrm{H}_{1}( \mathrm{i})}\bigr) \\ &\quad\Rightarrow\quad \hat{u}_{v} \leqslant \theta \end{aligned}$$

(by the weak comparison principle; see Pucci–Serrin [28], Theorem 3.4.1, p. 61).

So, we have proved that

$$\begin{aligned} \hat{u}_{v} \in [\overline{u}_{\lambda },\theta ]. \end{aligned}$$
(28)

From (28), (24) and (27), we infer that \(\hat{u}_{v}\) is a positive solution of (\(P_{\lambda }^{v}\)). Since \({\overline{u}_{\lambda }}^{-\eta } \in L^{s}(\Omega )\) (\(s >N\)) (see (20) and (14)), reasoning as in the proof of Proposition 5 (see the part of the proof after (20)), we infer that \(\hat{u}_{v} \in \operatorname{int} C_{+}\).

This proof is now complete. □

Let \(S_{v}^{\lambda }\) be the set of positive solutions of the “frozen problem” (\(P_{\lambda }^{v}\)). We have just proved that for \(v \in C^{1}_{0}(\overline{\Omega })\) and \(0 < \lambda \leqslant \hat{\lambda }_{*}\) we have

$$ \emptyset \neq S_{v}^{\lambda }\subseteq \operatorname{int} C_{+}. $$

We will show that each of these solution sets has a smallest element. In this way we have a canonical procedure to choose an element from \(S_{v}^{\lambda }\) as \(v \in C^{1}_{0}(\overline{\Omega })\) varies. So, we define the minimal solution map on which we will use Theorem 1 (the Leray–Schauder alternative principle).

To produce a minimal element for the solution set \(S_{v}^{\lambda }\), we need the following result providing a lower bound for the set \(S_{v}^{\lambda }\).

Proposition 7

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold and \(0 < \lambda \leqslant \hat{\lambda }_{*}\), then \(\overline{u}_{\lambda }\leqslant u\)for all \(u \in S_{v}^{\lambda }\).

Proof

Let \(u \in S_{v}^{\lambda }\) and consider the Carathéodory function \(\hat{\mu }_{\lambda }(z,x)\) defined on \(\Omega \times \mathring{\mathbb {R}}_{+}=\Omega \times (0,\infty )\) by

$$\begin{aligned} \hat{\mu }_{\lambda }(z,x)= \textstyle\begin{cases} \lambda x^{-\eta } & \text{if } 0 < x \leqslant u(z), \\ \lambda {u(z)}^{-\eta } & \text{if } u(z) < x. \end{cases}\displaystyle \end{aligned}$$
(29)

We consider the following singular Dirichlet problem

$$\begin{aligned} -\operatorname{div} a(Du) = \hat{\mu }_{\lambda }(z,u) \quad \text{in } \Omega , u| _{ \partial \Omega } =0, u \geqslant 0. \end{aligned}$$
(30)

Reasoning as in the proof of Proposition 5, we show that problem (30) has a solution \(\widetilde{u}_{\lambda }\in \operatorname{int} C_{+}\) and using (29), we show that

$$ \widetilde{u}_{\lambda }\in [0,u]. $$

But then (29) and Proposition 5 imply that

$$\begin{aligned} & \widetilde{u}_{\lambda }=\overline{u}_{\lambda } \\ &\quad\Rightarrow\quad \overline{u}_{\lambda }\leqslant u \quad \text{for all } u \in S_{v}^{\lambda }. \end{aligned}$$

This proof is now complete. □

Using this lower bound, we show the existence of a minimal element for the set \(S_{v}^{\lambda }\).

Proposition 8

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold and \(0 < \lambda \leqslant \hat{\lambda }_{*}\), then problem (\(P_{\lambda }^{v}\)) has a smallest positive solution \(u^{*}_{v} \in \operatorname{int} C_{+}\)such that

$$ 0 \leqslant u_{v}^{*}(z) \leqslant \theta \quad \textit{for all } z \in \overline{\Omega }. $$

Proof

From Proposition 18 of Papageorgiou–Rădulescu–Repovš [23], we know that \(S_{v}^{\lambda }\) is downward directed. Then using Lemma 3.10, p. 178, of Hu–Papageorgiou [14], we can find \(\{u_{n}\}_{n \in \mathbb {N}} \subseteq S_{v}^{\lambda }\) decreasing such that

$$ \inf_{n \in \mathbb {N}} u_{n}=\inf S_{v}^{\lambda }. $$

From the proof of Proposition 6 we know that \(S_{v}^{\lambda }\cap [0,\theta ] \neq \emptyset \). So, without any loss of generality, we may assume that \(\{u_{n}\}_{n \in \mathbb {N}} \subseteq [0,\theta ]\). We have

$$\begin{aligned} & \bigl\langle A(u_{n}),h \bigr\rangle = \int _{\Omega } \bigl(\lambda {u_{n}}^{- \eta } + f(z,u_{n},Dv) \bigr) h \,dz \quad \text{for all } h \in W^{1,p}_{0}( \Omega ), \text{all } n \in \mathbb {N}, \end{aligned}$$
(31)
$$\begin{aligned} & \overline{u}_{\lambda }\leqslant u_{n} \leqslant u_{1} \quad \text{for all } n \in \mathbb {N}\ (\text{see Proposition 7}). \end{aligned}$$
(32)

In (31) we choose \(h=u_{n} \in W^{1,p}_{0}(\Omega )\). Using Lemma 2(b), (32) and hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\), we infer that

$$ \{u_{n}\}_{n \in \mathbb {N}} \subseteq W^{1,p}_{0}( \Omega ) \text{ is bounded}. $$

So, we may assume that

$$\begin{aligned} u_{n} \xrightarrow{w} u_{v}^{*} \quad \text{in } W^{1,p}_{0}(\Omega ) \quad \text{and}\quad u_{n} \rightarrow u_{v}^{*} \quad \text{in } L^{p}(\Omega ). \end{aligned}$$
(33)

In (31) we choose \(h = u_{n} -u_{v}^{*} \in W^{1,p}_{0}(\Omega )\), pass to limit as \(n \rightarrow \infty \) and use (33). We obtain

$$\begin{aligned} &\lim_{n \rightarrow \infty } \bigl\langle A(u_{n}),u_{n} -u_{v}^{*} \bigr\rangle =0 \quad (\text{see the proof Proposition 5}) \\ &\quad\Rightarrow\quad u_{n} \rightarrow u_{v}^{*} \quad \text{in } W^{1,p}_{0}( \Omega ) \ (\text{see Proposition 4}). \end{aligned}$$
(34)

Then, if in (31) we pass to the limit as \(n \rightarrow \infty \) and use (34), we obtain

$$ \bigl\langle A\bigl(u_{v}^{*}\bigr),h \bigr\rangle = \int _{\Omega } \bigl(\lambda \bigl({u_{v}^{*}} \bigr)^{- \eta } + f\bigl(z,u_{v}^{*},Dv\bigr) \bigr) h \,dz \quad \text{for all } h \in W^{1,p}_{0}( \Omega )\text{ and } \overline{u}_{\lambda }\leqslant u_{v}^{*}. $$

It follows that

$$ u_{v}^{*} \in S_{v}^{\lambda }, \quad\quad u_{v}^{*} = \inf S_{v}^{\lambda }, \quad \quad 0 \leqslant u_{v}^{*}(z) \leqslant \theta \quad \text{for all } z \in \overline{\Omega }. $$

This proof is now complete. □

Using Proposition 8, we define the minimal solution map

$$ \xi _{\lambda }: C^{1}_{0}(\overline{\Omega }) \mapsto C^{1}_{0}( \overline{\Omega }), \quad \lambda \in (0, \hat{ \lambda }_{*}], $$

by setting

$$ \xi _{\lambda }(v)=u_{v}^{*} \quad \text{for all } v \in C^{1}_{0}( \overline{\Omega }). $$

Evidently a fixed point of \(\xi _{\lambda }(\cdot )\) is a solution of problem (\(P_{\lambda }\)). To produce a fixed point of (\(P_{\lambda }\)), we will use Theorem 1 (the Leray–Schauder alternative principle). This is done in the next section.

Positive solution

To apply Theorem 1, we need to know that \(\xi _{\lambda }(\cdot )\) is compact. The next proposition will be helpful in this respect.

Proposition 9

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold, \(v_{n} \rightarrow v\)in \(C^{1}_{0}(\overline{\Omega })\), \(u \in S^{\lambda }_{v} \cap [0,\theta ]\)with \(\lambda \in (0,\hat{\lambda }_{*}]\), then we can find \(u_{n} \in S^{\lambda }_{v_{n}}\), \(n \in \mathbb {N}\)such that

$$ u_{n} \rightarrow u\quad \textit{in } C^{1}_{0}( \overline{\Omega }). $$

Proof

Choosing \(M \geqslant \sup_{n \in \mathbb {N}}\Vert Dv_{n}\Vert _{\infty }\), we see that all the previous results are valid. We consider the following Dirichlet problem:

$$\begin{aligned} -\operatorname{div} a\bigl(D y(z)\bigr) = \tau _{v_{n}}^{\lambda }\bigl(z,u(z)\bigr) \quad \text{in } \Omega , y| _{\partial \Omega }=0 \ (\text{see (24)}). \end{aligned}$$
(35)

This problem has a unique solution \(y_{n} \in W^{1,p}_{0}(\Omega )\). We know that \({\overline{u}_{\lambda }}^{-\eta } \in L^{s}(\Omega )\), \(s > N\) (see (20) and (14)). Therefore \(\{ \tau _{v_{n}}^{\lambda }(\cdot ,u(\cdot )) \} _{n \in \mathbb {N}} \subseteq L^{s}(\Omega )\) is bounded (see (24)). Consider the linear Dirichlet problem

$$ -\Delta e(z)=\tau _{v_{n}}^{\lambda }\bigl(z,u(z)\bigr) \quad \text{in } \Omega , e| _{ \partial \Omega }=0. $$

For each \(n \in \mathbb {N}\), this problem has a unique solution \(e_{n} \in W^{2,s}(\Omega )\) (see Gilbarg–Trudinger [12], Theorem 9.15, p. 241) and in addition we have

$$\begin{aligned} \Vert e_{n} \Vert _{W^{2,s}(\Omega )} \leqslant c_{19} \quad \text{for some } c_{19} > 0, \text{all } n \in \mathbb {N} \end{aligned}$$
(36)

(see Gilbarg–Trudinger [12], Lemma 9.17, p. 242). From the Sobolev embedding theorem we know that, for \(\alpha =1-\frac{N}{s} \in (0,1)\), we have

$$ W^{2,s}(\Omega ) \hookrightarrow C^{1,\alpha }(\overline{\Omega }) \text{ compactly}. $$

Hence from (36) it follows that

$$ \{ \hat{\gamma }_{n}=D e_{n} \} _{n \in \mathbb {N}} \subseteq C^{0, \alpha }(\overline{\Omega }) \text{ is bounded}. $$

We rewrite (35) as

$$ -\operatorname{div} \bigl(a(Dy)-\hat{\gamma }_{n}\bigr)=0 \quad \text{on } \partial \Omega , y| _{\partial \Omega }=0. $$

Then the nonlinear regularity theory of Lieberman [17] implies that we can find \(\mu \in (0,1)\) and \(c_{20} > 0\) such that

$$\begin{aligned} y_{n} \in C^{1,\mu }_{0}(\overline{ \Omega }) \quad \text{and}\quad \Vert y_{n} \Vert _{C^{1, \mu }_{0}(\overline{\Omega })} \leqslant c_{20} \quad \text{for all } n \in \mathbb {N}. \end{aligned}$$
(37)

From (37) and the compact embedding of \(C^{1,\mu }_{0}(\overline{\Omega })\) into \(C^{1}_{0}(\overline{\Omega })\), it follows that

$$ \{ y_{n} \} _{n \in \mathbb {N}} \subseteq C^{1}_{0}( \overline{\Omega }) \text{ is relatively compact}. $$

Note that \(\tau _{v_{n}}^{\lambda }(\cdot ,u(\cdot )) \mapsto \tau _{v}^{\lambda }( \cdot ,u(\cdot ))\) in \(L^{s}(\Omega )\). So, for the whole sequence, we have

$$ y_{n} \rightarrow u \quad \text{in } C^{1}_{0}( \overline{\Omega }) \text{ as } n \rightarrow \infty . $$

We set \(y_{n} = y_{n}^{0}\) and consider the following Dirichlet problem:

$$ -\operatorname{div} a\bigl(Dy(z)\bigr) = \tau _{v_{n}}^{\lambda } \bigl(z,y_{n}^{0}(z)\bigr) \quad \text{in } \Omega , y| _{\partial \Omega } =0. $$

As above this problem has a unique solution \(y_{n}^{1} \in C^{1}_{0}(\overline{\Omega })\) and we have

$$ y_{n}^{1} \rightarrow u \quad \text{in } C^{1}_{0}(\overline{\Omega }) \text{ as } n \rightarrow \infty . $$

We continue this way and generate a sequence \(\{ y_{n}^{k} \} _{k, n \in \mathbb {N}} \subseteq C^{1}_{0}( \overline{\Omega })\) such that

$$\begin{aligned} &-\operatorname{div} a\bigl(Dy_{n}^{k}(z) \bigr) = \tau _{v_{n}}^{\lambda }\bigl(z,y_{n}^{k-1}(z) \bigr) \quad \text{in } \Omega , y_{n}^{k}| _{\partial \Omega } =0, \end{aligned}$$
(38)
$$\begin{aligned} &y_{n}^{k} \rightarrow u \quad \text{in } C^{1}_{0}(\overline{\Omega }) \text{ as } n \rightarrow \infty \text{ for all } k \in \mathbb {N}. \end{aligned}$$
(39)

We will show that for every \(n \in \mathbb {N}\)

$$\begin{aligned} \bigl\{ y_{n}^{k} \bigr\} _{k \in \mathbb {N}} \subseteq W^{1,p}_{0}( \Omega )\text{ is bounded}. \end{aligned}$$
(40)

Arguing by contradiction, suppose that (40) is not true. By passing to a subsequence if necessary, we may assume that

$$\begin{aligned} \bigl\Vert y_{n}^{k} \bigr\Vert \rightarrow \infty \quad \text{as } k \rightarrow \infty . \end{aligned}$$
(41)

We set \(w_{k}=\frac{y_{n}^{k}}{\Vert y_{n}^{k}\Vert }\), \(k \in \mathbb {N}\). Then \(\Vert w_{k}\Vert =1\) for all \(k \in \mathbb {N}\) and so we may assume that

$$\begin{aligned} w_{k} \xrightarrow{w} w \quad \text{in } W^{1,p}_{0}(\Omega )\text{ and } w_{k} \rightarrow w \text{ in } L^{p}(\Omega ) \text{ as } k \rightarrow \infty . \end{aligned}$$
(42)

From (38) we have

$$\begin{aligned} & \int _{\Omega } \biggl(\frac{a(Dy_{n}^{k})}{ \Vert y_{n}^{k} \Vert ^{p-1}},Dw_{k} \biggr)_{\mathbb {R}^{N}} \,dz = \int _{\Omega }\frac{\tau _{v_{n}}^{\lambda }(z,y_{n}^{k-1})}{ \Vert y_{n}^{k} \Vert ^{p-1}} w_{k} \,dz \\ &\quad\Rightarrow\quad \frac{c_{1}}{p-1} \Vert Dw_{k} \Vert _{p}^{p} \leqslant \int _{\Omega }\frac{\tau _{v_{n}}^{\lambda }(z,y_{n}^{k-1})}{ \Vert y_{n}^{k} \Vert ^{p-1}} w_{k} \,dz \quad \bigl( \text{see Lemma 2(b)}\bigr). \end{aligned}$$
(43)

On account of (24), (41) and (42), we have

$$\begin{aligned} & \int _{\Omega }\frac{\tau _{v_{n}}^{\lambda }(z,y_{n}^{k-1})}{ \Vert y_{n}^{k} \Vert ^{p-1}} w_{k} \,dz \rightarrow 0 \quad \text{as } k \rightarrow \infty \\ &\quad\Rightarrow\quad w_{k} \rightarrow 0 \text{ in } W^{1,p}_{0}(\Omega ) \text{ as } k \rightarrow \infty \quad ( \text{see (43)}), \end{aligned}$$

which contradicts the fact that \(\Vert w_{k}\Vert =1\) for all \(k \in \mathbb {N}\).

Therefore (40) is true.

Then on account of (40) and the nonlinear regularity theory (see [17] and [21]), we infer that \(\{ y_{n}^{k} \} _{k \in \mathbb {N}} \subseteq C^{1}_{0}( \overline{\Omega })\) is relatively compact. Hence we may assume that

$$\begin{aligned} y_{n}^{k} \rightarrow u_{n} \quad \text{in } C^{1}_{0}(\overline{\Omega }) \text{ as } k \rightarrow \infty . \end{aligned}$$
(44)

From (38) and hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\), we have

$$\begin{aligned} & -\operatorname{div} a(Du_{n}) = \tau _{v_{n}}^{\lambda }(z,u_{n}) \quad \text{in } \Omega , u_{n}| _{\partial \Omega }=0, \end{aligned}$$
(45)
$$\begin{aligned} & \overline{u}_{\lambda }\leqslant u_{n} \leqslant \theta \quad \text{for all } n \in \mathbb {N}\ (\text{see (24)}) \\ &\quad\Rightarrow\quad u_{n} \in S_{v_{n}}^{\lambda } \quad \text{for all } n \in \mathbb {N}. \end{aligned}$$
(46)

From (45), (46) and the nonlinear regularity theory of Lieberman [17], it follows that \(\{ u_{n} \} _{n \in \mathbb {N}} \subseteq C^{1}_{0}( \overline{\Omega })\) is relatively compact. Then (39), (44) and the double limit lemma (see Gasinski–Papageorgiou [7], Problem 1.175, p. 61) imply that

$$ u_{n} \rightarrow u \quad \text{in } C^{1}_{0}( \overline{\Omega }) \text{ with } u_{n} \in S_{v_{n}}^{\lambda } \text{ for all } n \in \mathbb {N}. $$

This proof is now complete. □

Using this proposition, we can establish the compactness of the minimal solution map \(\xi _{\lambda }(\cdot )\).

Proposition 10

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold and \(\lambda \in (0,\hat{\lambda }_{*}]\), then the minimal solution map \(\xi _{\lambda }: C^{1}_{0}(\overline{\Omega }) \mapsto C^{1}_{0}( \overline{\Omega })\)is compact.

Proof

Let \(B \subseteq C^{1}_{0}(\overline{\Omega })\) be bounded. Let \(v \in B\) and \(u=\xi _{\lambda }(v)\). We have

$$\begin{aligned} & -\operatorname{div} a(Du)=\lambda u^{-\eta } + f(z,u,Dv) \quad \text{in } \Omega , u| _{\partial \Omega }=0, \overline{u}_{\lambda }\leqslant u \leqslant \theta \\ &\quad\Rightarrow\quad \frac{c_{1}}{p-1} \Vert Du \Vert _{p}^{p} \leqslant c_{21} \quad \text{for some } c_{21} > 0 \\ &\quad\Rightarrow\quad \xi _{\lambda }(B) \subseteq W^{1,p}_{0}( \Omega ) \text{ is bounded}. \end{aligned}$$

From this as before using the nonlinear regularity theory of Lieberman [17], we infer that

$$\begin{aligned} \xi _{\lambda }(B) \subseteq C^{1}_{0}( \overline{\Omega }) \text{ is relatively compact}. \end{aligned}$$
(47)

Next, we show that \(\xi _{\lambda }(\cdot )\) is continuous. So, let \(v_{n} \rightarrow v\) in \(C^{1}_{0}(\overline{\Omega })\) and let \(u_{n}=\xi _{\lambda }(v_{n})\), \(n \in \mathbb {N}\) and \(u=\xi _{\lambda }(v)\). From (47) we see that \(\{ u_{n} \} _{n \in \mathbb {N}} \subseteq C^{1}_{0}( \overline{\Omega })\) is relatively compact. Hence we may assume that

$$\begin{aligned} u_{n} \rightarrow \widetilde{u} \quad \text{in }C^{1}_{0}(\overline{\Omega }) \text{ as } n \rightarrow \infty . \end{aligned}$$
(48)

Evidently \(\widetilde{u} \in S_{v}^{\lambda }\) and so we have

$$\begin{aligned} u \leqslant \widetilde{u}. \end{aligned}$$
(49)

On the other hand, from Proposition 9, we know that we can find \(\widetilde{u}_{n} \in S_{v_{n}}^{\lambda }\), \(n \in \mathbb {N}\) such that

$$\begin{aligned} \widetilde{u}_{n} \rightarrow u \quad \text{in } C^{1}_{0}( \overline{\Omega }) \text{ as } n \rightarrow \infty . \end{aligned}$$
(50)

We have \(u_{n} \leqslant \widetilde{u}_{n}\) for all \(n \in \mathbb {N}\) and so from (48) and (50), we obtain

$$\begin{aligned} & \widetilde{u} \leqslant u \\ &\quad\Rightarrow\quad u=\widetilde{u} \quad (\text{see (49)}). \end{aligned}$$

So, for the whole sequence we have

$$\begin{aligned} & u_{n} \rightarrow u \quad \text{in } C^{1}_{0}( \overline{\Omega }) \\ &\quad\Rightarrow\quad \xi _{\lambda }(\cdot ) \text{ is continuous} \\ &\quad\Rightarrow\quad \xi _{\lambda }(\cdot ) \text{ is compact}. \end{aligned}$$

This proof is now complete. □

Let \(\mathcal{D}_{\lambda }= \{ u \in C^{1}_{0}(\overline{\Omega }) : u=t \xi _{\lambda }(u), 0< t < 1\}\).

Proposition 11

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold and \(\lambda \in (0,\hat{\lambda }_{*}]\), then \(\mathcal{D}_{\lambda }\subseteq C^{1}_{0}(\overline{\Omega })\)is bounded.

Proof

Let \(u \in \mathcal{D}_{\lambda }\). We have

$$ \frac{1}{t} u = \xi _{\lambda }(u) \quad \text{with } 0 < t < 1. $$

It follows that

$$\begin{aligned} \biggl\langle A \biggl(\frac{1}{t}u \biggr),h \biggr\rangle = \int _{\Omega } \biggl(\lambda \biggl(\frac{1}{t}u \biggr)^{-\eta } + f \biggl(z, \frac{1}{t}u,Du \biggr) \biggr) h \,dz \end{aligned}$$
(51)

for all \(h \in W^{1,p}_{0}(\Omega )\).

From Proposition 7, we know that

$$\begin{aligned} & \overline{u}_{\lambda }\leqslant \frac{1}{t} u \\ &\quad\Rightarrow\quad t \overline{u}_{\lambda }\leqslant u \leqslant \theta \quad \bigl(\text{since } t \in (0,1)\bigr). \end{aligned}$$
(52)

In (51) we choose \(h = u \in W^{1,p}_{0}(\Omega )\). Using Lemma 2(b), (52) (recall that \({\overline{u}_{\lambda }}^{-\eta }\in L^{s}(\Omega )\)) and hypothesis \({\mathrm{H}_{1}(\mathrm{i})}\), we obtain

$$\begin{aligned}& \begin{aligned} \frac{c_{1}}{p-1} \Vert Du \Vert _{p}^{p} &\leqslant c_{22} \biggl(1 + \int _{\Omega } \bigl\vert f(z,u,Du) \bigr\vert \,dz \biggr) \quad \text{for some } c_{22} >0 \\ & \leqslant c_{23} \bigl(1 + \Vert Du \Vert ^{p-1} \bigr) \quad \text{for some } c_{23} > 0 \end{aligned} \\& \quad\Rightarrow\quad \mathcal{D}_{\lambda }\subseteq W^{1,p}_{0}(\Omega ) \text{ is bounded}. \end{aligned}$$

Then as before via the nonlinear regularity theory of Lieberman [17], we see that \(\mathcal{D}_{\lambda }\subseteq C^{1}_{0}(\overline{\Omega })\) is bounded (in fact relatively compact).

This proof is now complete. □

So we can use Theorem 1 on the map \(\xi _{\lambda }(\cdot )\) and produce a fixed point which is a positive solution of problem (\(P_{\lambda }\)), for \(\lambda \in (0,\hat{\lambda }_{*}]\). Concluding we can state the following existence theorem for problem (\(P_{\lambda }\)).

Theorem 12

If hypotheses \({\mathrm{H}_{0}}\), \({\mathrm{H}_{1}}\)hold, then there exists \(\hat{\lambda }_{*} > 0\)such that for all \(\lambda \in (0, \hat{\lambda }_{*}]\)problem (\(P_{\lambda }\)) admits a positive solution

$$ \hat{u}_{\lambda }\in [\overline{u}_{\lambda },\theta ] \cap \operatorname{int} C_{+}. $$

References

  1. 1.

    Bai, Y.: Existence of solutions to nonhomogeneous Dirichlet problems with dependence on the gradient. Electron. J. Differ. Equ. 2018, 101 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bai, Y., Gasinski, L., Papageorgiou, N.S.: Nonlinear nonhomogeneous Robin problems with dependence on the gradient. Bound. Value Probl. 2018, 17 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Bobkov, V., Tanaka, M.: On sign-changing solutions for \((p,q)\)-Laplace equations with two parameters. Adv. Nonlinear Anal. 8, 101–129 (2019)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Candito, P., Gasinski, L., Papageorgiou, N.S.: Nonlinear nonhomogeneous Robin problems with convection. Ann. Acad. Sci. Fenn., Math. 44, 755–767 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Faraci, F., Motreanu, D., Puglisi, D.: Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc. Var. Partial Differ. Equ. 54, 525–538 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Gasinski, L., Krech, I., Papageorgiou, N.S.: Nonlinear nonhomogeneous Robin problems with gradient dependent reaction. Nonlinear Anal., Real World Appl. 55, 103135 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis. Part 1. Springer, Cham (2014)

    Google Scholar 

  8. 8.

    Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2016)

    Google Scholar 

  9. 9.

    Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis. Part 2: Nonlinear Analysis. Springer, Cham (2016)

    Google Scholar 

  10. 10.

    Gasinski, L., Papageorgiou, N.S.: Positive solutions for nonlinear elliptic problems with dependence on the gradient. J. Differ. Equ. 263, 1451–1476 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Gasinski, L., Winkert, P.: Existence and uniqueness results for double phase problems with convection term. J. Differ. Equ. 268, 4183–4193 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1998)

    Google Scholar 

  13. 13.

    Guo, Z., Webb, J.R.L.: Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. Proc. R. Soc. Edinb., Sect. A 124, 189–198 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  15. 15.

    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    Google Scholar 

  16. 16.

    Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Lieberman, G.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)

    MATH  Article  Google Scholar 

  18. 18.

    Liu, Z., Papageorgiou, N.S.: Positive solutions for resonant \((p,q)\)-equations with convection. Adv. Nonlinear Anal. 10, 217–232 (2021)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Papageorgiou, N.S., Rădulescu, V.D.: Coercive and noncoercive nonlinear Neumann problems with indefinite potential. Forum Math. 28, 545–571 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Positive solutions for nonvariational Robin problems. Asymptot. Anal. 108, 243–255 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonlinear Analysis-Theory and Methods. Springer, Cham (2019)

    Google Scholar 

  22. 22.

    Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Positive solutions for nonlinear Neumann problems with singular terms and convection. J. Math. Pures Appl. 136, 1–21 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonlinear nonhomogeneous singular problems. Calc. Var. Partial Differ. Equ. 59, 9 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Papageorgiou, N.S., Vetro, C., Vetro, F.: Existence of positive solutions for nonlinear Dirichlet problems with gradient dependence and arbitrary growth. Electron. J. Qual. Theory Differ. Equ. 18, 9 (2018)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Papageorgiou, N.S., Winkert, P.: Singular p-Laplacian equations with superlinear perturbation. J. Differ. Equ. 266, 1462–1487 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Papageorgiou, N.S., Zhang, C.: Noncoercive resonant \((p,2)\)-equations with concave terms. Adv. Nonlinear Anal. 9, 228–249 (2020)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Papageorgiou, N.S., Zhang, Y.: Constant sign and nodal solutions for superlinear \((p,q)\)-equations with indefinite potential and concave boundary term. Adv. Nonlinear Anal. 10, 76–101 (2021)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and Their Applications, vol. 73. Birkhäuser, Basel (2007)

    Google Scholar 

  29. 29.

    Rădulescu, V.D.: Isotropic and anisotropic double-phase problems: old and new. Opusc. Math. 39, 259–279 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9, 710–728 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Tanaka, M.: Existence of positive solution for quasilinear elliptic equations with nonlinearity including the gradient. Bound. Value Probl. 2013, 173 (2013)

    MathSciNet  MATH  Article  Google Scholar 

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This paper focuses on theoretical analysis, not involving experiments and data.

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This research is partially supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2019zzts211). This paper has been completed while Youpei Zhang was visiting University of Craiova (Romania) with the financial support of China Scholarship Council (No. 201906370079). Youpei Zhang would like to thank the China Scholarship Council and the Embassy of the People’s Republic of China in Romania.

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Papageorgiou, N.S., Zhang, Y. Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms. Bound Value Probl 2020, 153 (2020). https://doi.org/10.1186/s13661-020-01450-0

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MSC

  • 35B50
  • 35J75
  • 35J92

Keywords

  • Frozen variable method
  • Nonlinear regularity
  • Minimal positive solution
  • Leray–Schauder alternative principle
  • Truncation
  • Fixed point
  • Convection term