Combined effects of concave–convex nonlinearities and indefinite potential in some elliptic problems

Abstract

We consider a nonlinear Dirichlet problem driven by the p-Laplacian and a reaction which exhibits the combined effects of concave (that is, sublinear) terms and of convex (that is, superlinear) terms. The concave term is indefinite and the convex term need not satisfy the usual in such cases Ambrosetti–Rabinowitz condition. We prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies.

Keywords

indefinite concave term superlinear perturbation nonlinear regularity positive solutions

1. Introduction

Let $Ω \subseteq R^{N}$ be a bounded domain with a $C^{2}$ -boundary $\partial Ω$ . In this paper, we study the following nonlinear parametric elliptic problem $\{\begin{matrix} - Δ_{p} u (z) = ϑ (z) u {(z)}^{q - 1} + f (z, u (z), λ) & in Ω, \\ u = 0 & on \partial Ω, \\ u > 0 & in Ω, \end{matrix}$ (P_λ) where $λ > 0$ and $1 < q < p$ . Here $Δ_{p}$ denotes the p-Laplace differential operator defined by $Δ_{p} u = div ({| D u |}^{p - 2} D u) for all u \in W_{0}^{1, p} (Ω), 1 < p < \infty .$ The perturbation $(z, x) \mapsto f (z, x, λ)$ is a Carathéodory function (that is, for all $x \in R$ , $z \mapsto f (z, x, λ)$ is measurable and for a.a. $z \in Ω$ , $x \mapsto f (z, x, λ)$ is continuous), which exhibits $(p - 1)$ -superlinear growth near $+ \infty$ , without satisfying the usual in such cases (unilateral) Ambrosetti–Rabinowitz condition ( $AR$ -condition for short). So, in problem ( P λ ) we have the combined effects of a concave (that is, of a $(p - 1)$ -sublinear) nonlinearity which is expressed by the term $ϑ (z) u^{q - 1}$ (recall $1 < q < p$ ) and of a convex (that is, of a $(p - 1)$ -superlinear) nonlinearity, expressed by the term $f (z, u, λ)$ . Hence, we are dealing with a “concave–convex problem”. The interesting feature of our work here, is that the concave term $ϑ (z) u^{q - 1}$ is indefinite, namely the weight function $ϑ (\cdot)$ may change sign.

Problems with combined nonlinearities, were first investigated by Ambrosetti, Brezis and Cerami [2], where $p = 2$ (semilinear problem) and the parametric reaction has the form $λ x^{q - 1} + x^{r - 1} for all x ⩾ 0, with 1 < q < 2 < r < 2^{*} = \{\begin{matrix} \frac{2 N}{N - 2} & if 2 < N, \\ + \infty & if N = 1, 2 . \end{matrix}$ They proved bifurcation type results describing the dependence of the set of positive solutions on the parameter $λ > 0$ . Their work was extended to nonlinear problems driven by the p-Laplacian, by Garcia Azorero, Manfredi and Peral Alonso [8] and Guo and Zhang [10]. Problems with more general reactions, were studied by Hu and Papageorgiou [11] and Marano and Papageorgiou [14]. Problems with indefinite concave nonlinearities were investigated by de Paiva [6], Li, Wu and Zhou [12], and Papageorgiou and Rădulescu [18] only in the context of semilinear equations (that is, $p = 2$ ) and with a particular reaction of the form $x \mapsto ϑ (z) x^{q - 1} + λ x^{r - 1}$ for all $x ⩾ 0$ , with $ϑ \in L^{\infty} (Ω)$ and $1 < q < 2 < r < 2^{*}$ . We also refer to the related papers by de Figueiredo, Gossez and Ubilla [5] and Narukawa and Takajo [16].

Using variational methods based on the critical point theory, combined with suitable truncation and comparison techniques, we establish the existence, nonexistence and multiplicity of positive solutions for problem ( P λ ) as the parameter $λ > 0$ varies.

2. Mathematical background

Let X be a Banach space and $X^{*}$ be its topological dual. By $⟨ \cdot, \cdot ⟩$ we denote the duality brackets for the pair $(X^{*}, X)$ . Given $φ \in C^{1} (X)$ , we say that φ satisfies the “Cerami condition” (the “C-condition” for short), if the following is true:

“Every sequence ${u_{n}}_{n ⩾ 1} \subseteq X$ such that ${φ (u_{n})}_{n ⩾ 1} \subseteq R$ is bounded and $(1 + ∥ u_{n} ∥) φ^{'} (u_{n}) \to 0 in X^{*} as n \to \infty,$ admits a strongly convergent subsequence”.

This is a compactness type condition on the functional φ, which is needed since the ambient space X need not be locally compact (since, in general X is infinite dimensional). The C-condition is the main tool in proving a deformation theorem, from which one can derive the minimax theory for the critical values of φ. One of the main results in this theory, is the so-called “mountain pass theorem” due to Ambrosetti and Rabinowitz [3], stated here is a slightly more general form (see Gasinski and Papageorgiou [9]).

Theorem 1.
Assume that $φ \in C^{1} (X)$ satisfies the C-condition, $u_{0}, u_{1} \in X$ , $∥ u_{1} - u_{0} ∥ > ρ > 0$ , $max {φ (u_{0}), φ (u_{1})} < inf [φ (u) : ∥ u - u_{0} ∥ = ρ] = η_{ρ}$ and $c = inf_{γ \in Γ} max_{0 ⩽ t ⩽ 1} φ (γ (t)) with Γ = {γ \in C ([0, 1], X) : γ (0) = u_{0}, γ (1) = u_{1}} .$ Then $c ⩾ η_{ρ}$ and c is a critical value of φ.

In the analysis of problem ( P λ ), in addition to the Sobolev space $W_{0}^{1, p} (Ω)$ we will also use the Banach space $C_{0}^{1} (\bar{Ω}) = {u \in C^{1} (\bar{Ω}) : u |_{\partial Ω} = 0}$ . This is an ordered Banach space with positive cone $C_{+} = {u \in C_{0}^{1} (\bar{Ω}) : u (z) ⩾ 0 for all z \in \bar{Ω}}$ . This cone has a nonempty interior given by $int C_{+} = {u \in C_{+} : u (z) > 0 for all z \in Ω, \frac{\partial u}{\partial n} |_{\partial Ω} < 0} .$ Here by $n (\cdot)$ we denote the outward unit normal on $\partial Ω$ .

Let $f_{0} : Ω \times R \to R$ be a Carathéodory function with subcritical growth in $x \in R$ , that is, $\begin{array}{rcl} | f_{0} (z, x) | ⩽ a_{0} (z) (1 + {| x |}^{r - 1}) for a.a. z \in Ω, all x \in R, \\ with a_{0} \in L^{\infty} {(Ω)}_{+} and 1 < r < p^{} = \{\begin{matrix} \frac{N p}{N - p} & if p < N, \\ + \infty & if N ⩽ p . \end{matrix} \end{array}$

We set $F_{0} (z, x) = \int_{0}^{x} f_{0} (z, s) d s$ and consider the $C^{1}$ -functional $φ_{0} : W_{0}^{1, p} (Ω) \to R$ defined by $φ_{0} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} F_{0} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$

The next result can be found in Garcia Azero, Manfredi and Peral Alonso [8] and essentially is a consequence of the nonlinear regularity theory of Lieberman [13].
Proposition 2.
Assume that* $u_{0} \in W^{1, p} (Ω)$ is a local $C_{0}^{1} (\bar{Ω})$ -minimizer of $φ_{0}$ , that is, there exists $ρ_{0} > 0$ such that $φ_{0} (u_{0}) ⩽ φ (u_{0} + h) for all h \in C_{0}^{1} (\bar{Ω}) with {∥ h ∥}_{C_{0}^{1} (\bar{Ω})} ⩽ ρ_{0} .$ Then $u_{0} \in C_{0}^{1, α} (\bar{Ω})$ for some $α \in (0, 1)$ and it is also a local $W_{0}^{1, p} (Ω)$ -minimizer of $φ_{0}$ , that is, there exists $ρ_{1} > 0$ such that $φ_{0} (u_{0}) ⩽ φ (u_{0} + h) for all h \in W_{0}^{1, p} (Ω) with ∥ h ∥ ⩽ ρ_{1} .$

Hereafter by $∥ \cdot ∥$ we denote the norm of $W_{0}^{1, p} (Ω)$ . By virtue of the Poincaré inequality, we have $∥ u ∥ = {∥ D u ∥}_{p} for all u \in W_{0}^{1, p} (Ω) .$

Let $A : W_{0}^{1, p} (Ω) \to W^{- 1, p^{'}} (Ω) = W_{0}^{1, p} {(Ω)}^{}$ $(\frac{1}{p} + \frac{1}{p^{'}} = 1)$ be the nonlinear map defined by $\begin{array}{rcl} ⟨ A (u), y ⟩ = \int_{Ω} {| D u |}^{p - 2} {(D u, D y)}_{R^{N}} d z for all u, y \in W_{0}^{1, p} (Ω) . \end{array}$

The next proposition summarizes the main properties of this map (see, for example, Papageorgiou and Kyritsi [17, p. 314]).
Proposition 3.
The map* $A : W_{0}^{1, p} (Ω) \to W^{- 1, p^{'}} (Ω)$ is bounded (that is, maps bounded sets to bounded sets), demicontinuous, strictly monotone, hence maximal monotone too and of type ${(S)}_{+}$ (that is, if $u_{n} \overset{w}{\to} u$ in $W_{0}^{1, p} (Ω)$ and $\begin{array}{rcl} \underset{n \to \infty}{lim sup} ⟨ A (u_{n}), u_{n} - u ⟩ ⩽ 0, \end{array}$ then $u_{n} \to u$ in $W_{0}^{1, p} (Ω)$ as $n \to \infty$ ) .

We recall that the Dirichlet p-Laplacian ( $- Δ_{p}, W_{0}^{1, p} (Ω)$ ) admits a smallest eigenvalue ${\hat{λ}}_{1} > 0$ . Sometimes we write ${\hat{λ}}_{1} (Ω) > 0$ to emphasize the domain Ω. This eigenvalue is isolated, simple with eigenfunctions of constant sign. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [9, pp. 737–738]), imply that every positive eigenfunction corresponding to ${\hat{λ}}_{1} > 0$ belongs in $int C_{+}$ .

Finally let us fix our notation. So, for $x \in R$ , we set $x^{\pm} = max {\pm x, 0}$ . Then given $u \in W_{0}^{1, p} (Ω)$ , we set $u^{\pm} (\cdot) = u {(\cdot)}^{\pm}$ . We have $u^{\pm} \in W_{0}^{1, p} (Ω), u = u^{+} - u^{-}, | u | = u^{+} + u^{-} .$ Given any measurable function $h : Ω \times R \to R$ (for example, a Carathéodory function), we define $N_{h} (u) (\cdot) = h (\cdot, u (\cdot)) for all u \in W_{0}^{1, p} (Ω)$ (the Nemytski map corresponding to h). Evidently $z \mapsto N_{h} (u) (z)$ is measurable on Ω. By ${| \cdot |}_{N}$ we denote the Lebesgue measure on $R^{N}$ .
3. Positive solutions

The hypotheses on the data of problem ( P λ ) are the following:

$ϑ \in L^{\infty} (Ω)$ and if $D_{+} = {z \in Ω : ϑ (z) > 0}$ , then $D_{+} \neq \emptyset$ and there exists an open set $Ω_{+}$ such that ${\bar{Ω}}_{+} \subseteq D_{+}$ , $\partial Ω_{+}$ is $C^{2}$ and ${ess inf}_{Ω_{+}} ϑ = m_{+} > 0$ .

$f : Ω \times R \times (0, \infty) \to R$ is a function such that for all $λ > 0$ , $(z, x) \mapsto f (z, x, λ)$ is Carathéodory, $f (z, 0, λ) = 0$ for a.a. $z \in Ω$ and

$f (z, x, λ) ⩽ a (z, λ) (1 + x^{r (λ) - 1})$ for a.a. $z \in Ω$ with $r (λ) \in (p, p^{*}), a (\cdot, λ) \in L^{\infty} {(Ω)}_{+}, {∥ a (\cdot, λ) ∥}_{\infty} \to 0 as λ \to 0^{+};$

if $F (z, x, λ) = \int_{0}^{x} f (z, s, λ) d s$ , then $lim_{x \to + \infty} \frac{F (z, x, λ)}{x^{p}} = + \infty uniformly for a.a. z \in Ω$ and there exist $τ \in ((r (λ) - p) max {\frac{N}{p}, 1}, p^{*})$ , $τ > q$ and $η_{0} (λ) > 0$ with $λ \mapsto η_{0} (λ)$ nondecreasing such that $η_{0} (λ) ⩽ \underset{x \to + \infty}{lim inf} \frac{f (z, x, λ) x - p F (z, x, λ)}{x^{τ}} uniformly for a.a. z \in Ω;$

for every $ρ > 0$ , there exists $m_{ρ} (λ) > 0$ such that $m_{ρ} (λ) \to + \infty$ as $λ \to + \infty$ , $inf [f (z, x, λ) : x ⩾ ρ] = m_{ρ} (λ) > 0,$ for a.a. $z \in Ω$ , all $x ⩾ 0$ , the map $λ \mapsto f (z, x, λ)$ is nondecreasing and for every $ξ > 0$ , for a.a. $z \in Ω$ , all $x ⩾ ξ$ , all $λ^{'} > λ > 0$ , we have $f (z, x, λ) ⩾ m_{ξ} (λ)$ with $m_{ξ} (\cdot)$ nondecreasing, $m_{ξ} (λ) \to + \infty$ as $λ \to \infty$ and $f (z, x, λ^{'}) - f (z, x, λ) ⩾ η_{ξ} > 0$ ;

for every $ρ > 0$ , there exists $ξ_{ρ} (λ) > 0$ such that for a.a. $z \in Ω$ , the function $x \mapsto f (z, x, λ) + ξ_{ρ} (λ) x^{p - 1}$ is nondecreasing on $[0, ρ]$ .

Remark 1.

Since we are interested on positive solutions and all the above hypotheses concern the positive semiaxis $R_{+} = [0, + \infty)$ , without any loss of generality, we may assume that $f (z, x, λ) = 0$ for a.a. $z \in Ω$ , all $x ⩽ 0$ , all $λ > 0$ . Hypothesis $H_{2}$ (ii) implies that for a.a. $z \in Ω$ and all $λ > 0$ , the perturbation $x \mapsto f (z, x, λ)$ is $(p - 1)$ -superlinear near $+ \infty$ . However, we do not employ the usual in such cases $AR$ -condition. We recall that the $AR$ -condition (unilateral version, since we assume that $f (z, x, λ) = 0$ for a.a. $z \in Ω$ , all $x ⩽ 0$ , all $λ > 0$ ), says that there exist $μ = μ (λ) > p$ and $M = M (λ) > 0$ such that

$0 < μ F (z, x, λ) ⩽ f (z, x, λ) x$ for a.a. $z \in Ω$ , all $x ⩾ M$ ;

${ess inf}_{Ω} F (\cdot, M, λ) > 0$

(see Ambrosetti and Rabinowitz [3] and Mugnai [15]). Integrating (a) and using (b), we obtain the weaker condition

c_{1} x^{μ} ⩽ F (z, x, λ) for a.a. z \in Ω, all x ⩾ M and some c_{1} = c_{1} (λ) > 0 .

Evidently this unilateral growth estimate implies the much weaker condition

lim_{x \to + \infty} \frac{F (z, x, λ)}{x^{p}} = + \infty uniformly for a.a. z \in Ω .

Note that our hypothesis $H_{2}$ (ii) is weaker than the $AR$ -condition. Indeed, suppose that the $AR$ -condition holds (see (a) and (b) above). We may assume that $μ > (r (λ) - p) max {\frac{N}{p}, 1}$ . We have $\begin{array}{rcl} \frac{f (z, x, λ) x - p F (z, x, λ)}{x^{μ}} = \frac{f (z, x, λ) x - μ F (z, x, λ)}{x^{μ}} + (μ - p) \frac{F (z, x, λ)}{x^{μ}} \\ ⩾ (μ - p) \frac{F (z, x, λ)}{x^{μ}} for a.a. z \in Ω, all x ⩾ M (see (a)) \\ ⩾ (μ - p) c_{1} for a.a. z \in Ω, all x ⩾ M \\ ⟹ \underset{x \to + \infty}{lim inf} \frac{f (z, x, λ) x - p F (z, x, λ)}{x^{μ}} ⩾ (μ - p) c_{1} (λ) = η_{0} (λ) uniformly for a.a. z \in Ω . \end{array}$ So, hypothesis $H_{2}$ (ii) holds. See the examples that follow for functions which satisfy our hypothesis $H_{2}$ (ii) but not the $AR$ -condition.

Example 1.

The following functions satisfy hypotheses $H_{2}$ . For the sake of simplicity we drop the z-dependence: $\begin{array}{rcl} f_{1} (x, λ) = λ x^{r - 1} for all x ⩾ 0 with p < r < p^{*}, \\ f_{2} (x, λ) = ξ (λ) x^{p - 1} (ln (1 + x) + \frac{x}{p (x + 1)}) for all x ⩾ 0 \end{array}$ with $λ \mapsto ξ (λ)$ strictly increasing on $(0, + \infty)$ , ${lim}_{λ \to \infty} ξ (λ) = + \infty$ , ${lim}_{λ \to 0^{+}} ξ (λ) = 0$ .

Note that $f_{2} (\cdot, λ)$ does not satisfy the $AR$ -condition.

Let $\begin{array}{rcl} L = {λ > 0 : problem (P_{λ}) admits a positive solution}, \\ S (λ) = set of positive solutions of problem (P_{λ}) . \end{array}$

First we establish the nonemptiness and a structural property of the set $L$ of admissible parameters.

Proposition 4.

If hypotheses $H_{1}$ and $H_{2}$ hold, then $L \neq \emptyset$ , for every $u \in S (λ)$ we have $u (z) > 0$ for all $z \in Ω$ and $λ \in L$ implies $(0, λ] \subseteq L$ .

Proof.

We consider the following auxiliary Dirichlet problem $\begin{array}{rclr} - Δ_{p} e (z) = 1 in Ω, e |_{\partial Ω} = 0 . & (1) \end{array}$ Recalling that A is maximal monotone, strictly monotone and coercive (by virtue of the Poincaré inequality), we see that problem (1) has a unique solution $e \in W_{0}^{1, p} (Ω) ∖ {0}$ (see, for example, Gasinski and Papageorgiou [9, p. 319]). Acting on (1) with $- e^{-} \in W_{0}^{1, p} (Ω)$ , we obtain $\begin{array}{rcl} {∥ D e^{-} ∥}_{p}^{p} = - \int_{Ω} e^{-} d z ⩽ 0 \\ ⟹ e ⩾ 0, e \neq 0 . \end{array}$ From the nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [9, pp. 737–738]), we have $e \in int C_{+}$ . Claim 1.

There exists $\bar{λ} > 0$ such that for all $λ \in (0, \bar{λ})$ , we can find $ξ = ξ (λ) > 0$ for which we have $ξ^{q - 1} {∥ ϑ ∥}_{\infty} {∥ e ∥}_{\infty}^{q - 1} + {∥ a (\cdot, λ) ∥}_{\infty} (1 + ξ^{r - 1} {∥ e ∥}_{\infty}^{r - 1}) < ξ^{p - 1} .$

Arguing by contradiction, suppose that we can find $λ_{n} ↓ 0$ such that $ξ^{p - 1} ⩽ ξ^{q - 1} {∥ ϑ ∥}_{\infty} {∥ e ∥}_{\infty}^{q - 1} + {∥ a (\cdot, λ_{n}) ∥}_{\infty} (1 + ξ^{r - 1} {∥ e ∥}_{\infty}^{r - 1})$ for all $ξ > 0$ , all $n ⩾ 1$ .

Passing to the limit as $n \to \infty$ and using hypothesis $H_{2}$ (i), we obtain $ξ^{p - q} ⩽ {∥ ϑ ∥}_{\infty} {∥ e ∥}_{\infty}^{q - 1} for all ξ > 0,$ a contradiction. This proves the claim.

Let $\bar{u} = ξ e \in int C_{+}$ . Then we have $- Δ_{p} \bar{u} (z) = ξ^{p - 1} > ϑ (z) \bar{u} {(z)}^{q - 1} + f (z, \bar{u} (z), λ) for a.a. z \in Ω$ (2) (see Claim 1 and hypothesis $H_{2}$ (i)).

Fix $λ \in (0, \bar{λ})$ and consider the following Carathéodory function $g_{λ} (z, x) = \{\begin{matrix} 0 & if x < 0, \\ ϑ (z) x^{q - 1} + f (z, x, λ) & if 0 ⩽ x ⩽ \bar{u} (z), \\ ϑ (z) \bar{u} {(z)}^{q - 1} + f (z, \bar{u} (z), λ) & if \bar{u} (z) < x . \end{matrix}$ (3)

We set $G_{λ} (z, x) = \int_{0}^{x} g_{λ} (z, s) d s$ and consider the $C^{1}$ -functional $ψ_{λ} : W_{0}^{1, p} (Ω) \to R$ defined by $ψ_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} G_{λ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ From (3) it is clear that $ψ_{λ}$ is coercive. Also, using the Sobolev embedding theorem, we can see that $ψ_{λ}$ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find $u_{λ} \in W_{0}^{1, p} (Ω)$ such that $\begin{array}{rclr} ψ_{λ} (u_{λ}) = inf [ψ_{λ} (u) : u \in W_{0}^{1, p} (Ω)] . & (4) \end{array}$ Let $u \in C_{+} ∖ {0}$ with $supp u \subseteq Ω_{+}$ . Recall that $\bar{u} \in int C_{+}$ . Hence, we have that $\bar{u} |_{{\bar{Ω}}_{+}} > 0$ (see hypothesis $H_{1}$ ). Therefore, we can find $t \in (0, 1)$ small such that $t u ⩽ \bar{u}$ . So, using hypothesis $H_{2}$ (iii) and (3), we have $ψ_{λ} (t u) ⩽ \frac{t^{p}}{p} {∥ D u ∥}_{p}^{p} - \frac{t^{q}}{q} \int_{Ω} ϑ (z) u^{q} d z .$ Since $supp u \subseteq Ω_{+}$ and $q < p$ , choosing $t \in (0, 1)$ even smaller if necessary, we infer that $\begin{array}{rcl} ψ_{λ} (t u) < 0 \\ ⟹ ψ_{λ} (u_{λ}) < 0 = ψ_{λ} (0) (see (4)), hence u_{λ} \neq 0 . \end{array}$ From (4), we have $\begin{array}{rclr} ψ_{λ}^{'} (u_{λ}) = 0 \\ ⟹ A (u_{λ}) = N_{g_{λ}} (u_{λ}) . & (5) \end{array}$ On (5) first we act with $- u_{λ}^{-} \in W_{0}^{1, p} (Ω)$ and obtain $\begin{array}{rcl} {∥ D u_{λ}^{-} ∥}_{p}^{p} = 0 (see (3)) \\ ⟹ u_{λ} ⩾ 0, u_{λ} \neq 0 . \end{array}$ Also, on (5) we act with ${(u_{λ} - \bar{u})}^{+} \in W_{0}^{1, p} (Ω)$ and obtain $\begin{array}{rcl} ⟨ A (u_{λ}), {(u_{λ} - \bar{u})}^{+} ⟩ = \int_{Ω} g_{λ} (z, u_{λ}) {(u_{λ} - \bar{u})}^{+} d z \\ = \int_{Ω} [ϑ (z) {\bar{u}}^{q - 1} + f (z, \bar{u}, λ)] {(u_{λ} - \bar{u})}^{+} d z (see (3)) \\ ⩽ \int_{Ω} ξ^{p - 1} {(u_{λ} - \bar{u})}^{+} d z (see Claim 1) \\ {= ⟨ A (\bar{u}), (u_{λ} - \bar{u})}^{+} ⟩ (see (1)) \\ ⟹ \int_{{u_{λ} > \bar{u}}} {(| D u_{λ} |^{p - 2} D u_{λ} - | D \bar{u} |^{p - 2} D \bar{u}, D u_{λ} - D \bar{u})}_{R^{N}} d z ⩽ 0 \\ ⟹ {| {u_{λ} > \bar{u}} |}_{N} = 0, hence u_{λ} ⩽ \bar{u} . \end{array}$ So, we have proved that $u_{λ} \in [0, \bar{u}] = {u \in W_{0}^{1, p} (Ω) : 0 ⩽ u (z) ⩽ \bar{u} (z) for a.a. z \in Ω} .$ This fact and (3) imply that $u_{λ} \in S (λ)$ . The nonlinear regularity theory (see Lieberman [13]), implies that $u_{λ} \in C_{+} ∖ {0}$ . Invoking Harnack’s inequality (see Pucci and Serrin [19, p. 163]), we infer that $u_{λ} (z) > 0$ for all $z \in Ω$ .

Now let $λ \in L$ and let $μ \in (0, λ)$ , $u_{λ} \in S (λ)$ . We have $\begin{array}{rclr} - Δ_{p} u_{λ} (z) & = & ϑ (z) u_{λ} {(z)}^{q - 1} + f (z, u_{λ} (z), λ) \\ ⩾ & ϑ (z) u_{λ} {(z)}^{q - 1} + f (z, u_{λ} (z), μ) for μ a.a. z \in Ω & (6) \\ (see hypothesis H_{2} (iii)) . \end{array}$ We introduce the Carathéodory function $γ_{μ} (z, x)$ defined by $\begin{array}{rclr} γ_{μ} (z, x) = \{\begin{matrix} 0 & if x < 0, \\ ϑ (z) x^{q - 1} + f (z, x, μ) & if 0 ⩽ x ⩽ u_{λ} (z), \\ ϑ (z) u_{λ} (z) + f (z, u_{λ} (z), μ) & if u_{λ} (z) < x . \end{matrix} & (7) \end{array}$ We set $Γ_{μ} (z, x) = \int_{0}^{x} γ_{μ} (z, s) d s$ and consider the $C^{1}$ -functional $τ_{μ} : W_{0}^{1, p} (Ω) \to R$ defined by $τ_{μ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} Γ_{μ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ From (7) it is clear that $τ_{μ}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_{μ} \in W_{0}^{1, p} (Ω)$ such that $τ_{μ} (u_{μ}) = inf [τ_{μ} (u) : u \in W_{0}^{1, p} (Ω)] .$ As before, we can show that $τ_{μ} (u_{μ}) < 0 = τ_{μ} (0), hence u_{μ} \neq 0 .$ Also, we have $\begin{array}{rcl} τ_{μ}^{'} (u_{μ}) = 0 \\ ⟹ A (u_{μ}) = N_{γ_{μ}} (u_{μ}) . \end{array}$ Acting with $- u_{μ}^{-} \in W_{0}^{1, p} (Ω)$ and with ${(u_{μ} - u_{λ})}^{+} \in W_{0}^{1, p} (Ω)$ , as before, we show that $\begin{array}{rcl} u_{μ} \in [0, u_{λ}] = {u \in W_{0}^{1, p} (Ω) : 0 ⩽ u (z) ⩽ u_{λ} (z) for a.a. z \in Ω} \\ ⟹ u_{μ} \in S (μ) (see (7)) \\ ⟹ μ \in L and so (0, λ] \subseteq L . \end{array}$ This completes the proof. □

Let $λ^{*} = sup L$ .

Proposition 5.

If hypotheses $H_{1}$ and $H_{2}$ hold, then $λ^{*} < \infty$ .

Proof.

Let ${\hat{λ}}_{1}^{+} = {\hat{λ}}_{1} (Ω_{+})$ (see Section 2) and take $β > {\hat{λ}}_{1}^{+}$ . Claim 2.

There exists $λ_{0} > 0$ big such that $ϑ (z) x^{q - 1} + f (z, x, λ) ⩾ β x^{p - 1} for a.a. z \in Ω_{+}, all x ⩾ 0, all λ ⩾ λ_{0} .$

Since $q < p$ , we can find $δ > 0$ small such that $\begin{array}{rclr} ϑ (z) x^{q - 1} ⩾ β x^{p - 1} for a.a. z \in Ω_{+}, all 0 ⩽ x ⩽ δ (see hypothesis H_{1}) . & (8) \end{array}$ Note that hypothesis $H_{2}$ (ii) implies that for all $λ > 0$ , we have $lim_{x \to + \infty} \frac{f (z, x, λ)}{x^{p - 1}} = + \infty uniformly for a.a. z \in Ω .$ So, we can find $M = M (λ) > 0$ such that $\begin{array}{rclr} f (z, x, λ) ⩾ β x^{p - 1} for a.a. z \in Ω all x ⩾ M . & (9) \end{array}$ Let $δ > 0$ be as in (8) and $λ > 0$ . By virtue of hypothesis $H_{2}$ (iii), we have $inf [f (z, x, λ) : x ⩾ δ] = m_{δ} (λ) > 0,$ with $λ \mapsto m_{δ} (λ)$ nondecreasing and $m_{δ} (λ) \to + \infty$ as $λ \to + \infty$ . So, we can choose $λ_{0} ⩾ 1$ big such that $\begin{array}{rclr} m_{δ} (λ) ⩾ β M^{p - 1} for all λ ⩾ λ_{0} \\ ⟹ f (z, x, λ) ⩾ β x^{p - 1} for a.a. z \in Ω, all δ ⩽ x ⩽ M, all λ ⩾ λ_{0} . & (10) \end{array}$ Combining (8), (9), (10), we conclude that Claim 2 holds.

Take $λ > λ_{0}$ and assume that $λ \in L$ . Then we can find $u_{λ} \in S (λ)$ such that $- Δ_{p} u_{λ} (z) = ϑ (z) u_{λ} {{(z)}^{q - 1} + f (z, u_{λ} (z), λ) ⩾ β u_{λ} (z)}^{q - 1} for a.a. z \in Ω_{+}$ (see Claim 2).

Let ${\hat{u}}_{+} \in int C_{+} (Ω_{+})$ be the principal, $L^{p}$ -normalized (that is, ${∥ {\hat{u}}_{+} ∥}_{L^{p} (Ω_{+})} = 1$ ) positive eigenfunction for $(- Δ_{p}, W_{0}^{1, p} (Ω_{+}))$ . From Proposition 4, we know that $u_{λ} |_{{\bar{Ω}}_{+}} > 0 .$ So, we can find $t \in (0, 1)$ small such that $\begin{array}{rclr} t {\hat{u}}_{+} (z) ⩽ u_{λ} (z) for all z \in {\bar{Ω}}_{+} . & (11) \end{array}$ We have $\begin{array}{rclr} - Δ_{p} (t {\hat{u}}_{+}) (z) = {\hat{λ}}_{1}^{+} (t {\hat{u}}_{+}) {{(z)}^{p - 1} < β (t {\hat{u}}_{+}) (z)}^{p - 1} for a.a. z \in Ω_{+} & (12) \\ (recall that {\hat{λ}}_{1}^{+} < β) . \end{array}$ We consider the following Carathéodory function $k_{β} (z, x) = \{\begin{matrix} β (t {\hat{u}}_{+}) {(z)}^{p - 1} & if x < t {\hat{u}}_{+} (z), \\ β x^{p - 1} & if t {\hat{u}}_{+} (z) ⩽ x ⩽ u_{λ} (z), (z, x) \in Ω_{+} \times R, \\ β u_{λ} {(z)}^{p - 1} & if u_{λ} (z) < x \end{matrix}$ (13) (see (11)).

We set $K_{β} (z, x) = \int_{0}^{x} k_{β} (z, s) d s$ and consider the $C^{1}$ -functional $χ_{β} : W_{0}^{1, p} (Ω_{+}) \to R$ defined by $χ_{β} (u) = \frac{1}{p} {∥ D u ∥}_{L^{p} (Ω_{+}, R^{N})} - \int_{Ω_{+}} K_{β} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω_{+}) .$ From (13) it is clear that $χ_{β}$ is coercive and also it is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find ${\tilde{u}}_{β} \in W_{0}^{1, p} (Ω_{+})$ such that $\begin{array}{rclr} χ_{β} ({\tilde{u}}_{β}) = inf [χ_{β} (u) : u \in W_{0}^{1, p} (Ω_{+})] \\ ⟹ χ_{β}^{'} ({\tilde{u}}_{β}) = 0 \\ ⟹ \hat{A} ({\tilde{u}}_{β}) = N_{k_{β}} ({\tilde{u}}_{β}) (with \hat{A} = A |_{W_{0}^{1, p} (Ω_{+})}) . & (14) \end{array}$ On (14), first we act with ${(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} \in W_{0}^{1, p} (Ω_{+})$ . Then $\begin{array}{rcl} ⟨ A ({\tilde{u}}_{β}), {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} ⟩ = \int_{Ω_{+}} k_{β} (z, {\tilde{u}}_{β}) {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} d z \\ {= \int_{Ω_{+}} β (t {\hat{u}}_{+})}^{p - 1} {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} d z \\ {⩾ \int_{Ω_{+}} {\hat{λ}}_{1}^{+} (t {\hat{u}}_{+})}^{p - 1} {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} d z (see (12)) \\ = ⟨ \hat{A} (t {\hat{u}}_{+}), {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} ⟩ \\ ⟹ ⟨ \hat{A} (t {\hat{u}}_{+}) - \hat{A} ({\tilde{u}}_{β}), {(t {\hat{u}}_{+} - {\tilde{u}}_{β})}^{+} ⟩ ⩽ 0 \\ ⟹ {| {t {\hat{u}}_{+} > {\tilde{u}}_{β}}_{Ω_{+}} |}_{N} = 0, hence t {\hat{u}}_{+} ⩽ {\tilde{u}}_{β} in Ω_{+} . \end{array}$ Next, on (14) we act with ${({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} \in W_{0}^{1, p} (Ω_{+})$ where ${\tilde{u}}_{λ} = u_{λ} |_{Ω_{+}}$ (recall that $u_{λ} |_{{\bar{Ω}}_{+}} > 0$ ). We have $\begin{array}{rcl} ⟨ \hat{A} ({\tilde{u}}_{β}), {({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} ⟩ \\ = \int_{Ω_{+}} k_{β} (z, {\tilde{u}}_{β}) {({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} d z \\ = \int_{Ω_{+}} β {\tilde{u}}_{λ}^{p - 1} {({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} d z \\ ⩽ \int_{Ω_{+}} [ϑ (z) {\tilde{u}}_{λ}^{q - 1} + f (z, {\tilde{u}}_{λ}, λ)] {({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} d z (see Claim 2) \\ = ⟨ \hat{A} ({\tilde{u}}_{β}) - \hat{A} ({\tilde{u}}_{λ}), {({\tilde{u}}_{β} - {\tilde{u}}_{λ})}^{+} ⟩ (since u_{λ} \in S (λ)) \\ ⟹ {| {{\tilde{u}}_{β} > {\tilde{u}}_{λ}}_{Ω_{+}} |}_{N} = 0, hence {\tilde{u}}_{β} ⩽ {\tilde{u}}_{λ} in Ω_{+} . \end{array}$ So, finally we have $t {\hat{u}}_{+} (z) ⩽ {\tilde{u}}_{β} (z) ⩽ u_{λ} (z) for all z \in {\bar{Ω}}_{+} .$ From (13) and (14) it follows that $- Δ_{p} {\tilde{u}}_{β} (z) = β {\tilde{u}}_{β} {(z)}^{p - 1} for a.a. z \in Ω_{+}, {\tilde{u}}_{β} |_{Ω_{+}} = 0, {\tilde{u}}_{β} ⩾ 0,$ a contradiction since $β > {\hat{λ}}_{1}^{+}$ (recall that every nonprincipal eigenvalue of $(- Δ_{p}, W_{0}^{1, p} (Ω_{+}))$ ), has nodal (that is, sign changing) eigenfunctions, see [9].

This means that $λ^{*} ⩽ λ_{0} < \infty$ . □

In what follows, for every $λ > 0$ , by $φ_{λ} : W_{0}^{1, p} (Ω) \to R$ we denote the energy functional for problem ( P λ ) defined by $φ_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \frac{1}{q} \int_{Ω} ϑ (z) u^{+} {(z)}^{q} d z - \int_{Ω} F (z, u (z), λ) d z$ for all $u \in W_{0}^{1, p} (Ω)$ .

Evidently $φ_{λ} \in C^{1} (W_{0}^{1, p} (Ω))$ .

Proposition 6.

If hypotheses $H_{1}$ and $H_{2}$ hold, then $λ^{*} \in L$ .

Proof.

Let ${λ_{n}}_{n ⩾ 1} \subseteq L$ such that $λ_{n} \to {(λ^{*})}^{-}$ as $n \to \infty$ and for every $n ⩾ 1$ , let $u_{n} \in S (λ_{n})$ . We may assume that $\begin{array}{rclr} φ_{λ_{n}} (u_{n}) < 0 for all n ⩾ 1 . & (15) \end{array}$ Indeed, if $λ < \tilde{λ} < λ^{*}$ and $\tilde{u} \in S (\tilde{λ})$ , then by virtue of hypothesis $H_{2}$ (iii), we have $\begin{array}{rclr} - Δ_{p} \tilde{u} (z) & = & ϑ (z) \tilde{u} {(z)}^{q - 1} + f (z, \tilde{u} (z), \tilde{λ}) \\ ⩾ & ϑ (z) \tilde{u} {(z)}^{q - 1} + f (z, \tilde{u} (z), λ) for a.a. z \in Ω & (16) \end{array}$ (see hypothesis $H_{2}$ (iii)).

Then reasoning as in the proof of Proposition 4, we introduce the following truncation of the reaction of problem ( P λ ): $w_{λ} (z, x) = \{\begin{matrix} 0 & if x < 0, \\ ϑ (z) x^{q - 1} + f (z, x, λ) & if 0 ⩽ x ⩽ \tilde{u} (z), \\ ϑ (z) \tilde{u} {(z)}^{q - 1} + f (z, \tilde{u} (z), λ) & if \tilde{u} (z) < x . \end{matrix}$ (17) This is a Carathéodory function. We set $W_{λ} (z, x) = \int_{0}^{x} w_{λ} (z, s) d s$ and consider the $C^{1}$ -functional ${\hat{ψ}}_{λ} : W_{0}^{1, p} (Ω) \to R$ defined by ${\hat{ψ}}_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} W_{λ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ Again, ${\hat{ψ}}_{λ}$ is coercive (see (17)) and sequentially weakly lower semicontinuous. So, we can find $u_{λ} \in W_{0}^{1, p} (Ω)$ such that $\begin{array}{rclr} {\hat{ψ}}_{λ} (u_{λ}) = inf [\hat{ψ_{λ}} (u) : u \in W_{0}^{1, p} (Ω)] . & (18) \end{array}$ As in the proof of Proposition 4, we show that ${\hat{ψ}}_{λ} (u_{λ}) < 0 = {\hat{ψ}}_{λ} (0), hence u_{λ} \neq 0 .$ From (18), we have $\begin{array}{rclr} {\hat{ψ}}_{λ}^{'} (u_{λ}) = 0 \\ ⟹ A (u_{λ}) = N_{w_{λ}} (u_{λ}) . & (19) \end{array}$ On (19) first we act with $- u_{λ}^{-} \in W_{0}^{1, p} (Ω)$ and obtain $u_{λ} ⩾ 0$ , $u_{λ} \neq 0$ (see (17)). Then we act with ${(u_{λ} - \tilde{u})}^{+} \in W_{0}^{1, p} (Ω)$ . We obtain $\begin{array}{rcl} ⟨ A (u_{λ}), {(u_{λ} - \tilde{u})}^{+} ⟩ \\ = \int_{Ω_{+}} w_{λ} (z, u_{λ}) {(u_{λ} - \tilde{u})}^{+} d z \\ = \int_{Ω_{+}} [ϑ (z) {\tilde{u}}^{q - 1} + f (z, \tilde{u}, λ)] {(u_{λ} - \tilde{u})}^{+} d z (see (17)) \\ ⩽ ⟨ A (\tilde{u}), {(u_{λ} - \tilde{u})}^{+} ⟩ (see (16)) \\ ⟹ ⟨ A (u_{λ}) - A (\tilde{u}), {(u_{λ} - \tilde{u})}^{+} ⟩ ⩽ 0 \\ ⟹ {| {u_{λ} > \tilde{u}} |}_{N} = 0, hence u_{λ} ⩽ \tilde{u} . \end{array}$ So, we have $\begin{array}{rcl} u_{λ} \in [0, \tilde{u}] = {u \in W_{0}^{1, p} (Ω) : 0 ⩽ u (z) ⩽ \tilde{u} (z) for a.a. z \in Ω} \\ ⟹ u_{λ} \in S (λ) (see (17)) and φ_{λ} (u_{λ}) < 0 (since φ_{λ} |_{[0, \tilde{u}]} = {\hat{ψ}}_{λ} |_{[0, \tilde{u}]}) . \end{array}$ This proves that we can always assume that (15) holds.

From (15) we have $\begin{array}{rclr} {∥ D u_{n} ∥}_{p}^{p} - \frac{p}{q} \int_{Ω} ϑ (z) u_{n}^{q} d z - \int_{Ω} p F (z, u_{n}, λ_{n}) d z < 0 for all n ⩾ 1 . & (20) \end{array}$ Also, since $u_{n} \in S (λ_{n})$ for all $n ⩾ 1$ , we have $\begin{array}{rclr} A (u_{n}) = ϑ (z) u_{n}^{q - 1} + N_{f_{λ_{n}}} (u_{n}), n ⩾ 1, & (21) \end{array}$ where $f_{λ_{n}} (z, x) = f (z, x, λ_{n})$ . On (21) we act with $u_{n} \in W_{0}^{1, p} (Ω)$ and obtain $\begin{array}{rclr} - {∥ D u_{n} ∥}_{p}^{p} + \int_{Ω} ϑ (z) u_{n}^{q} d z + \int_{Ω} f (z, u_{n}, λ_{n}) u_{n} d z = 0 for all n ⩾ 1 . & (22) \end{array}$ Adding (20) and (22), we obtain $\begin{array}{rclr} \int_{Ω} [f (z, u_{n}, λ_{n}) u_{n} - p F (z, u_{n}, λ_{n})] d z ⩽ (\frac{p}{q} - 1) \int_{Ω} ϑ (z) u_{n}^{q} d z for all n ⩾ 1 . & (23) \end{array}$ It is clear that in hypothesis $H_{2}$ (i) without any loss of generality, we may assume that $λ \to r (λ)$ and $λ \to {∥ a (\cdot, λ) ∥}_{\infty}$ are both nondecreasing in $(0, + \infty)$ . Then hypotheses $H_{2}$ (i)(ii), imply that we find $c_{2}, c_{3} > 0$ such that $\begin{array}{rclr} c_{3} x^{τ} - c_{4} ⩽ f (z, x, λ_{n}) x - p F (z, x, λ_{n}) for a.a. z \in Ω, all x ⩾ 0, all n ⩾ 1 . & (24) \end{array}$ Using (24) in (23) and recalling that $τ > q$ (see hypothesis $H_{2}$ (ii)), we infer that $\begin{array}{rclr} {u_{n}}_{n ⩾ 1} \subseteq L^{τ} (Ω) is bounded . & (25) \end{array}$ It is clear from hypothesis $H_{2}$ (iii) that without any loss of generality, we may assume that $τ < r_{1} = r (λ_{1}) ⩽ r (λ_{n})$ for all $n ⩾ 1$ (see hypothesis $H_{1}$ (i)). Suppose $N \neq p$ . Then we can find $t \in (0, 1)$ such that $\begin{array}{rclr} \frac{1}{r_{1}} = \frac{1 - t}{τ} + \frac{t}{p^{*}} . & (26) \end{array}$ Invoking the interpolation inequality (see, for example, Gasinski and Papageorgiou [9, p. 905]), we have $\begin{array}{rclr} {∥ u_{n} ∥}_{r} ⩽ {∥ u_{n} ∥}_{τ}^{1 - t} {∥ u_{n} ∥}_{p^{*}}^{t} for all n ⩾ 1 \\ ⟹ {∥ u_{n} ∥}_{r}^{r} ⩽ M_{1} {∥ u_{n} ∥}^{t r} for some M_{1} > 0, all n ⩾ 1 & (27) \end{array}$ (see (25) and use the Sobolev embedding theorem).

From hypothesis $H_{2}$ (i), we have $f (z, x, λ_{n}) x ⩽ c_{4} (1 + x^{r}) for a.a. z \in Ω, all x ⩾ 0, some c_{4} > 0 .$ From (22), (25) and since $τ > q$ (see $H_{2}$ (ii)) and $ϑ \in L^{\infty} (Ω)$ (see $H_{1}$ ) $\begin{array}{rclr} {∥ D u_{n} ∥}_{p}^{p} & ⩽ & c_{5} (1 + {∥ u_{n} ∥}_{r}^{r}) for some c_{5} > 0, all n ⩾ 1 \\ ⩽ & c_{6} (1 + {∥ u_{n} ∥}^{t r}) for some c_{6} > 0, all n ⩾ 1 (see (27)) . & (28) \end{array}$ From (26) and our hypothesis on τ (see $H_{2}$ (iii)), it follows that $t r < p$ . Hence from (28) we infer that ${u_{n}}_{n ⩾ 1} \subseteq W_{0}^{1, p} (Ω) is bounded .$ If $N = p$ , then since $p^{*} = + \infty$ and $W_{0}^{1, p} (Ω)$ is compactly embedded into $L^{q} (Ω)$ for all $q \in [1, \infty)$ , then the above argument works, if we replace $p^{*}$ by $η > r > p$ big. Then again we reach the same conclusion.

So, we may assume that $\begin{array}{rclr} u_{n} \overset{w}{\to} u in W_{0}^{1, p} (Ω) and u_{n} \to u in L^{r_{*}} (Ω) (r_{*} = r (λ^{*})) as n \to \infty . & (29) \end{array}$ On (21) we act with $u_{n} - u \in W_{0}^{1, p} (Ω)$ , pass to the limit as $n \to \infty$ and use (29). Then $\begin{array}{rclr} lim_{n \to \infty} ⟨ A (u_{n}), u_{n} - u ⟩ = 0 \\ ⟹ u_{n} \to u in W_{0}^{1, p} (Ω) as n \to \infty & (30) \end{array}$ (see Proposition 3).

Passing to the limit as $n \to \infty$ in (21) and using (30), we obtain $\begin{array}{rclr} A (u) = ϑ (z) u^{q - 1} + N_{f_{λ^{*}}} (u) . & (31) \end{array}$ We need to show that $u \neq 0$ , because then $u \in S (λ^{*})$ , that is $λ^{*} \in L$ . To this end, we consider the following auxiliary Dirichlet problem $\begin{array}{rclr} - Δ_{p} u (z) = ϑ (z) u {(z)}^{q - 1} in Ω_{+}, u |_{\partial Ω_{+}} = 0, u > 0 . & (32) \end{array}$ Since $q < p$ , a straightforward application of the direct method (as before), establishes that problem (32) admits a nontrivial solution $\bar{u} \in W_{0}^{1, p} (Ω)$ , $\bar{u} ⩾ 0$ . The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [9, pp. 737–738]), imply that $\bar{u} \in int C_{+}$ . Moreover, Theorem 2 of Diaz and Saa [7], implies that $\bar{u} \in int C_{+}$ is the unique positive solution of (32).

Now, let $λ \in L$ and $u_{λ} \in S (λ)$ . We introduce the following Carathéodory function $\begin{array}{rclr} j (z, x) = \{\begin{matrix} 0 & if x < 0, \\ ϑ (z) x^{q - 1} & if 0 ⩽ x ⩽ u_{λ} (z), (z, x) \in Ω_{+} \times R, \\ ϑ (z) u_{λ} {(z)}^{q - 1} & if u_{λ} (z) < x . \end{matrix} & (33) \end{array}$ Let $J (z, x) = \int_{0}^{x} j (z, s) d s$ and consider the $C^{1}$ -functional $J : W_{0}^{1, p} (Ω_{+}) \to R$ defined by $J (u) = \frac{1}{p} {∥ D u ∥}_{L^{p} (Ω_{+}, R^{N})}^{p} - \int_{Ω_{+}} J (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω_{+}) .$ As before, (33) implies that $J (\cdot)$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $\tilde{u} \in W^{1, p} (Ω_{+})$ such that $\begin{array}{rclr} J (\tilde{u}) = inf [J (u) : u \in W_{0}^{1, p} (Ω_{+})] . & (34) \end{array}$ Since $q < p$ and $u_{λ} |_{{\bar{Ω}}_{+}} > 0$ (recall $u_{λ} \in int C_{+}$ ), as in previous similar cases, we have $J (\tilde{u}) < 0 = J (0), hence \tilde{u} \neq 0 .$ From (34), we have $\begin{array}{rclr} J^{'} (\tilde{u}) = 0, \\ ⟹ A (\tilde{u}) = N_{j} (\tilde{u}) . & (35) \end{array}$ On (35) we act with $- {\tilde{u}}^{-} \in W_{0}^{1, p} (Ω_{+})$ and with ${(\tilde{u} - u_{λ})}^{+} \in W_{0}^{1, p} (Ω_{+})$ (here we use the fact that $u_{λ} |_{{\bar{Ω}}_{+}} > 0$ ) and we obtain $\begin{array}{rcl} 0 < \tilde{u} (z) ⩽ u_{λ} (z) for all z \in {\bar{Ω}}_{+}, \tilde{u} \neq 0 \\ ⟹ \tilde{u} is a positive solution of (32) \\ ⟹ \tilde{u} = \bar{u} (recall that \bar{u} is the unique positive solution of (32)) \\ ⟹ \bar{u} ⩽ u_{λ} in Ω_{+} . \end{array}$ So, we have $\begin{array}{rcl} \bar{u} (z) ⩽ u_{n} (z) for all z \in {\bar{Ω}}_{+}, all n ⩾ 1 \\ ⟹ \bar{u} (z) ⩽ u (z) for all z \in Ω_{+} (see (30)) \\ ⟹ u \neq 0 . \end{array}$ Therefore, $u \in S (λ^{*})$ and so $λ^{*} \in L$ . □

Next, we look for additional positive solutions for problem ( P λ ). To this end, we consider the following auxiliary Dirichlet problem: $\{\begin{matrix} - Δ_{p} u (z) = ϑ^{+} (z) u {(z)}^{q - 1} + f (z, u (z), λ) & in Ω, \\ u = 0 & on \partial Ω, \\ u > 0 & in Ω . \end{matrix}$ (Au_λ) Reasoning as in the proofs of Propositions 4, 5 and 6 (with $Ω_{+}$ replaced by Ω) we obtain the following proposition.

Proposition 7.

If hypotheses $H_{1}$ and $H_{2}$ hold, then there exists $λ_{0}^{*} \in (0, λ^{*}]$ such that for all $λ \in (0, λ_{0}^{*}]$ problem ( Au λ ) has at least one positive solution ${\tilde{u}}_{λ} \in int C_{+}$ .

Remark 2.

Note that in this case, the solution ${\tilde{u}}_{λ} \in C_{+} ∖ {0}$ satisfies $\begin{array}{rcl} - Δ_{p} {\tilde{u}}_{λ} (z) = ϑ^{+} (z) {\tilde{u}}_{λ} {(z)}^{q - 1} + f (z, {\tilde{u}}_{λ} (z), λ) ⩾ 0 for a.a. z \in Ω (see hypotheses H_{2} (iii)) \\ ⟹ {\tilde{u}}_{λ} \in int C_{+} (see Gasinski and Papageorgiou [9, p. 738]) . \end{array}$

We can use Proposition 7, to produce a multiplicity result for the positive solutions of problem ( P λ ) with $λ \in (0, λ_{0}^{*}]$ .

Proposition 8.

If hypotheses $H_{1}$ and $H_{2}$ hold and $λ \in (0, λ_{0}^{*}]$ , then problem ( P λ ) has at least two positive solutions $\begin{array}{rcl} u_{λ}, {\hat{u}}_{λ} \in C_{+} ∖ {0}, u_{λ} \neq {\hat{u}}_{λ}, \\ 0 < u_{λ} (z) ⩽ {\hat{u}}_{λ} (z) for all z \in Ω . \end{array}$

Proof.

Let $μ \in (λ, λ_{0}^{*})$ . From Proposition 4, we know that $λ, μ \in L$ and we can find $u_{λ} \in S (λ) \subseteq C_{+}$ and ${\hat{u}}_{λ} \in int C_{+}$ solution of $(A u_{μ})$ . We claim that we can have $\begin{array}{rclr} u_{λ} ⩽ {\tilde{u}}_{μ} . & (36) \end{array}$ Indeed note that $\begin{array}{rclr} - Δ_{p} {\tilde{u}}_{μ} (z) & = & ϑ^{+} (z) {\tilde{u}}_{μ} {(z)}^{q - 1} + f (z, {\tilde{u}}_{μ} (z), μ) \\ ⩾ & ϑ (z) {\tilde{u}}_{μ} {(z)}^{q - 1} + f (z, {\tilde{u}}_{μ} (z), λ) for a.a. z \in Ω & (37) \end{array}$ (see hypothesis $H_{2}$ (iii)).

We consider the following truncation of the reaction of problem ( P λ ). ${\hat{g}}_{λ} (z, x) = \{\begin{matrix} 0 & if x < 0, \\ ϑ (z) x^{q - 1} + f (z, x, λ) & if 0 ⩽ x ⩽ {\tilde{u}}_{μ} (z), \\ ϑ (z) {\tilde{u}}_{μ} {(z)}^{q - 1} + f (z, {\tilde{u}}_{μ} (z), λ) & if {\tilde{u}}_{μ} (z) < x . \end{matrix}$ (38) This is a Carathéodory function.

We set ${\hat{G}}_{λ} (z, x) = \int_{0}^{x} {\hat{g}}_{λ} (z, s) d s$ and introduce the $C^{1}$ -functional ${\hat{ψ}}_{λ} : W_{0}^{1, p} (Ω) \to R$ defined by ${\hat{ψ}}_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} {\hat{G}}_{λ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ Evidently ${\hat{ψ}}_{λ}$ is coercive (see (38)) and sequentially weakly lower semicontinuous. So, we can find $u_{λ} \in W_{0}^{1, p} (Ω)$ such that $\begin{array}{rclr} {\hat{ψ}}_{λ} (u_{λ}) = inf [{\hat{ψ}}_{λ} (u) : u \in W_{0}^{1, p} (Ω)] . & (39) \end{array}$ Since $q < p$ , as before (see, for example, the proof of Proposition 4), we have ${\hat{ψ}}_{λ} (u_{λ}) < 0 = {\hat{ψ}}_{λ} (0), hence u_{λ} \neq 0 .$ From (39) we have $\begin{array}{rclr} {\hat{ψ}}_{λ}^{'} (u_{λ}) = 0 \\ ⟹ A (u_{λ}) = N_{{\hat{g}}_{λ}} (u_{λ}) . & (40) \end{array}$ On (40) we act with $- u_{λ}^{-} \in W_{0}^{1, p} (Ω)$ and with ${(u_{λ} - {\tilde{u}}_{μ})}^{+} \in W_{0}^{1, p} (Ω)$ . As in the proof of Proposition 4, using this time (37), we show that $u_{λ} \in [0, {\tilde{u}}_{μ}] = {u \in W_{0}^{1, p} (Ω) : 0 ⩽ u (z) ⩽ {\tilde{u}}_{μ} (z) for a.a. z \in Ω} .$ Hence $u_{λ} \in S (λ) \subseteq C_{+}$ (see (38)) and $u_{λ} (z) > 0$ for all $z \in Ω$ (by Harnack’s inequality, see Pucci and Serrin [19, p. 163]). Therefore (36) holds.

We introduce the following truncation of the reaction of problem ( P λ ): $k_{λ} (z, x) = \{\begin{matrix} ϑ (z) u_{λ} {(z)}^{q - 1} + f (z, u_{λ} (z), λ) & if x ⩽ u_{λ} (z), \\ ϑ (z) x^{q - 1} + f (z, x, λ) & if u_{λ} (z) < x . \end{matrix}$ (41) This is a Carathéodory function.

We set $K_{λ} (z, x) = \int_{0}^{x} k_{λ} (z, s) d s$ and consider the $C^{1}$ -functional $τ_{λ} : W_{0}^{1, p} (Ω) \to R$ defined by $τ_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} K_{λ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ From the proof of Proposition 6, we know that $\begin{array}{rclr} τ_{λ} satisfies the C -condition . & (42) \end{array}$ We truncate $k_{λ} (z, \cdot)$ as follows: ${\hat{k}}_{λ} (z, x) = \{\begin{matrix} k_{λ} (z, x) & if x ⩽ {\tilde{u}}_{μ} (z), \\ k_{λ} (z, {\tilde{u}}_{μ} (z)) & if {\tilde{u}}_{μ} (z) < x . \end{matrix}$ (43) This too is a Carathéodory function. We set ${\hat{K}}_{λ} (z, x) = \int_{0}^{x} {\hat{k}}_{λ} (z, s) d s$ and consider the $C^{1}$ -functional ${\hat{τ}}_{λ} : W_{0}^{1, p} (Ω) \to R$ defined by ${\hat{τ}}_{λ} (u) = \frac{1}{p} {∥ D u ∥}_{p}^{p} - \int_{Ω} {\hat{K}}_{λ} (z, u (z)) d z for all u \in W_{0}^{1, p} (Ω) .$ From (43) it is clear that ${\hat{τ}}_{λ}$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ${\bar{u}}_{λ} \in W_{0}^{1, p} (Ω)$ such that $\begin{array}{rclr} {\hat{τ}}_{λ} ({\bar{u}}_{λ}) = inf [{\hat{τ}}_{λ} (u) : u \in W_{0}^{1, p} (Ω)] \\ ⟹ {\hat{τ}}_{λ}^{'} ({\bar{u}}_{λ}) = 0 \\ ⟹ A ({\bar{u}}_{λ}) = N_{{\hat{k}}_{λ}} ({\bar{u}}_{λ}) . & (44) \end{array}$ On (44) we act, first with $(u_{λ} - {\bar{u}}_{λ}) \in W_{0}^{1, p} (Ω)$ and then with ${({\bar{u}}_{λ} - {\tilde{u}}_{μ})}^{+} \in W_{0}^{1, p} (Ω)$ . As in the proof of Proposition 5, we show that ${\bar{u}}_{λ} \in [u_{λ}, {\tilde{u}}_{μ}] = {u \in W_{0}^{1, p} (Ω) : u_{λ} (z) ⩽ u (z) ⩽ {\tilde{u}}_{μ} (z) for a.a. z \in Ω} .$ If ${\bar{u}}_{λ} \neq u_{λ}$ , then this is the desired second positive solution of problem ( P λ ) and $u_{λ} ⩽ {\bar{u}}_{λ} = {\hat{u}}_{λ}$ .

So, we may assume that ${\bar{u}}_{λ} = u_{λ}$ . Let $ρ = {∥ {\tilde{u}}_{μ} ∥}_{\infty}$ (recall that $\tilde{u} \in int C_{+}$ , see Proposition 7) and let $ξ_{ρ} = ξ_{ρ} (μ) > 0$ be as postulated by hypothesis $H_{2}$ (iv) for the perturbation $f (z, \cdot, μ)$ . We have $\begin{array}{rclr} - Δ_{p} u_{λ} (z) + ξ_{ρ} u_{λ} {(z)}^{p - 1} \\ = ϑ (z) u_{λ} {{(z)}^{q - 1} + f (z, u_{λ} (z), λ) + ξ_{ρ} u_{λ} (z)}^{p - 1} \\ = ϑ (z) u_{λ} {{(z)}^{q - 1} + f (z, u_{λ} (z), μ) + ξ_{ρ} u_{λ} (z)}^{p - 1} + [f (z, u_{λ} (z), λ) - f (z, u_{λ} (z), μ)] \\ ⩽ ϑ^{+} (z) u_{λ} {{(z)}^{q - 1} + f (z, u_{λ} (z), μ) + ξ_{ρ} u_{λ} (z)}^{p - 1} - δ_{(λ^{'}, λ)} (z) \\ with δ_{(λ^{'}, λ)} (z) = f (z, u_{λ} (z), μ) - f (z, u_{λ} (z), λ) \\ ⩽ ϑ^{+} (z) {\tilde{u}}_{μ} (z) + f (z, {\tilde{u}}_{μ} (z), μ) + ξ_{ρ} {\tilde{u}}_{μ} {(z)}^{p - 1} (see hypothesis H_{2} (iv)) \\ = - Δ_{p} {\tilde{u}}_{μ} (z) + ξ_{ρ} {\tilde{u}}_{μ} {(z)}^{p - 1} for a.a. z \in Ω . & (45) \end{array}$ For every $K \subseteq Ω$ compact, we have $u_{λ} |_{K} ⩾ ξ > 0$ (recall $u_{λ} (z) > 0$ for all $z \in Ω$ , see Proposition 4). So, by hypothesis $H_{2}$ (iii) we have $δ_{λ, μ} |_{K} ⩾ m_{ξ} > 0 .$ Then from (45) and Proposition 2.6 of Arcoya and Ruiz [4] (recall that ${\tilde{u}}_{μ} \in int C_{+}$ ), we infer that $\begin{array}{rclr} {\tilde{u}}_{μ} - u_{λ} \in int C_{+} . & (46) \end{array}$ We claim that $u_{λ}$ is a local $C_{0}^{1} (\bar{Ω})$ -minimizer of the functional $τ_{λ}$ . Indeed, if this is not the case, we can find ${u_{n}}_{n ⩾ 1} \subseteq C_{0}^{1} (\bar{Ω})$ such that $\begin{array}{rclr} u_{n} \to u_{λ} in C_{0}^{1} (\bar{Ω}) and τ_{λ} (u_{n}) < τ_{λ} (u_{λ}) for all n ⩾ 1 . & (47) \end{array}$ From (46) and (47) it follows that we can find $n_{0} \in N$ such that $\begin{array}{rcl} u_{n} ⩽ {\tilde{u}}_{μ} for all n ⩾ n_{0} \\ ⟹ u_{n}^{+} ⩽ {\tilde{u}}_{μ} for all n ⩾ n_{0} (recall {\tilde{u}}_{μ} \in int C_{+}) . \end{array}$ Also, we have ${\hat{τ}}_{λ} (u_{n}^{+}) = τ_{λ} (u_{n}^{+}) ⩽ τ_{λ} (u_{n}) < τ_{λ} (u_{λ}) for all n ⩾ n_{0}$ which contradicts the fact that $u_{λ} = {\bar{u}}_{λ}$ is a global minimizer of ${\hat{τ}}_{λ}$ .

Since $u_{λ}$ is a local $C_{0}^{1} (\bar{Ω})$ -minimizer of $τ_{λ}$ . We may assume that $u_{λ}$ is an isolated critical point of $τ_{λ}$ , or otherwise we already have a sequence of distinct positive solutions, since the critical set of $τ_{λ}$ is in $[u_{λ}) = {u \in W_{0}^{1, p} (Ω) : u_{λ} (z) ⩽ u (z) for a.a. z \in Ω}$ (see (41)). Therefore, we can find $ρ \in (0, 1)$ small such that $\begin{array}{rclr} τ_{λ} (u_{λ}) < inf [τ_{λ} (u) : ∥ u - u_{λ} ∥ = ρ] = m_{λ} & (48) \end{array}$ (see Aizicovici, Papageorgiou and Staicu [1, Proof of Proposition 29]).

Moreover, by virtue of hypothesis $H_{2}$ (ii), if $u \in int C_{+}$ , then $\begin{array}{rclr} τ_{λ} (t u) \to - \infty as t \to + \infty . & (49) \end{array}$ Because of (42), (48) and (49), we can apply Theorem 1 (the mountain pass theorem) and find ${\hat{u}}_{λ} \in W_{0}^{1, p} (Ω)$ such that $\begin{array}{rclr} τ_{λ}^{'} ({\hat{u}}_{λ}) = 0 and m_{λ} ⩽ τ ({\hat{u}}_{λ}) . & (50) \end{array}$ From (49), (50) and since the critical set of $τ_{λ}$ is in $[u_{λ})$ , it follows that ${\hat{u}}_{λ} \in S (λ) \subseteq C_{+} and {\hat{u}}_{λ} \neq u_{λ}, u_{λ} ⩽ {\hat{u}}_{λ} .$ Finally from Proposition 4, we have $0 < u_{λ} (z) ⩽ {\hat{u}}_{λ} (z) for all z \in Ω .$ The proof is now complete. □

So, summarizing the situation for problem ( P λ ), we can state the following result describing the set of positive solutions as the parameter $λ > 0$ varies.

Theorem 9.

If hypotheses $H_{1}$ and $H_{2}$ hold, then

there exists $λ^{*} > 0$ such that for all $λ \in (0, λ^{*}]$ problem ( P λ ) has at least one positive solution $u_{λ} \in C_{+}$ with $u_{λ} (z) > 0$ for all $z \in Ω$ and for $λ > λ^{*}$ there are no positive solutions;

there exists $λ_{0}^{*} \in (0, λ^{*}]$ such that for all $λ \in (0, λ_{0}^{*})$ problem ( P λ ) has at least two positive solutions $u_{λ}, {\hat{u}}_{λ} \in C_{+}, u_{λ} ⩽ {\hat{u}}_{λ}, u_{λ} \neq {\hat{u}}_{λ}, u_{λ} (z) > 0 for all z \in Ω .$

Remark 3.

It will be interesting to know if $λ_{0}^{*} = λ^{*}$ . Also, it is not clear to us if this result can be extended to Neumann problems. A careful inspection of the proofs reveals that they fail in the Neumann case.

Footnotes

Acknowledgement

V. Rădulescu has been supported by Grant CNCS PCE-47/2011.

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