Abstract
We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a \((p-1)\)-superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions.
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1 Introduction
Let \(\Omega \subset \mathbb {R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega \) and let \(1<p<+\infty \). In this paper we study the following nonlinear Dirichlet problem with a singular reaction term:
In this problem \(\Delta _p\) stands for the p-Laplace differential operator defined by
for \(1<p<+\infty \). Also \(\mu \in (0,1)\) and \(f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) is a Carathéodory perturbation of the singular term (that is, for all \(x\in \mathbb {R}\), \(z\longmapsto f(z,x)\) is measurable and for almost all \(z\in \Omega \), \(x\longmapsto f(z,x)\) is continuous). We assume that \(f(z,\cdot )\) is \((p-1)\)-superlinear near \(+\infty \) but need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition.
We are looking for positive solutions and we prove the existence of at least two positive smooth solutions. Our approach is variational based on the critical point theory, together with truncation and comparison techniques.
In the past multiplicity theorems for positive solutions of singular problems were proved by Hirano et al. [20], Sun et al. [31] (semilinear problems driven by the Dirichlet Laplacian) and Giacomoni et al. [18], Kyritsi–Papageorgiou [21], Papageorgiou et al. [27], Papageorgiou–Smyrlis [28, 29], Perera–Zhang [30], Zhao et al. [32]. In all aforementioned works, there is a parameter \(\lambda >0\) in the reaction term. The presence of the parameter \(\lambda >0\) permits a better control of the right-hand side nonlinearity as the parameter becomes small. In particular in [29] the authors also deal with superlinear singular problems. However, the assumptions lead to a different geometry. More precisely, in [29] the perturbation function f(z, x) has a fixed sign, that is, \(f(z,x)>0\). We do not assume this here. In fact our conditions here force \(f(z,\cdot )\) to be sign-changing by requiring an oscillatory behaviour near zero (see hypothesis H(f)(i)). Our work here complements that of [27], where the authors deal with the resonant case, that is, in [27] the perturbation \(f(z,\cdot )\) is \((p-1)\)-linear. The present work and [27] cover a broad class of parametric nonlinear singular Dirichlet problems. We mention also the parametric work of Aizicovici et al. [2] on singular Neumann problems. For other parametric problems see also Gasiński–Papageorgiou [7,8,9,10,11,12,13,14,15,16]. Nonparametric singular Dirichlet problems were examined by Canino–Degiovanni [4], Gasiński–Papageorgiou [6] and Mohammed [25]. In [4, 25] we have existence but not multiplicity while in [6] we have also multiplicity results (the methods of proofs in all these papers are different).
2 Preliminaries and Hypotheses
Let X be a Banach space and \(X^*\) its topological dual. By \(\langle \cdot ,\cdot \rangle \) we denote the duality brackets for the pair \((X^*,X)\). Given \(\varphi \in C^1(X)\) we say that \(\varphi \) satisfies the Cerami condition, if the following property holds:
“Every sequence \(\{u_n\}_{n\geqslant 1}\subseteq X\) such that \(\{\varphi (u_n)\}_{n\geqslant 1}\) is bounded and
$$\begin{aligned} (1+\Vert u_n\Vert )\varphi '(u_n)\longrightarrow 0\quad \text {in}\ X^*\quad \text {as}\ n\rightarrow +\infty , \end{aligned}$$admits a strongly convergent subsequence.”
Evidently this is a kind of compactness-type condition on the functional \(\varphi \). Using the Cerami condition one can prove a deformation theorem from which follows the minimax theory of the critical values of \(\varphi \). A basic result in that theory is the mountain pass theorem which we will use in the sequel.
Theorem 2.1
If \(\varphi \in C^1(X)\) satisfies the Cerami condition, \(u_0,u_1\in X\), \(0<r<\Vert u_1-u_0\Vert \),
and
with \(\Gamma =\{\gamma \in C([0,1];X):\ \gamma (0)=u_0,\ \gamma (1)=u_1\}\), then \(c\geqslant m_r\) and c is a critical value of \(\varphi \) (that is, there exists \(u\in X\) such that \(\varphi (u)=c\) and \(\varphi '(u)=0\)).
The Sobolev space \(W^{1,p}_0(\Omega )\) and the Banach space \(C^1_0(\overline{\Omega })=\{u\in C^1(\overline{\Omega }):\ u|_{\partial \Omega }=0\}\) will be the two main spaces of this work. By \(\Vert \cdot \Vert \) we will denote the norm of \(W^{1,p}_0(\Omega )\). On account of Poincaré’s inequality, we have
The Banach space \(C^1_0(\overline{\Omega })\) is an ordered Banach space with positive (order) cone
This cone has a nonempty interior given by
Here \(\frac{\partial u}{\partial n}\) denotes the normal derivative of u defined by
with n being the outward unit normal on \(\partial \Omega \).
Let \(A:W^{1,p}_0(\Omega )\longrightarrow W^{1,p}_0(\Omega )^*=W^{-1,p'}(\Omega )\) (\(\frac{1}{p}+\frac{1}{p'}=1\)) be the nonlinear map defined by
In the next proposition, we recall the main properties of this map (see Motreanu et al. [26, p. 40]).
Proposition 2.2
The map \(A:W^{1,p}_0(\Omega )\longrightarrow W^{-1,p'}(\Omega )\) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone) and of type \((S)_+\), that is,
“if \(u_n{\mathop {\longrightarrow }\limits ^{w}}u\) in \(W^{1,p}_0(\Omega )\) and \(\limsup \limits _{n\rightarrow +\infty }\langle A(u_n),u_n-u\rangle \leqslant 0\), then \(u_n\longrightarrow u\) in \(W^{1,p}_0(\Omega )\).”
By \(p^*\) we denote the critical Sobolev exponent corresponding to p, i.e.,
The hypotheses on the perturbation term f are the following:
\(\underline{H(f)}\): \(f:\Omega \times \mathbb {R}\longrightarrow \mathbb {R}\) is a Carathéodory function such that \(f(z,0)=0\) for a.a. \(z\in \Omega \) and
- (i):
-
there exist \(a\in L^{\infty }(\Omega )\) and \(r\in (p,p^*)\) such that
$$\begin{aligned} |f(z,x)|\leqslant a(z)(1+x^{r-1})\quad \text {for a.a.}\ z\in \Omega , \text {all}\ x\geqslant 0 \end{aligned}$$and there exists \(w\in C^1(\overline{\Omega })\) such that
$$\begin{aligned} w(z)\geqslant \widehat{c}>0\ \text {for all}\ z\in \overline{\Omega },\quad \Delta _p w\in L^{\infty }(\Omega ),\quad \Delta _p w\leqslant 0\ \text {for a.a.}\ z\in \Omega \end{aligned}$$and for every compact set \(K\subseteq \Omega \), there exists \(c_K>0\) such that
$$\begin{aligned} w(z)^{-\mu }+f(z,w(z))\leqslant -c_K<0\quad \text {for a.a.}\ z\in K; \end{aligned}$$ - (ii):
-
if \(F(z,x)=\int _0^x f(z,s)\,ds\) and for every \(\lambda >0\) we define
$$\begin{aligned} \xi _{\lambda }(z,x)=\bigg (\frac{p}{1-\mu }-1\bigg )x^{1-\mu }+\lambda (f(z,x)x-p F(z,x)), \end{aligned}$$then
$$\begin{aligned} \lim _{x\rightarrow +\infty }\frac{F(z,x)}{x^p}=+\infty \quad \text {uniformly for a.a.}\ z\in \Omega , \end{aligned}$$and there exists \(\beta _{\lambda }\in L^1(\Omega )\), \(\beta _{\lambda }(z)\geqslant 0\) for a.a. \(z\in \Omega \) such that
$$\begin{aligned} \xi _{\lambda }(z,x)\leqslant \xi _{\lambda }(z,y)+\beta _{\lambda }(z)\quad \text {for a.a.}\ z\in \Omega , \text {all}\ 0\leqslant x\leqslant y; \end{aligned}$$ - (iii):
-
there exists \(\delta \in (0,\widehat{c}]\) such that
$$\begin{aligned} f(z,x)\geqslant 0\quad \text {for a.a.}\ z\in \Omega , \text {all} 0\leqslant x\leqslant \delta ; \end{aligned}$$ - (iv):
-
for every \(\varrho >0\), there exists \(\widehat{\xi }_{\varrho }>0\) such that for a.a. \(z\in \Omega \) the function
$$\begin{aligned} x\longmapsto f(z,x)+\widehat{\xi }_{\varrho }x^{p-1} \end{aligned}$$is nondecreasing on \([0,\varrho ]\).
Remark 2.3
Since we look for positive solutions and the above hypotheses concern the positive semiaxes \(\mathbb {R}_+=[0,+\infty )\), without any loss of generality, we assume that
Hypothesis H(f)(ii) implies that for a.a. \(z\in \Omega \), \(f(z,\cdot )\) is \((p-1)\)-superlinear, that is,
We stress that for the superlinearity of \(f(z,\cdot )\) we do not use the Ambrosetti–Rabinowitz condition which says that there exist \(r>p\) and \(M>0\) such that
This condition implies that \(f(z,\cdot )\) has at least \(x^{r-1}\)-growth near \(+\infty \), that is
for some \(c_0>0\). This excludes from consideration \((p-1)\)-superlinear nonlinearities with “slower” growth near \(+\infty \) (see Example 2.4). Here we replace the Ambrosetti–Rabinowitz condition with a quasimonotonicity condition on \(\xi (z,\cdot )\) (see hypothesis H(f)(ii)), which incorporates in our framework more superlinear nonlinearities. Hypothesis H(f)(ii) is a slight generalization of a condition used by Li–Yang [23]. It is satisfied, if there is \(M>0\) such that for a.a. \(z\in \Omega \), the function \(x\longmapsto \frac{f(z,x)}{x^{p-1}}\) is nondecreasing on \([M,+\infty )\) and this in turn is equivalent to saying that for a.a. \(z\in \Omega \), \(\xi (z,\cdot )\) is nondecreasing on \([M,+\infty )\). For details see Li–Yang [23]. Hypotheses H(f)(i) and (iii) imply that for a.a. \(z\in \Omega \), \(f(z,\cdot )\) exhibits a kind of oscillatory behaviour near zero. In hypothesis H(f)(i), the condition \(\Delta _p w(z)\leqslant 0\) for a.a. \(z\in \Omega \), implies that
Evidently the condition with \(w(\cdot )\) in hypothesis H(f)(i) is satisfied if \(w(z)\equiv c_+>0\) for all \(z\in \overline{\Omega }\) and \(\mathop {\text {ess inf}}\limits _{\Omega }f(\cdot ,c_+)<-\frac{1}{c_+^{\mu }}\). So, hypotheses H(f)(i) and (ii) dictate an oscillatory behaviour for \(f(z,\cdot )\) near zero.
Example 2.4
The following function satisfies hypotheses H(f). For the sake of simplicity we drop the z-dependence:
with \(1<q<p<r<+\infty \) and \(c>2\) [see (2.1)]. Note that f although \((p-1)\)-superlinear, it fails to satisfy the Ambrosetti–Rabinowitz condition.
Finally let us fix our notation. If \(x\in \mathbb {R}\), we set \(x^{\pm }=\max \{\pm \, x,0\}\). Then given \(u\in W^{1,p}_0(\Omega )\) we define \(u^{\pm }(\cdot )=u(\cdot )^{\pm }\) and we have
Set \(\widehat{C}_+=\{u\in C^1(\overline{\Omega }):\ u|_{\overline{\Omega }}\geqslant 0,\ \frac{\partial u}{\partial n}\leqslant 0\ \text {on}\ \partial \Omega \cap u^{-1}(0)\}\). We also mention that when we want to emphasize the domain D on which the cones \(C_+\) and \(\mathrm {int}\, C_+\) are considered, we write \(C_+(D)\) and \(\mathrm {int}\, C_+(D)\).
Moreover, by \(|\cdot |_N\) we denote the Lebesgue measure on \(\mathbb {R}^N\) and if \(\varphi \in C^1(X)\), then
(the “critical set” of \(\varphi \)).
3 Positive Solutions
In this section we prove the existence of two positive smooth solution for problem (1.1).
Proposition 3.1
If hypotheses H(f)(i) and (iii) hold, then there exists \(\underline{u}\in \mathrm {int}\, C_+\) such that
Proof
We consider the following auxiliary singular Dirichlet problem
From Proposition 5 of Papageorgiou–Smyrlis [29], we know that this problem has a unique positive solution \(\widetilde{u}\in \mathrm {int}\, C_+\).
With \(\widehat{c}>0\) and \(\delta >0\) as postulated by hypotheses H(f)(i) and (iii) respectively, we choose
We set \(\underline{u}=t \widetilde{u}\in \mathrm {int}\, C_+\). We have
(recall that \(t\leqslant 1\) and see hypothesis H(f)(iii) and Papageorgiou–Smyrlis [29]). Moreover, we have \(\underline{u}\leqslant w\). \(\square \)
Using \(\underline{u}\in \mathrm {int}\, C_+\), from Proposition 3.1 and \(w\in C^1(\overline{\Omega })\) from hypothesis H(f)(i), we introduce the following truncation of \(f(z,\cdot )\):
Given \(y,v\in W^{1,p}(\Omega )\), \(y\leqslant v\), we define
Also by \(\mathrm {int}_{C^1_0(\overline{\Omega })}[y,v]\) we denote the interior in the \(C^1_0(\overline{\Omega })\)-norm topology of \([y,v]\cap C^1_0(\overline{\Omega })\).
Proposition 3.2
If hypotheses H(f)(i) and (iii) hold, then problem (1.1) admits a solution \(u_0\in [\underline{u},w]\cap C^1_0(\overline{\Omega })\).
Proof
Let
and consider the functional \(\widehat{\varphi }:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}\) defined by
Proposition 3 of Papageorgiou–Smyrlis [29] implies that \(\widehat{\varphi }\in C^1(W^{1,p}_0(\Omega ))\) and we have
From (3.1) it is clear that \(\widehat{\varphi }\) is coercive. Also, the Sobolev embedding theorem implies that \(\widehat{\varphi }\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(u_0\in W^{1,p}_0(\Omega )\) such that
so \(\widehat{\varphi }'(u_0)=0\), hence
In (3.2) first we choose \(h=(\underline{u}-u_0)^+\in W^{1,p}_0(\Omega )\). We have
[see (3.1)] and Proposition 3.1), so
hence \(\underline{u}\leqslant u_0\).
Next in (3.2) we choose \(h=(u_0-w)^+\in W^{1,p}_0(\Omega )\) (see hypothesis H(f)(i)). Then we have
[see (3.1)] and hypothesis H(f)(i)), so
hence \(u_0\leqslant w\). So, we have proved that
From (3.1), (3.2) and (3.3), we have
Let \(d(z)=d(z,\partial \Omega )\) for \(z\in \overline{\Omega }\) (the distance from the boundary \(\partial \Omega \)). Then Lemma 14.16 of Gilbarg-Trudinger [19, p. 355] implies that there exists \(\delta _0>0\) such that
where \(\Omega _{\delta _0}=\{z\in \Omega :\ d(z)=d(z,\partial \Omega )<\delta _0\}\). Let \(D=\overline{\Omega }\setminus \Omega _{\delta _0}\) and consider the ordered Banach space C(D) with positive (order) cone \(C(D)_+\). Since \(u_0(z)\geqslant \widetilde{c}>0\) for all \(z\in D\), it follows that
Recall that \(\underline{u}\in \mathrm {int}\, C_+\) (see Proposition 3.1). So, on account of Proposition 2.1 of Marano–Papageorgiou [24], we can find \(0<c_1<c_2\) such that
For all \(h\in W^{1,p}_0(\Omega )\) we have
for some \(c_3,c_4>0\) (since \(\Omega \subseteq \mathbb {R}^N\) is bounded, \(\mu \in (0,1)\) and using Hardy’s inequality; see Brézis [3, p. 313]).
Therefore from (3.4) and since \(u_0^{-\mu }\in L^1(\Omega )\) (see Lazer-McKenna [22, Lemma]), it follows that
Invoking Theorem B.1 of Giacomoni-Schindler-Takáč [18], we have that \(u_0\in \mathrm {int}\, C_+\). Therefore finally we can say that \(u_0\in [\underline{u},w]\cap C_0^1(\overline{\Omega })\). \(\square \)
If we strengthen the conditions on the perturbation term f(z, x) we can improve the condition of Proposition 3.2.
Proposition 3.3
If hypotheses H(f)(i), (iii) and (iv) hold, then
Proof
From Proposition 3.2 we already know that
Let \(\varrho =\Vert w\Vert _{\infty }\) and let \(\widehat{\xi }_{\varrho }>0\) be as postulated by hypothesis H(f)(iv). We have
[see (3.3)], hypotheses H(f)(iv), (iii) and Proposition 3.1). Then (3.6) and Proposition 4 of Papageorgiou–Smyrlis [29], imply that
Let \(D_0=\{z\in \Omega :\ u_0(z)=w(z)\}\). The hypothesis on the function w (see hypothesis H(f)(i)), implies that \(D_0\subseteq \Omega \) is compact. So, we can find an open set \(U\subseteq \Omega \) with \(C^2\)-boundary \(\partial U\) such that
We have
[see (3.3) and hypotheses H(f)(i) and (iv)]. Then Proposition 5 of Papageorgiou–Smyrlis [29] (the “singular” strong comparison principle) implies that
Since \(D_0\subseteq U\), it follows that \(D_0=\emptyset \) and so we have
Therefore, we conclude that \(u_0\in \mathrm {int}_{C_0^1(\overline{\Omega })}[\underline{u},w]\). \(\square \)
Next we produce a second positive solution for problem (1.1).
Proposition 3.4
If hypotheses H(f) hold, then problem (1.1) admits a second positive solution \(\widehat{u}\in \mathrm {int}\, C_+\).
Proof
We introduce the following truncation of the reaction term in problem (1.1):
Clearly this is a Carathéodory function. We set \(E(z,x)=\int _0^x e(z,s)\,ds\) and consider the functional \(\varphi _*:W^{1,p}_0(\Omega )\longrightarrow \mathbb {R}\) defined by
We know that \(\varphi _*\in C^1(W^{1,p}_0(\Omega ))\) (see Papageorgiou–Smyrlis [29, Proposition 3]).
Claim:\(\varphi _*\) satisfies the Cerami condition.
Let \(\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) be a sequence such that
From (3.9) we have
with \(\varepsilon _n\rightarrow 0^+\). In (3.10) we choose \(h=-u_n^-\in W^{1,p}_0(\Omega )\). Then
[see (3.7)], so
for some \(c_5>0\), thus
We use (3.11) in (3.8) and we have
for some \(M_2>0\). Also, if in (3.10) we choose \(h=u_n^+\in W^{1,p}_0(\Omega )\), then
We add (3.12) and (3.13) and obtain
for some \(M_3>0\), so
for some \(M_4>0\) [see (3.7)], thus
Suppose that the sequence \(\{u_n^+\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) is not bounded. By passing to a subsequence if necessary, we may assume that
Let \(y_n=\frac{u_n^+}{\Vert u_n^+\Vert }\) for \(n\in \mathbb {N}\). Then \(\Vert y_n\Vert =1\), \(y_n\geqslant 0\) for all \(n\in \mathbb {N}\). So, passing to a next subsequence if necessary, we may assume that
with \(y\geqslant 0\).
First assume that \(y\ne 0\). Let \(\Omega _+=\{z\in \Omega : y(z)>0\}\). We have \(|\Omega _+|_N>0\) [see (3.15)] and
Hypothesis H(f)(ii) implies that
From (3.16) and Fatou’s lemma we have
On the other hand hypothesis H(f)(ii) implies that we can find \(M_5>0\) such that
It follows that
for some \(c_6>0\). From (3.17) and (3.18) we infer that
On the other hand, from (3.12) we have
for some \(c_7>0\), so
for some \(c_8>0\). Comparing (3.19) and (3.20), we have a contradiction. This proves the Claim when \(y\ne 0\).
Next assume that \(y=0\). For \(k>0\), let \(v_n=(kp)^{\frac{1}{p}}y_n\) for \(n\in \mathbb {N}\). Then from (3.15) we have
We can find \(n_0\in \mathbb {N}\) such that
Let \(t_n\in [0,1]\) be such that
From (3.21) and Krasonoselskii’s theorem (see Gasiński–Papageorgiou [5, Theorem 3.4.4, p.407]), we have
From (3.22) and (3.23), we have
[see (3.24)]. But \(k>0\) is arbitrary. So, we infer that
We know that
for some \(M_6>0\) [see (3.8) and (3.11)]. From (3.25), (3.26) and (3.23) it follows that
Then we have
(by the chain rule), so
We have
[see (3.7) and hypothesis H(f)(ii)].
We use (3.28) in (3.27) and recall that \(\underline{u}(z)^{-\mu }+f(z,\underline{u}(z))\geqslant 0\) for a.a. \(z\in \Omega \) (see hypothesis H(f)(iii)). We have
(see hypothesis H(f)(ii)), so
for some \(c_9, c_{10}>\Vert \beta \Vert _1\). Comparing (3.25) and (3.29), we have a contradiction.
So, we have proved that
From (3.11) and (3.30) we infer that
So, passing to a subsequence if necessary, we may assume that
In (3.10) we choose \(h=u_n-u\in W^{1,p}_0(\Omega )\). We have
with \(\varepsilon _n'\rightarrow 0^+\). Note that
[see (3.7)]. Recall that \(\underline{u}\in \mathrm {int}\, C_+\). Hence we can find \(c_{11}>0\) such that
(see Proposition 2.1 of Marano–Papageorgiou [24]), so
thus
for some \(c_{12}>0\). From Lazer–McKenna [22, Lemma], we know that \(\widehat{u}_1^{-\frac{\mu }{p'}}\in L^{p'}(\Omega )\), so \(c_{12}\underline{u}^{-\mu }\in L^{p'}(\Omega )\). Therefore, we have
[see (3.31)]. Similarly, we have
We return to (3.32), pass to the limit as \(n\rightarrow +\infty \) and use (3.33), (3.34), (3.35). We obtain
so \(u_n\rightarrow u\) in \(W^{1,p}_0(\Omega )\) (see Proposition 2.2) and thus \(\varphi _*\) satisfies the Cerami condition. This proves the Claim.
From (3.1) and (3.7) we see that
(here \(\widehat{\varphi }\) is as in the proof of Proposition 3.2). From the proof of Proposition 3.2, we know that
while from Proposition 3.3, we know that
Then (3.36), (3.37) and (3.38) imply that
thus
(see Theorem 1.1 of Giacomoni–Saoudi [17]). Using (3.7) we can easily see that
Therefore we may assume that \(K_{\varphi _*}\) is finite or otherwise we already have an infinity of positive smooth solutions of (1.1) [see (3.7)] all bigger than \(u_0\) and so we are done. The finiteness of \(K_{\varphi _*}\) and (3.39) imply that we can find \(\varrho \in (0,1)\) small such that
(see Aizicovici et al. [1, proof of Proposition 29]). Hypothesis H(f)(ii) implies that if \(u\in \mathrm {int}\, C_+\), then
Then (3.41), (3.42) and the Claim permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find \(\widehat{u}\in W^{1,p}_0(\Omega )\) such that
From (3.40), (3.41), (3.43) and (3.7) we conclude that \(\widehat{u}\in \mathrm {int}\, C_+\), \(\widehat{u}\ne u_0\), \(\widehat{u}\) is a positive solution of (1.1) and \(\widehat{u}\geqslant u_0\). \(\square \)
We can state the following multiplicity theorem for problem (1.1).
Theorem 3.5
If hypotheses H(f) hold, then problem (1.1) has two positive smooth solutions
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The authors wish to thank a knowledgable referee for his/her corrections and helpful remarks.
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The Leszek Gasiński was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169.
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Bai, Y., Gasiński, L. & Papageorgiou, N.S. Positive solutions for nonlinear singular superlinear elliptic equations. Positivity 23, 761–778 (2019). https://doi.org/10.1007/s11117-018-0636-8
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DOI: https://doi.org/10.1007/s11117-018-0636-8
Keywords
- p-Laplacian
- Positive solutions
- Singular term
- \((p-1)\)-superlinear perturbation
- Nonlinear regularity
- Truncations