Abstract
This paper concerns semilinear elliptic equations involving sign-changing weight function and a nonlinearity of subcritical nature understood in a generalized sense. Using an Orlicz–Sobolev space setting, we consider superlinear nonlinearities which do not have a polynomial growth, and state sufficient conditions guaranteeing the Palais–Smale condition. We study the existence of a bifurcated branch of classical positive solutions, containing a turning point, and providing multiplicity of solutions.
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1 Introduction
In this paper we study the classical positive solutions to the Dirichlet problem for a class of semilinear elliptic equations whose nonlinear term is of subcritical nature in a generalized sense and involves indefinite nonlinearities. More precisely, given \(\Omega \subset {\mathbb {R}}^N \), \(N> 2,\) a bounded, connected open subset, with \(C^{2}\) boundary \(\partial \Omega \), we look for positive solutions to:
where \(\lambda \in {\mathbb {R}}\) is a real parameter, \(a\in C^1({\bar{\Omega }})\) changes sign in \(\Omega \),
\(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\alpha > 0\) is a fixed exponent, and \(f,g \in C^1({\mathbb {R}})\) satisfy
We will say that f satisfies hypothesis (H) whenever (H)\(_0\), (H)\(_\infty \), and (H)\(_{g'}\) are satisfied. Since we are interested in positive solutions, we
note that, since (H)\(_0\), \(f(0)=0\) and that
When \(\lambda =0\), \(a(x)\equiv 1\) and \(g(s)\equiv 0\), this kind of nonlinearity has been studied in [5,6,7, 16], and in [11] for the case of the \(p-\)laplacian operator, with \(\alpha > \frac{p}{N-p}\). It is known the existence of uniform \(L^\infty \) a-priori bounds for any positive classical solution, and as a consequence, the existence of positive solutions. When \(\alpha \rightarrow 0,\) there is a positive solution blowing up at a non-degenerate point of the Robin function as \(\alpha \rightarrow 0,\) see [9] for details.
Let \((\lambda _1,\varphi _1)\) stands for the first eigen-pair of the Dirichlet eigenvalue problem \(-\Delta \varphi =\lambda \varphi \text{ in } \Omega ,\ \varphi = 0 \text{ on } \partial \Omega \,.\) From [10] it is known that \((\lambda _1,0)\) is a bifurcation point of positive solutions \((\lambda ,u_\lambda )\) to the equation (1.1). If f behaves like \(\vert u\vert ^{p-2}u\) at zero with \(2 \le p \le 2^*\), the influence of the negative part of the weight a is displayed under the sign of \(\int _{\Omega } a(x)\varphi _1(x)^{p}\, dx\), where \(\varphi _1\) is the first positive eigenfunction for \(-\Delta \) in \(H_0^1(\Omega )\). Specifically, whenever
the bifurcation of positive solutions from the trivial solution set is ’on the right’ of the first eigenvalue, in other words, for values of \(\lambda >\lambda _1\). When
the bifurcation from the trivial solution set is ’on the left’ of the first eigenvalue, in other words, for values of \(\lambda < \lambda _1\).
Inspired by the work of Alama and Tarantello in [1], we will focus our attention to the case of a(x) changing sign and (1.5) is being satisfied, and, among other things, we will prove the existence of a turning point for a value of the parameter \(\Lambda >\lambda _1\), and in particular the existence of solutions when \(\lambda = \lambda _1\). We will use local bifurcation and variational techniques.
All throughout the paper, for \(v:\Omega \rightarrow {\mathbb {R}}\), \(v=v^+-v^-\) where
Let us also define
and assume that both \(\Omega ^+,\ \Omega ^-\) are non empty sets.
For this nonlinearity the Palais–Smale condition of the energy functional becomes a delicate issue, needing Orlicz spaces and a Orlicz–Sobolev embedding theorem.
In order to prove (PS) condition, Alama and Tarantello ([1]) assume that the zero set \(\Omega ^0\) has a non empty interior. This is also a common hypothesis for other authors when dealing with changing sign superlinear nonlinearities [8, 20, 23]. But this is a technical hypothesis. (PS)-condition will be proved in Proposition 3.1 without assuming that hypothesis. We neither use Ambrosetti-Rabinowitz condition.
Let us now denote
and remark that hypothesis (H) implies that \(C_0<+\infty \). Observe also that
Let u be a weak solution to (1.1). By a regularity result, see Lemma 2.1, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }}).\) So by a solution, we mean a classical solution.
Assume that u is a non-negative nontrivial solution. It is easy to see that the solution is strictly positive. Indeed, adding \(\pm C_0a(x)u\) to the r.h.s. of the equation, splitting \(a=a^+-a^-\), taking into account (1.4) and (1.7), and letting in each side the nonnegative terms, we can write
Now, the strong Maximum Principle implies that \(u>0\) in \(\Omega ,\) and \(\frac{\partial u}{\partial \nu }<0\) on \(\partial \Omega \).
Our main result is the following theorem.
Theorem 1.1
Assume that \(g \in C^1({\mathbb {R}})\) satisfies hypothesis (H). Let \(C_0>0\) be defined by (1.6). If a changes sign in \(\Omega \), and (1.5) holds, then there exists a \(\Lambda \in {\mathbb {R}},\)
and such that (1.1) has a classical positive solution if and only if \(\lambda \le \Lambda \).
Moreover, there exists a continuum (a closed and connected set) \( {\mathscr {C}}\) of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point \((\lambda ,u)=(\lambda _1,0)\) which is unbounded. Furthermore,
-
(a)
For every, \(\lambda \in \big (\lambda _1, \Lambda )\), (1.1) admits at least two classical ordered positive solutions.
-
(b)
For \(\lambda =\Lambda \), problem (1.1) admits at least one classical positive solution.
-
(c)
For every \(\lambda \le \lambda _1\), problem (1.1) admits at least one classical positive solution.
The paper is organized in the following way. Section 2 contains a regularity result and a non existence result. (PS)-condition and an existence of solutions result for \(\lambda <\lambda _1\) based in the Mountain Pass Theorem will be proved in Sect. 3. A bifurcation result for \(\lambda >\lambda _1\) is developed in Sect. 4. The main result is proved in Sect. 5. Appendix A contains some useful estimates. Orlicz spaces, and a Orlicz–Sobolev embeddings theorems, will be treated in Appendix B.
2 A Regularity Result and a Non Existence Result
Next, we recall a regularity Lemma stating that any weak solution is in fact a classical solution.
Lemma 2.1
If \(u\in H_0^1(\Omega )\) weakly solves (1.1) with a continuous function f with polynomial critical growth
then, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) and
for any \(r>N\) and \(\mu =1-N/r.\) Moreover, if \(\partial \Omega \in C^{2,\mu }\), then \(u\in C^{2,\mu }({\overline{\Omega }})\).
Proof
Due to an estimate of Brézis-Kato [3], based on Moser’s iteration technique [17], \(u\in L^{r}(\Omega )\) for any \(r>1\); and by elliptic regularity \(u\in W^{2,r}(\Omega ),\) for any \(r>1\) (see [22, Lemma B.3] and comments below).
Moreover, by Sobolev embeddings for \(r>N\) and interior elliptic regularity \(u\in C^{1,\alpha }({\overline{\Omega }})\cap C^{2}(\Omega )\). Furthermore, if \(\partial \Omega \in C^{2,\alpha }\), then \(u\in C^{2,\alpha }({\overline{\Omega }})\). \(\square \)
Proposition 2.2
Let f satisfy hypothesis (H) and let \(C_0\) be defined in (1.6). Assume that a changes sign in \(\Omega \).
-
1.
Problem (1.1) does not admit a positive solution \(u\in H_0^1 (\Omega )\) for any
$$\begin{aligned} \lambda \ge \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )+C_0\sup a^+. \end{aligned}$$ -
2.
If \(\mathrm{int}\,(\Omega ^0)\ne \emptyset ,\) then \(\lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )<+\infty \) and (1.1) does not admit a positive solution for any
$$\begin{aligned} \lambda \ge \lambda _1\big (\mathrm{int}\,(\Omega ^0)\big ). \end{aligned}$$
Proof
1. Let \(\lambda \ge \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big ) + C_0\sup a^+ ,\) and assume by contradiction that there exists a non-negative non-trivial solution \(u\in H_0^1 (\Omega )\) to (1.1) for the parameter \(\lambda \). Since the Maximum Principle \(u>0\) in \(\Omega \), see (1.8).
Let \({\hat{\varphi }} \) be the positive eigenfunction of \(\big ( -\Delta , H_0^1 (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )\big )\) of \(L^2\)-norm equal to 1. For simplicity, we will also denote by \({\hat{\varphi }}\) the extension by 0 of \({\hat{\varphi }}\) in all \(\Omega \). By Hopf’s maximum principle, we have \(\frac{\partial {\hat{\varphi }} }{\partial \nu } <0\) on \(\partial \big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )\), where \(\nu \) is the outward normal.
Again, if we multiply the equation (1.1) by \({\hat{\varphi }}\) and integrate along \(\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\) we find, after integrating by parts,
a contradiction.
2. Let \(\lambda \ge \lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )\) and, by contradiction, assume the existence of a positive solution \(u\in H_0^1 (\Omega )\) of problem (1.1) for the parameter \(\lambda \). Let \({\tilde{\varphi }}\) be a positive eigenfunction associated to \(\lambda _1 \big (\mathrm{int}\,(\Omega ^0)\big )<+\infty \). For simplicity, we will also denote by \({\tilde{\varphi }}\) the extension by 0 in all \(\Omega \). If we multiply equation (1.1) by \({\tilde{\varphi }}\) and integrate along \(\Omega ^0\) we find, after integrating by parts,
On the other hand
Hence
a contradiction. \(\square \)
3 An Existence Result for \(\lambda <\lambda _1\)
In this section, we prove the existence of a nontrivial solution to equation (1.1) for \(\lambda <\lambda _1\), through the Mountain Pass Theorem.
3.1 On Palais–Smale Sequences
In this subsection, we define the framework for the functional \(J_\lambda \) associated to the problem (1.1)\(_\lambda \). Hereafter, we denote by \(\Vert \cdot \Vert \) the usual norm of \(H_0^1 (\Omega )\):
Given \(f(s)=h(s)+g(s)\) defined by (1.2), let us denote by \(F(s):= \int _0^s f(t) \, dt.\) Observe that (1.7) implies the following
Consider the functional \(J_\lambda : H_0^{1}(\Omega )\rightarrow {\mathbb {R}}\) given by
Take note that for all \(v\in H_0^1(\Omega )\), \( J_\lambda \big [v^+\big ]\le J_\lambda [v]. \)
The functional \(J_\lambda \) is well defined and belongs to the class \(C^1\) with
for all \(\psi \in H_0^{1}(\Omega ).\) As a result, non-negative critical points of the functional \(J_\lambda \) correspond to non-negative weak solutions to (1.1).
The next Proposition establishes that Palais–Smale sequences are bounded whenever \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0)\), where \(\lambda _1 (\mathrm{int}\,\Omega ^0)\) may be infinite.
Proposition 3.1
Assume that \(g \in C^1({\mathbb {R}})\) fulfills hypothesis (H) and that \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0)\le +\infty \).
Then any (PS) sequence, that is, a sequence satisfying the conditions
- \((J_1)\):
-
\(J_\lambda [u_n] \le C\),
- \((J_2)\):
-
\(\vert J_\lambda ^{'} [u_n]\, \psi \vert \le \varepsilon _n\, \Vert \psi \Vert \), where \(\varepsilon _n \rightarrow 0\) as \(n \rightarrow +\infty \)
is a bounded sequence.
Proof
1. Let \(\{u_n\}_{n\in {\mathbb {N}}} \) be a (PS) sequence in \(H_0^{1}(\Omega )\) and, in contradiction, assume that \(\Vert u_n\Vert \rightarrow +\infty \). Let us first prove the following claim:
Claim. Let \(v\in H_0^1(\Omega )\) be the weak limit of \(v_n=\frac{u_n}{\Vert u_n\Vert }\) and assume that \(v_n\rightarrow v\), strongly in \(L^{2^*-1} (\Omega )\) and a.e. Then \(v= 0\) a.e. in \(\Omega \).
Assume that \(v\not \equiv 0\) and write \(\gamma _n =\Vert u_n\Vert \). Let \(\omega _n:=\{x\in \Omega : v_n^+(x)>1\} \), then for any \(\psi \in C_0^1 (\Omega ) \),
Let \(x\in \Omega \setminus \omega _n,\) based on the estimates (A.1),
Besides, by the reverse of the Lebesgue dominated convergence theorem, see for instance [2, Theorem 4.9, p. 94] , there exists \(h_i\in L^1(\Omega ) \), \(1\le i\le 3\) such that, up to a subsequence,
for all \(n\in {\mathbb {N}}\), and therefore
By Lebesgue’s dominated convergent theorem, we have
We have used here that if \(v^+(x)\not =0,\) then
and if \(v^+(x)=0,\) then
On the other hand
Hence, using \((J_2)\) for an arbitrary test function \(\psi \), multiplying by \(\frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}}\) and passing to the limit we find
In particular \(v^+= 0\) a.e. in \(\Omega \setminus \Omega ^0\).
Assume that \(\ \mathrm{int}\,\Omega ^0\ne \emptyset \), and that \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0) \). Thus, for any \(\psi \in C_0^1 (\mathrm{int}\,\Omega ^0)\) we have from \((J_2)\)
Dividing by \(\Vert u_n\Vert \) and passing to the limit we have
From the Maximum Principle, \(v\ge 0\) in \(\mathrm{int}\,\Omega ^0.\) Since \(\lambda <\lambda _1(\mathrm{int}\,\Omega ^0)\) then it must be \(v^+\equiv 0\) in \(\mathrm{int}\,\Omega ^0\). Hence \(v^+\equiv 0\) in \(\Omega \).
On the other hand, taking \(u_n^-\) as a test function in the condition \((J_2),\)
so \(\Vert u_n^-\Vert \rightarrow 0\) and then \(v^-\equiv 0,\) and we conclude the proof of the claim.
2. In order to achieve a contradiction, we use a Hölder inequality, and properties on convergence into an Orlicz space, cf. Appendix B.
To this end, the analysis of Lemma A.2 gives us the existence of \(\alpha ^* >0\) such that the function \(s\rightarrow \frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \) is increasing along \([0,+\infty [\) if \(\alpha \le \alpha ^*\). In this case, we will denote
If \(\alpha > \alpha ^*\) the function \(s\rightarrow \frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \) possesses a local maximum \(s_1\) in \([0,+\infty [\). Let us denote by \({\overline{s}}_1\) the unique solution \(s>s_1\) such that
and define the non-decreasing function
It follows that
By using
we get that
which implies that there exists \(K>0\) such that \(m(2s)\le Km(s)\) for all \(s\ge 0\) and consequently M satisfies the \(\Delta _2\)-condition (B.1).
Since \(v_n\rightharpoonup 0\) in \(H_0^1(\Omega )\) and strongly in \(L^2(\Omega )\), it follows from \((J_2) \) applied to \(\psi =u_n\) that
Since the Hölder inequality into Orlicz spaces, see Proposition B.11.(ii),
By Theorem B.3 and Theorem B.12 we have
Moreover, since there exists \(C>0\) such that \(m(s)\le C s^{2^*-1}\), \(M(s)\le Cs^{2^*}\) for all \(s\ge 0,\) and the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset H_0^1(\Omega )\), then, for each \(n\in {\mathbb {N}}\), there exists a \(C_n\) such that
By using definition B.8 of \(M^*\) and identities of Proposition B.9 we have
then, for each \(n\in {\mathbb {N}}\),
Observe that \(\ \vert f(s)\vert \le C(1+m(s))\), so then
see Proposition B.11.(iii) and (i), concluding that the l.h.s. is bounded for each n.
Consequently, \(a(x)\frac{f(u_n^+)}{\Vert u_n\Vert } \in L_{M^*} (\Omega )\), which is the dual of \(L_{M} (\Omega )\) (see [15], Theorem 14.2).
On the other hand, from \(J_2\), for all \(\psi \in C_c^\infty (\Omega ),\)
Taking the limit, and since \(C_c^\infty (\Omega )\) is dense in \( L_M(\Omega )\) (see [13]),
Moreover, since (3.7), \(v_n \rightarrow v=0\) in \(L_M(\Omega )\), [2, Proposition 3.13 (iv)], and (3.9) imply
which contradicts (3.5). This concludes the proof. \(\square \)
Theorem 3.2
Assume the hypothesis of Proposition 3.1 and let \(\{u_n\}_{n\in {\mathbb {N}}} \) be a (PS) sequence in \(H_0^{1}(\Omega )\).
Then, there exists a subsequence, denoted by \(\{u_n\}_{n\in {\mathbb {N}}}\), such that
Proof
From Proposition 3.1 we know that the sequence is bounded. Consequently, there exists a subsequence, denoted by \(\{u_{n}\}_{n\in {\mathbb {N}}}\), and some \(u\in H_0^1(\Omega )\) such that
By testing \((J_2)\) against \(\psi =u_n-u\) and using (3.10), and (3.11) we get
Claim.
In order to prove this claim, we use, as in the above proposition, a Hölder inequality and a compact embedding into some Orlicz space, c.f. Appendix B.
By Theorem B.3 and Theorem B.12 we have
where m, and M are defined by (3.2)–(3.4), as in the above proposition. On the other hand, because there exists \(C>0\) such that \(m(s)\le C s^{2^*-1}\) and \(M(s)\le Cs^{2^*}\) for all \(s\ge 0,\) and the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(H_0^1(\Omega )\), then
By using definition B.8 of \(M^*\) and identities of Proposition B.9 we have
then
for all \(n\in {\mathbb {N}}\). Finally, by inequality (B.5) of Proposition B.12 we get
Now, using Holder’s inequality (B.6) and that \(\frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \le m(s)\) for all \(s\ge 0\), we get
and it follows from (3.13) that \(\Vert u_n-u\Vert \rightarrow 0\). \(\square \)
3.2 An Existence Result for \(\lambda <\lambda _1\)
The next theorem provides a solution to (1.1) for \(\lambda <\lambda _1\) based on the Mountain Pass Theorem.
Theorem 3.3
Assume that \(\Omega \subset {\mathbb {R}}^N \) is a bounded domain with \(C^{2}\) boundary. Assume that the nonlinearity f defined by (1.2) satisfies (H), and that the weight \(a\in C^1({\overline{\Omega }})\). Then, the boundary value problem (1.1)\(_\lambda \) has at least one classical positive solution for any \(\lambda <\lambda _1\).
Proof
We verify the hypothesis of the Mountain Pass Theorem, see [14, Theorem 2, Section 8.5]. Observe that the derivative of the functional \(J_\lambda ':H_0^{1}(\Omega )\rightarrow H_0^{1}(\Omega )\) is Lipschitz continuous on bounded sets of \(H_0^{1}(\Omega )\); also the (PS) condition is satisfied, see Proposition 3.1. Clearly \(J_\lambda [0]=0\).
-
1.
Let now \(u\in H_0^{1}(\Omega )\) with \(\Vert u\Vert =r\), for \(r>0\) to be chosen below. Then,
$$\begin{aligned} J_\lambda [u]=\frac{r^2}{2}-\frac{\lambda }{2}\int _{\Omega } (u^+)^{2}\, dx-\int _{\Omega }\,a(x)F(u^+) \,\, \, dx. \end{aligned}$$(3.14)From hypothesis (H) we have
$$\begin{aligned} \left| \int _{\Omega }\,a(x)G(u^+) \,\, \, dx\right|&\le C\int _{\Omega } \big (\vert u\vert ^{p}+ \vert u\vert ^{q}\big )\, dx\le C\left( r^{p}+r^{q}\right) . \end{aligned}$$where \(G(s):= \int _0^s g(t) \, dt.\) Now, definition (1.2) implies that
$$\begin{aligned} \left| \int _{\Omega }\,a(x)F(u^+) \,\, \, dx\right| \le C\left( r^{p}+r^{q}+r^{2^*}\right) . \end{aligned}$$In view of (3.14), and as a result of the Poincaré inequality, we get
$$\begin{aligned} J_\lambda [u]\ge \frac{1}{2}\left( 1-\frac{\vert \lambda \vert }{\lambda _1} \right) \, r^2 -C\left( r^{p}+r^{q}+r^{2^*}\right) \ge C_1 r^2, \end{aligned}$$taking \(\vert \lambda \vert <\lambda _1\), \(r>0\) small enough, and using that \(p,\ q,\ 2^*>2\).
-
2.
Now, fix some element \(0\le u_0\in H_{0}^{1}(\Omega )\), \( u_0> 0\) in \(\Omega ^+\), \( u_0\equiv 0\) in \(\Omega ^-\). Let \(v=tu_0\) for a certain \(t=t_0>0\) to be selected a posteriori. Since
$$\begin{aligned} f(tu_0)=\vert t\vert ^{2^* -2}t\, f(u_0) \left( \dfrac{\ln (e+\vert u_0\vert )}{\ln (e+\vert tu_0\vert )}\right) ^\alpha +g(tu_0), \end{aligned}$$(3.15)then \(f(tu_0)/t\rightarrow +\infty \) as \(t\rightarrow +\infty \) in \(\Omega ^+\).
From definition, and integrating by parts,
It can be easily seen that \(\lim _{s \rightarrow +\infty } \frac{G(s)}{sf(s)}=0.\)
Therefore, using l’Hôpital’s rule we can write
hence
Let \(C_0\ge 0\) be such that \(F(s)+\frac{1}{2} C_0 s^2\ge 0\) for all \(s\ge 0\) (see (1.7)), and let
By definition, \(u_0\equiv 0\) in \(\Omega ^-\), so, introducing \(\pm \frac{1}{2} C_0 (tu_0)^2\), splitting the integral, and using (3.17)–(3.18) we obtain
Hence, there exists a positive constant \(C>0\) such that
for \(t=t_0>0\) big enough.
Step 3. We have at last checked that all the hypothesis of the Mountain Pass Theorem are accomplished. Let
then, there exists \(c\ge C_1\,r^2>0\) such that
is a critical value of \(J_\lambda \), that is, the set \({\mathscr {K}}_c:=\{v\in H_{0}^{1}(\Omega ): \, J_\lambda [v]=c,\ J_\lambda '[v]=0\}\ne \emptyset \). Thus there exists \(u\in H_0^1 (\Omega )\), \(u\ge 0\), \(u\ne 0\) such that for each \(\psi \in H_0^{1}(\Omega )\), we have
and thereby, u is a nontrivial weak solution to (3.19). By Lemma 2.1, u is a classical solution, and by (1.8), \(u>0\) in \(\Omega .\) \(\square \)
4 A Bifurcation Result for \(\lambda >\lambda _1\)
Next Proposition uses Crandall-Rabinowitz’s local bifurcation theory, see [10], and Rabinowitz’s global bifurcation theory, see [19].
Proposition 4.1
Let us define
If (1.5) holds then,
where \(C_0>0\) is such that \(f(s)+C_0s \ge 0\) for all \(s\ge 0,\) (see definition (1.6)).
Moreover, there exists an unbounded continuum (a closed and connected set) \( {\mathscr {C}}\) of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point \((\lambda ,u)=(\lambda _1,0)\).
Proof
Proposition 2.2 establish the upper bounds for \(\Lambda \). Next, we concentrate our attention in proving that \(\Lambda >\lambda _1.\) Choosing \(\lambda \) as the bifurcation parameter, we check that the conditions of Crandall - Rabinowitz’s Theorem [10] are satisfied. For \(r>N\), we define the set \(W^{2,r}_+:=\{ u\in W^{2,r}(\Omega ): \, u>0\ \text {in}\ \Omega \}\), and consider \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\) endowed with the topology of \(W^{2,r}(\Omega )\). If \(r>N\), we have that \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\hookrightarrow C_0^{1,\mu }(\Omega )\) for \(\mu =1-\frac{N}{r}\in (0,1)\). Moreover, from Hopf’s lemma, we know that if \({\tilde{u}}\) is a positive solution to (1.1) then \({\tilde{u}}\) lies in the interior of \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\).
We consider the map \({\mathscr {F}}:{\mathbb {R}}\times W^{2,r}_+(\Omega ) \cap W^{1,r}_{0}(\Omega )\rightarrow L^{r}(\Omega )\) for \(r>N\),
The map \({\mathscr {F}}\) is a continuously differentiable map. Since hypothesis (i), \(g(0)=0\), and so \(a(x)F(0)=0\), \({\mathscr {F}} (\lambda ,0)=0\) for all \(\lambda \in {\mathbb {R}}\), and since \(F_u(x,0)=0\),
Observe that
where \(N(\cdot )\) is the kernel, and \(R(\cdot )\) denotes the range of a linear operator.
Hence, the hypotheses of Crandall-Rabinowitz’s Theorem are satisfied and \((\lambda _1,0)\) is a bifurcation point. Thus, decomposing
where \(Z=span[\varphi _1]^{\bot }\), there exists a neighborhood \({\mathscr {U}}\) of \((\lambda _1,0)\) in \({\mathbb {R}} \times C_0^{1,\mu } ({\overline{\Omega }})\), and continuous functions \(\lambda (s), {\tilde{w}}(s),\) \( s\in (-\varepsilon ,\varepsilon ),\) \(\lambda : (-\varepsilon ,\varepsilon )\rightarrow {\mathbb {R}},\) \({\tilde{w}}: (-\varepsilon ,\varepsilon )\rightarrow Z\) such that \(\lambda (0)=\lambda _1,\) \({\tilde{w}}(0)=0,\) with \(\int _{\Omega } {\tilde{w}} \varphi _1\, dx =0,\) and the only nontrivial solutions to (1.1) in \({\mathscr {U}}\), are
Set \(u=u(s)=s\varphi _1 +s\,{\tilde{w}}(s)\). Note that by continuity \({\tilde{w}}(s)\rightarrow 0\) as \(s\rightarrow 0\), which guarantees that \(u(s)>0\) in \(\Omega \) for all \(s\in (0,\varepsilon )\) small enough.
Next, we show that \(\lambda (s)>\lambda _1\) for all s small enough. Since (3.15), and hypothesis (H)\(_0\) on f, note that \(\frac{a(x)f(su)}{s^{p-1}u^{p-1}}\rightarrow L_1a(x)\) as \(s\rightarrow 0\). In fact, as \({\tilde{w}}(s)\rightarrow 0\) uniformly as \(s\rightarrow 0\), hypothesis (H)\(_0\) yields
Hence, multiplying and dividing by \(\big (\varphi _1+{\tilde{w}}(s)\big )^{p-1}\), we deduce
Now we prove that \(\lambda (s)>\lambda _1\) arguing by contradiction. Assume that there is a sequence \((\lambda _n,u_n)=\big (\lambda (s_n),u(s_n)\big )\) of bifurcated solutions to (1.1) in \({\mathscr {U}}\), with \(\lambda (s_n)\le \lambda _1\). Multiplying (1.1)\(_{\lambda _n}\) by \(\varphi _1\) and integrating by parts
which yields a contradiction, and consequently, \(\Lambda >\lambda _1\).
Finally, Rabinowitz’s global bifurcation Theorem [19] states that, in fact, the set \( {\mathscr {C}}\) of positive solutions to (1.1) emanating from \((\lambda _1,0)\) is a continuum (a closed and connected set) which is either unbounded, or contains another bifurcation point, or contains a pair of points \((\lambda ,u)\), \((\lambda ,-u)\) with \(u\ne 0\). Since (1.8), any non-negative non-trivial solution is strictly positive, and moreover \((\lambda _1,0)\) is the only bifurcation point to positive solutions, so \( {\mathscr {C}}\) can not reach another bifurcation point. Since (1.3), neither \( {\mathscr {C}}\) contains a pair of points \((\lambda ,u)\), \((\lambda ,-u)\) with \(u\ne 0\), which states that \( {\mathscr {C}}\) is unbounded, ending the proof. \(\square \)
5 Proof of Theorem 1.1
First we prove an auxiliary result.
Proposition 5.1
For each \(\lambda \in (\lambda _1,\Lambda )\), the following holds:
-
(i)
Problem (1.1)\(_\lambda \) admits a positive solution
$$\begin{aligned} u_\lambda =\inf \big \{u(x): \, u>0 \text { solving } (1.1) _\lambda \big \}, \end{aligned}$$in other words \(u_\lambda \) is minimal.
-
(ii)
Moreover, the map \(\lambda \rightarrow u_\lambda \) is strictly monotone increasing, that is, if \(\lambda<\mu <\Lambda \), then \(u_\lambda (x)<u_\mu (x)\) for all \(x\in \Omega \), and \(\frac{\partial u_\lambda }{\partial \nu }(x)>\frac{\partial u_\mu }{\partial \nu }(x)\) for all \(x\in \partial \Omega \).
-
(iii)
Furthermore, \(u_\lambda \) is a local minimum of the functional \(J_\lambda \).
Proof
(i.a) Step 1. Existence of positive solutions for any \(\lambda \in (\lambda _1,\Lambda )\).
Let \(\lambda \in (\lambda _1,\Lambda )\) be fixed. By definition of \(\Lambda \), there exists a \(\lambda _0\in (\lambda ,\Lambda )\) such that the problem (1.1)\(_{\lambda _0}\) admits a positive solution \(u_0\). It is easy to verify that \(u_0>0\) is a supersolution to (1.1)\(_\lambda \). Indeed, for any \(\psi \in H_0^1(\Omega )\) with \(\psi \ge 0\) in \(\Omega \)
Moreover, for every \(\delta >0\) satisfying
the function \({\underline{u}}=\delta \varphi _1\) is a subsolution for (1.1)\(_\lambda \) whenever \(\lambda >\lambda _1\). Let \(\delta >0\) satisfying (5.1) and such that \(g(s)\ge 0\) for any \(s\in [0,\delta \Vert \varphi _1\Vert _{L^\infty (\Omega )}]\). For any \(\psi \in H_0^1(\Omega )\), \(\psi >0\) with in \(\Omega \) we deduce
This allows us to take \({\underline{u}}=\delta \varphi _1\) as a subsolution for (1.1)\(_\lambda \) with \({\underline{u}}<u_0\). The sub- and supersolution method now guarantees a positive solution u to (1.1)\(_\lambda \), with \({\underline{u}}\le u\le u_0\).
(i.b) Step 2. Existence of a minimal positive solution \(u_\lambda \) for any \(\lambda \in (\lambda _1,\Lambda )\).
To show that there is in fact a minimal solution, for each \(x\in \Omega \) we define
Firstly, we claim that \({\underline{u}}_\lambda \ge 0,\) \({\underline{u}}_\lambda \not \equiv 0\). Assume that \({\underline{u}}_\lambda \equiv 0\) by contradiction. This would yield a sequence \(u_n\) of positive solutions to (1.1)\(_\lambda \) such that \(\Vert u_{n}\Vert _{C({\overline{\Omega }})}\rightarrow 0\) as \(n\rightarrow \infty \), or in other words, \((\lambda ,0)\) is a bifurcation point from the trivial solution set to positive solutions. Set \(v_n:=\frac{u_{n}}{\Vert u_{n}\Vert _{C({\overline{\Omega }})}}\). Observe that \(v_{n}\) is a weak solution to the problem
It follows from \(\mathrm {(H)}_0\) that \( \frac{a(x)f(u_{n})}{\Vert u_{n}\Vert _{C({\overline{\Omega }})}}\rightarrow 0\) in \(C({\overline{\Omega }}) \) as \(n\rightarrow \infty \). Therefore, the right-hand side of (5.2) is bounded in \(C({{\overline{\Omega }}})\). Hence, by the elliptic regularity, \(v_n\in W^{2,r}(\Omega )\) for any \(r>1\), in particular for \(r>N\). Then, the Sobolev embedding theorem implies that \(||v_n||_{C^{1,\alpha }({\overline{\Omega }})}\) is bounded by a constant C that is independent of n. Then, the compact embedding of \(C^{1,\mu }({\overline{\Omega }})\) into \(C^{1,\beta }({\overline{\Omega }})\) for \(0<\beta <\mu \) yields, up to a subsequence, \(v_{n}\rightarrow \Phi \ge 0\) in \(C^{1,\beta }({\overline{\Omega }})\). Since \(\Vert v_{n}\Vert _{C({\overline{\Omega }})}=1\), we have that \(\Vert \Phi \Vert _{C({\overline{\Omega }})}=1\). Hence, \(\Phi \ge 0,\) \(\Phi \not \equiv 0.\)
Using the weak formulation of equation (5.2), passing to the limit, and taking into account that \(\lambda \) is fixed and \(v_{n}\rightarrow \Phi \), we obtain that \(\Phi \ge 0,\) \(\Phi \not \equiv 0,\) is a weak solution to the equation
Then, by the maximum principle, it follows that \(\Phi =\varphi _1>0\), the first eigenfunction, and \(\lambda =\lambda _1\) is its corresponding eigenvalue, which contradicts that \(\lambda >\lambda _1\).
Secondly, we show that \({\underline{u}}_\lambda \) solves (1.1)\(_\lambda \). We argue on the contrary. Observe that the minimum of any two positive solutions to (1.1)\(_\lambda \) furnishes a supersolution to (1.1)\(_\lambda \). Assume that there are a finite number of solutions to (1.1)\(_\lambda \), then \({\underline{u}}_\lambda (x):=\min \big \{u(x): u>0 \text { solves } (1.1) _\lambda \big \}\) and \({\underline{u}}_\lambda \) is a supersolution. Choosing \(\varepsilon _0\) small enough so that \(\varepsilon _0\varphi _1 <{\underline{u}}_\lambda \), the sub- supersolution method provides a solution \(\varepsilon _0\varphi _1 \le v\le {\underline{u}}_\lambda \). Since v is a solution and \({\underline{u}}_\lambda \) is not, then \( v\le {\underline{u}}_\lambda \), \(v\ne u\), contradicting the definition of \({\underline{u}}_\lambda \), and achieving this part of the proof.
Assume now that there is a sequence \(u_n\) of positive solutions to (1.1)\(_\lambda \) such that, for each \(x\in \Omega \), \(\inf u_n(x)={\underline{u}}_\lambda (x)\ge 0,\) \({\underline{u}}_\lambda \not \equiv 0.\) Let \({\underline{u}}_1:=\min \{u_1,u_2\}\). Choosing \(\varepsilon _1\) small enough so that \(\varepsilon _1\varphi _1 <{\underline{u}}_1\), the sub- supersolution method provides a solution \(\varepsilon _1\varphi _1 \le v_1\le {\underline{u}}_1 \). We reason by induction.
Let \({\underline{u}}_n:=\min \{v_{n-1},u_{n+1}\}\). Choosing \(\varepsilon _n\) small enough so that \(\varepsilon _n\varphi _1 <{\underline{u}}_n\), the sub- supersolution method provides a solution \(\varepsilon _n\varphi _1 \le v_n\le {\underline{u}}_n\le v_{n-1}\). With this induction procedure, we build a monotone sequence of solutions \(v_n\), such that
Since monotonicity and Lemma 2.1, \(\Vert v_n\Vert _{C({\overline{\Omega }})}\le \Vert v_1\Vert _{C({\overline{\Omega }})}\), by elliptic regularity, \(\Vert v_n\Vert _{C^{1,\mu }({\overline{\Omega }})}\le C\) for any \(\mu <1\), and by compact embedding \(v_n\rightarrow v\) in \(C^{1,\beta }({\overline{\Omega }})\) for any \(\beta <\alpha \). Using the weak formulation of equation (1.1)\(_\lambda \), passing to the limit, and taking into account that \(\lambda \) is fixed, we obtain that v is a weak solution to the equation (1.1)\(_\lambda \). Hence \(v(x)\ge {\underline{u}}_\lambda >0\). Moreover, since (5.3), \(v_n(x)\downarrow v(x)\) pointwise for \(x\in \Omega \), so \(\inf v_n(x)=v(x)\). Also, and due to (5.3), \({\underline{u}}_n(x)\downarrow v(x)\) pointwise for \(x\in \Omega \), and \(\inf {\underline{u}}_n(x)=v(x)\).
On the other hand, by construction \({\underline{u}}_n\le u_{n+1}\), so, for each \(x\in \Omega \), \(v(x)=\inf {\underline{u}}_n(x)\le \inf u_n(x)={\underline{u}}_\lambda (x)\). Therefore, and by definition of \({\underline{u}}_\lambda \), necessarily \(v= {\underline{u}}_\lambda \), proving that \({\underline{u}}_\lambda \) solves (1.1)\(_\lambda \), and achieving the proof of step 2.
(ii) The monotonicity of the minimal solutions is concluded from a sub- supersolution method. Reasoning as in step 1, \(u_\mu \) is a strict supersolution to (1.1)\(_\lambda \), so \(w:= u_\mu (x)- u_\lambda (x)\ge 0\), \(w\not \equiv 0\). Moreover, \(w= 0\) on \(\partial \Omega ,\) and we can always choose \(c_0:=C_0\Vert a\Vert _{\infty }>0\) where \(C_0\) is defined by (1.6), so that \(a^-(x)f'(s)+ c_0\ge 0\) and \(a^+(x)f'(s)+ c_0\ge 0\) for all \(s\ge 0\), then
finally, the Maximum Principle implies that \(w>0\) in \(\Omega ,\) and \(\frac{\partial w}{\partial \nu }<0\) on \(\partial \Omega ,\) ending the proof of step 3.
(iii) Since [4, Theorem 2] if there exists an ordered pair of \(L^\infty \) bounded sub and supersolution \({\underline{u}}\le {\overline{u}}\) to (1.1)\(_\lambda \), and neither \({\underline{u}}\) nor \({\overline{u}}\) is a solution to (1.1)\(_\lambda ,\) then there exist a solution \({\underline{u}}<u<{\overline{u}}\) to (1.1)\(_\lambda \) such that u is a local minimum of \(J_\lambda \) at \(H_0^1(\Omega )\).
Reasoning as in (i), \({\overline{u}}:=u_\mu \) with \(\mu >\lambda \) is a strict supersolution to (1.1)\(_\lambda \), and \({\underline{u}}:=\delta \varphi _1\) is a strict sub-solution for \(\delta >0\) small enough, such that \({\underline{u}}(x)<{\overline{u}}(x)\) for each \(x\in \Omega \). This achieves the proof. \(\square \)
Proof of Theorem 1.1
Theorem 3.3 provides the existence of positive solutions for \(\lambda <\lambda _1\), and Proposition 5.1 provide the existence of minimal positive solutions for \(\lambda \in (\lambda _1,\Lambda ).\)
(a) Step 1. Existence of a second positive solution for \(\lambda \in (\lambda _1,\Lambda ).\)
Fix an arbitrary \(\lambda \in (\lambda _1,\Lambda )\), and let \(u_{\lambda }\) be the minimal solution to (1.1)\(_{\lambda }\) given by Proposition 5.1, minimizing \(J_\lambda \). A second solution follows seeking a solution through variational arguments [12, Theorem 5.10] and the Mountain Pass procedure shown below.
First, reasoning as in Proposition 5.1(iii), we get a local minimum \({\tilde{u}}_\lambda >0\) of \(J_\lambda \). If \({\tilde{u}}_\lambda \ne u_\lambda \), then \({\tilde{u}}_\lambda \) is the second positive solution, ending the proof. Assume that \({\tilde{u}}_\lambda = u_\lambda \).
Now we reason as in [12, Theorem 5.10] on the nature of local minima. Thus, either
-
(i)
there exists \(\varepsilon _0>0\), such that \(\inf \big \{J_\lambda (u) : \Vert u-{\tilde{u}}_\lambda \Vert = \varepsilon _0 \big \} >J_\lambda ({\tilde{u}}_\lambda ),\) in other words, \({\tilde{u}}_\lambda \) is a strict local minimum, or
-
(ii)
for each \(\varepsilon >0,\) there exists \(u_\varepsilon \in H_0^1(\Omega )\) such that \(J_\lambda \) has a local minimum at a point \(u_\varepsilon \) with \(\Vert u_\varepsilon -{\tilde{u}}_\lambda \Vert =\varepsilon \) and \(J_\lambda (u_\varepsilon )=J_\lambda ({\tilde{u}}_\lambda ).\)
Let us assume that (i) holds, since otherwise case (ii) implies the existence of a second solution.
Consider now the functional \(I_\lambda : H_0^{1} (\Omega ) \rightarrow {\mathbb {R}}\) given by \(I_\lambda [v]=J_\lambda [u_\lambda +v]-J_\lambda [u_\lambda ]\), more specifically
where
Obviously \(I_\lambda [v^+]\le I_\lambda [v],\) and observe that \(I_\lambda '[v]=0\iff J_\lambda '[u_\lambda +v]=0.\)
Fix now some element \(0\le v_0\in H_{0}^{1}(\Omega )\cap L^\infty (\Omega )\), \( v_0> 0\) in \(\Omega ^+\), \( v_0\equiv 0\) in \(\Omega ^-\). Let \(v=tv_0\) for a certain \(t=t_0>0\) to be selected a posteriori, and evaluate
Reasoning as in the proof of Theorem 3.3 for large positive t, since \(F(t)/t^2\rightarrow \infty \) as \(t\rightarrow \infty ,\) and using also (3.1) we obtain that
so
for \(t=t_0\) big enough, and where \({\widetilde{\Omega }}_{\delta }^+\) is defined by (3.18). Thus, the Mountain Pass Theorem implies that if
then, there exists \(c>0\) such that
is a critical value of \(I_\lambda \), and thereby \({\mathscr {K}}_c:=\{v\in H_{0}^{1}(\Omega ) : \, I_\lambda [v]=c,\ I_\lambda '[v]=0\}\) is non empty.
Since for any \({\mathbf {g}}\in \Gamma \) we have \(I_\lambda [{\mathbf {g}}^+(t)] \le I_\lambda [{\mathbf {g}}(t)]\) for all \(t\in [0,1]\), it follows that \({\mathbf {g}}^+\in \Gamma \), and we derive the existence of a sequence \(v_n\) such that
On the other hand, \(w_n:=u_\lambda +v_n\) is a (PS) sequence for the original functional \(J_\lambda .\) Since Theorem 3.2, if \(\lambda <\lambda _1( \mathrm{int}\,\Omega ^0),\) \(v_n\rightarrow v_\lambda \) en \(H_0^1(\Omega ),\) so \(I_\lambda '[v]=0\) and \(I_\lambda [v]=c>0,\) hence \(v_\lambda \ge 0\) is a nontrivial critical point of \(I_\lambda \). Consequently, \(w_\lambda :=u_\lambda +v_\lambda \) is a positive critical point of \(J_\lambda ,\) such that, for each \(\psi \in H_0^{1}(\Omega )\), we have
and thereby \(w_\lambda :=u_\lambda +v_\lambda \ge u_\lambda \), \(w_\lambda \ne u_\lambda \) is a second positive solution to (1.1)\(_\lambda \).
(b) Step 2. Existence of a classical positive solution for \(\lambda =\Lambda \).
We prove the existence of a solution for \(\lambda =\Lambda \). For each \(\lambda \in (\lambda _1,\Lambda )\), problem (1.1) admits a minimal positive weak solution \(u_\lambda \) and \(\lambda \rightarrow u_\lambda \) is increasing, see Proposition 5.1. Taking the monotone pointwise limit, let us define
We next see that \(\Vert u_{\, \Lambda }\Vert <+\infty \), reasoning on the contrary. Assume that there exists a sequence of solutions \(u_n:=u_{\, {\lambda _n}}\) such that \(\Vert u_{\, \lambda _n}\Vert \rightarrow +\infty \) as \(\lambda _n\rightarrow \Lambda \). Set \(v_n:=u_n/\Vert u_{n}\Vert ,\) then there exists a subsequence, again denoted by \(v_n\) such that \(v_n \rightharpoonup v\) in \(H_0^1(\Omega )\), and \(v_n \rightarrow v\) in \(L^p(\Omega )\) for any \(p<2^*\) and a.e. Arguing as in the claim of Proposition 3.1, \(v\equiv 0\). Moreover
On the other hand, from the weak formulation, for all \(\psi \in C_c^\infty (\Omega ),\)
Taking the limit, and since \(C_c^\infty (\Omega )\) is dense in \( L^2(\Omega )\)
Since Lemma 2.1, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) and so \(a(x)\frac{f(u_n)}{\Vert u_n\Vert } \in L^2 (\Omega )\). Moreover \(v_n \rightarrow v=0\) in \(L^2(\Omega )\). Hence [2, Proposition 3.13 (iv)], and (5.6) imply
which contradicts (5.4) and yields \(\Vert u_{\, \Lambda }\Vert <+\infty \).
By Sobolev embedding and the Lebesgue dominated convergence theorem, \(u_{n}\rightarrow u_{\, \Lambda }\) in \(L^{2^*}(\Omega )\).
Now, by substituting \(\psi =u_n\) in (5.5), using Hölder inequality and Sobolev embeddings we obtain
By compactness, for a subsequence again denoted by \(u_n\), \(u_n \rightharpoonup u^*\) in \(H_0^1(\Omega )\), \(u_n \rightarrow u^*\) in \(L^p(\Omega )\) for any \(p<2^*\) and a.e. By uniqueness of the limit, \(u_\Lambda =u^*.\) Finally, by taking limits in the weak formulation of \(u_{n}\) as \(\lambda _n \rightarrow \Lambda \), we get
Hence \(u_{\, \Lambda }\) is a positive weak solution to (1.1)\(_{\Lambda }\). Lemma 2.1 yields that \(u_\Lambda \in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) is a classical solution.
(c) Step 3. Existence of a classical positive solution for \(\lambda \le \lambda _1\).
The existence of a classical positive solution for \(\lambda <\lambda _1\) is done in Theorem 3.3. Let’s look for a solution when \(\lambda =\lambda _1.\)
Since step 1, for any \(\lambda \in (\lambda _1,\Lambda )\) there exists a second positive solution to (1.1)\(_\lambda \). Let’s denote it by \({\tilde{u}}_\lambda \ne u_\lambda .\) Now, define the pointwise limit
Reasoning as in step 2, \(\Vert {\tilde{u}}_{\, \lambda _1}\Vert <+\infty \) and \({\tilde{u}}_{\, \lambda _1}\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) is a classical solution to (1.1)\(_{\lambda _1}\).
Moreover, \({\tilde{u}}_{\, \lambda _1}> 0.\) Assume on the contrary that \({\tilde{u}}_{\, \lambda _1}= 0.\) By the Crandall-Rabinowitz’s Theorem [10], the only nontrivial solutions to (1.1) in a neighbourhood of the bifurcation point \((\lambda _1,0)\) are given by (4.1)). Since Proposition 5.1, those are the minimal solutions \(u_\lambda \), and due to \({\tilde{u}}_\lambda \ne u_\lambda ,\) \({\tilde{u}}_\lambda \) are not in a neighbourhood of \((\lambda _1,0)\), contradicting the definition of \({\tilde{u}}_{\, \lambda _1}(x)\), (5.7)
Hence, \({\tilde{u}}_{\, \lambda _1}\ge 0\), and reasoning as in (1.8), the Maximum Principle implies that \({\tilde{u}}_{\, \lambda _1}> 0.\) \(\square \)
Change history
23 September 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00032-023-00386-1
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Acknowledgements
We would like to thank Professors Xavier Cabré, Carlos Mora and Guido Sweers for helpful discussion and references about Orlicz–Sobolev spaces. This work was started during Pardo’s visit to the LMPA, Université du Littoral Côte d’Opale ULCO, whose invitation and hospitality she thanks.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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Rosa Pardo is supported by grants PID2019-103860GB-I00, MICINN, Spain, and by UCM-BSCH, Spain, GR58/08, Grupo 920894.
Appendices
Some Estimates
First, we prove an useful estimate of \(\frac{\ln (e+s)}{\ln (e+as)}\).
Lemma A.1
Let \(0<a\le 1\) be fixed. Then for all \(s\ge 0\),
Proof
Denote \(\ell (s)=\frac{\ln (e+s)}{\ln (e+as)}\) for all \(s\ge 0\). Then \(1\le \ell (s)\le \ell (s_0)\) where \(s_0 >0\) is the unique value where \(\ell '(s)=0\). When computing \(s_0\) we find
and therefore
Notice that we have \(\ell (s_0)\le \frac{1}{a}\). In order to find a better upper bound of \(\ln (\frac{e+as_0}{e+s_0}) \) let us denote for all \(s\ge 0\)
Then, there exists \(\chi \in (0,s_0)\) such that
Then
and
Since \(\frac{e+as}{a(e+s)}\) is decreasing,
and the first inequality of (A.1) is achieved. The second one is obvious. \(\square \)
Next lemma is about the variations of \(h(s)=\frac{s^{2^*-1}}{[\ln (e+s)]^\alpha }\) for \(s\ge 0\).
Lemma A.2
There exists \(\alpha ^* >2(2^*-1)\) such that h is an increasing function on \(]0,+\infty [\) if and only if \(\alpha \le \alpha ^*\). Moreover, if \(\alpha >\alpha ^*\) there exists \(s_1<s_2\) such that h is increasing in \([0,+\infty [\,\setminus \, ]s_1,s_2[\).
Proof
We have
Let us define for \(s\ge 0\),
so
We have:
Hence:
-
(1)
If \(\frac{\alpha }{2^*-1}\le 1\) then \(\theta '(s)\ge 0\) for all \(s\ge 0\) and in particular \(\theta (s)\ge 0\) and therefore \(h'(s)\ge 0\) for all \(s\ge 0\);
-
(2)
if \(\frac{\alpha }{2^*-1}> 1\) then
$$\begin{aligned} \theta '(s_0)=0\ \hbox { for } s_0= e \left( \frac{\alpha }{2^*-1} -1\right) . \end{aligned}$$Let us compute \(\theta (s_0)\):
$$\begin{aligned} \theta (s_0)=\ln \left( \frac{\alpha }{2^*-1}\right) -\frac{\alpha }{2^*-1} +2, \end{aligned}$$and hence:
-
(i)
if \(\theta (s_0)\ge 0\) then \(\theta (s)\ge 0\) for all \(s\ge 0\) and therefore \(h'(s)\ge 0\) for all \(s\ge 0\);
-
(ii)
if \(\theta (s_0)<0\) then there exists \(s_1<s_2\) such that
$$\begin{aligned} \theta (s)>0 \quad \forall s\in [0,+\infty [\,\setminus \, ]s_1,s_2[ \implies h'(s)>0 \quad \forall s\in [0,+\infty [\,\setminus \, ]s_1,s_2[. \end{aligned}$$Notice that \(t\rightarrow \ln t\) is greater that \(t\rightarrow t-2\) somewhere between some \(t_1<1\) and the value \(t^*=\)the unique solution \(>2\) of the equation
$$\begin{aligned} \ln t^*=t^* -2. \end{aligned}$$Finally the statement of the lemma holds for \(\alpha ^*= t^* (2^*-1)\). \(\square \)
-
(i)
A Compact Embedding Using Orlicz Spaces
For references on Orlicz spaces see [15, 21]. Throughout \(\Omega \subset {\mathbb {R}}^N\) is an bounded open set. We will denote
Definition B.1
We will say that a function \(M:[0,+\infty [\rightarrow [0,+\infty [\) is a N -function if and only if
-
(N1)
M is convex, increasing and continuous,
-
(N2)
\(\displaystyle {\lim _{s\rightarrow 0^+} \frac{M(s)}{s}=0}\),
-
(N3)
\(\displaystyle {\lim _{s\rightarrow +\infty } \frac{M(s)}{s}=+\infty }\).
The proof of the following property is trivial, we just quoted it for the sake of completeness.
Proposition B.2
Any N-function M admits a representation of the form
where \(m:[0,+\infty [\rightarrow [0,+\infty [\) is a non-decreasing right-continuous function satisfying \(m(0)=0\) and
Thus, m is the right-derivative of M.
Our first aim is to prove the following result:
Theorem B.3
Let \(M:[0,+\infty [\rightarrow {\mathbb {R}}\) be a N-function such that
Assume also that M satisfies the \(\Delta _2\)-condition, that is,
Let \(\{u_n\}_{n\in {\mathbb {N}}}\) in \(H_0^1(\Omega )\) be a sequence satisfying
-
1.
\(\sup _{n\in {\mathbb {N}}}\Vert u_n\Vert _{2^{*}} <\infty \),
-
2.
there exists \(u\in H_0^1 (\Omega )\) such that \(\lim _{n\rightarrow +\infty } u_n (x)=u(x)\) a.e.
Then there exists a subsequence \(\{u_{n_k}\}_{k\in {\mathbb {N}}}\) such that
In order to proof this theorem we need some definitions.
Definition B.4
Let \({\mathcal {K}}\subset {{\mathcal {L}}}(\Omega )\). We say that \({{\mathcal {K}}}\) has equi-absolutely continuous integrals if and only if \(\forall \varepsilon >0\) there exists \(h>0\) such that
Lemma B.5
Let \(M:[0,+\infty [\rightarrow {\mathbb {R}}\) be a N-function satisfying the \(\Delta _2\) condition (B.1). Let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a sequence of measurable functions converging a.e. to some function u and such that the set
has equi-absolutely continuous integrals. Then (B.2) holds.
Proof
Let fix \(\varepsilon >0\) and let \( \delta >0 \) be such that
Using Fatou’s lemma we infer that also
Let \(\Omega _n=\{x\in \Omega :\, \vert u_n (x)-u(x)\vert >M^{-1}(\varepsilon )\}\). As a consequence of Egoroff’s theorem, the sequence \((u_n)_{n\in {\mathbb {N}}}\) converge in measure to u so there exists \(n_0\in {\mathbb {N}}\) such that
Then, using the convexity of M and (B.1) it comes
\(\square \)
In order to prove that, for the sequence of our theorem, the set
has equi-absolutely continuous integrals we are going to use the following lemma :
Lemma B.6
Let \({{\mathcal {K}}}\subset {{\mathcal {L}}}(\Omega )\) and let \(\Phi :[0,+\infty [\rightarrow [0,+\infty [\) be an increasing function satisfying
Suppose that there exists \(D>0\) such that
Then all the functions \(u \in {{\mathcal {K}}}\) are integrable and \({{\mathcal {K}}}\) has equi-absolutely continuous integrals (Valle Poussin’s theorem).
Moreover, if \(M:[0,+\infty [\rightarrow [0,+\infty [\) is a continuous increasing function satisfying
then the family \({{\mathcal {K}}}_1 =\{M \big (\vert u\vert \big ) :\; u\in {{\mathcal {K}}}\} \) has equi-absolutely continuous integrals.
Proof
For the Valle Poussin’s theorem see [18] page 159. To prove the second statement remark that the function \({\tilde{\Phi }}=\Phi \circ M^{-1}\) satisfies (B.3). Here \(M^{-1}\) stand for the right-hand inverse. \(\square \)
Proof of theorem B.3
Let us take \(\Phi (s)=\vert s\vert ^{2^*}\). From hypothesis (1) of the theorem, the set \({{\mathcal {K}}}=\{u_n : \, n\in {\mathbb {N}}\}\) satisfies (B.4) for some \(D>0\). Then the conclusion follows from lemma B.5 and Lemma B.6 . \(\square \)
Remark B.7
Whenever (B.2) is satisfied we say that the sequence \(\{u_{n_k}\}_{k\in {\mathbb {N}}}\) converges in M-mean to u.
One can formulate Theorem B.3 as a compact embedding of \(H_0^1(\Omega )\) in some vector space endowed of the Luxembourg norm associate to M (see [15, 21]). Instead, we are going to use the Orlicz-norm which is more suitable to our purposes. We will see later in Theorem B.12 that the convergence in M-mean implies the convergence with respect to the Orlicz-norm, provided that the \(\Delta _2\)-condition is satisfied.
Definition B.8
Let M be a N-function. The complementary of M defined for all \(s\ge 0 \) is the function
As before, we give the following trivial result for the sake of completeness:
Proposition B.9
If m is the right derivative of M then
is the right derivative of \(M^*\) and \(M^*\) is a N-function. Furthermore, for all \(s\ge 0\) we have
Next, let us introduce the Orlicz norm associated to M :
Definition B.10
Let M be a N-function and let \(M^*\) be its complementary. Let us denote for any \(v\in {{\mathcal {L}}}(\Omega )\)
and define the Orlicz norm of any \(u\in {{\mathcal {L}}}(\Omega )\) by
\(\Vert \cdot \Vert _M\) is a norm in the real vector space
(see [15] for the details). Let us prove the following less trivial properties:
Proposition B.11
-
(i)
For all \(u \in {{\mathcal {L}}}(\Omega )\),
$$\begin{aligned} \Vert u\Vert _M\le \int _\Omega M(\vert u\vert )\, dx +1. \end{aligned}$$(B.5) -
(ii)
For any u and v in \( {{\mathcal {L}}}(\Omega )\) it holds
$$\begin{aligned} \left| \int _\Omega uv\, dx\right| \le \Vert u\Vert _M\, \Vert v\Vert _{M^*} \hbox { (Holder's inequality)}. \end{aligned}$$(B.6) -
(iii)
For any u and v in \( {{\mathcal {L}}}(\Omega )\) we have \(\Vert u\Vert _M\, \le \Vert v\Vert _M \) if \( \vert u\vert \le \vert v\vert \) a.e.
Proof
-
(i)
This follows from the definition of \(\Vert \cdot \Vert _M\) and the inequality \(\vert uv\vert \le M(\vert u\vert )+M^*\big (\vert v\vert \big )\).
-
(ii)
The divide the proof in 3 steps. Step 1: For all \(v\in {{\mathcal {L}}}(\Omega ), \)
$$\begin{aligned} \left| \int _\Omega uv\, dx \right| \le \left\{ \begin{array}{ll} \Vert u\Vert _M &{} \hbox { if }\rho (v,M^*)\le 1\\ \rho (v,M^*) \Vert u\Vert _M &{} \hbox { if } \rho (v,M^*)> 1 \end{array}\right. \end{aligned}$$Indeed, the first case follows directly from the definition. If \(\rho (v,M^*)>1\) then by convexity
$$\begin{aligned} M^*\left( \frac{\vert v\vert }{\rho (v,M^*)}\right) \le \frac{M^* \big (\vert v\vert \big )}{\rho (v,M^*)} \end{aligned}$$and therefore
$$\begin{aligned} \rho \left( \frac{\vert v\vert }{\rho (v,M^*)}, M^*\right) \le \frac{1}{\rho (v,M^*)}\int _{\Omega }M^*\big (\vert v\vert \big )dx=1 \end{aligned}$$and
$$\begin{aligned} \left| \int _\Omega u \frac{v}{\rho (v,M^*)}\, dx\right| \le \Vert u\Vert _M. \end{aligned}$$Step 2: If \(\Vert u\Vert _M \le 1\) then \(\rho \big (m(\vert u\vert ), M^*\big )\le 1\). Set \(\ u_n =u\chi _{\{\vert u\vert \le n\}}\ \) for all \(\ n\in {\mathbb {N}}\). Since \(u_n\) is bounded then \(\rho (m(\vert u_n\vert ), M^*)<+\infty \). Assume by contradiction that \( \int _\Omega M^* \big (m(\vert u\vert )\big )\, dx>1\) and let \(n_0\in {\mathbb {N}}\) be such that \(\int _\Omega M^* \big (m(\vert u_{n_0}\vert )\big )\, dx>1.\) We have
$$\begin{aligned} M^* \big (m(\vert u_{n_0}\vert )\big ) <M \big (\vert u_{n_0}\vert \big )+M^* \big (m(\vert u_{n_0}\vert ) \big ) =\vert u_{n_0}\vert \, m(\vert u_{n_0}\vert ) \end{aligned}$$and therefore, by (i),
$$\begin{aligned} \rho \big (m(\vert u_{n_0}\vert ), M^*\big )< \int _\Omega \vert u_{n_0}\vert \, m(\vert u_{n_0}\vert )\, dx \le \Vert u_{n_0}\Vert _M\,\rho \big (m(\vert u_{n_0}\vert ), M^*\big ) \end{aligned}$$which contradicts \(\Vert u_{n_0}\Vert _M\le \Vert u\Vert _M\le 1.\) This is trivial from the definition of \(\Vert u\Vert _M\), step 1 and the fact that \(\vert u\vert m(\vert u\vert )=M(\vert u\vert )+M^* \big (m(\vert u\vert )\big ).\) Step 3: If \(\Vert u\Vert _M\le 1\) then \(\rho (u,M)\le \Vert u\Vert _M.\) Let us remark that for all \(s\ge 0\)
$$\begin{aligned} M^*(m(s))+M(s)=sm(s). \end{aligned}$$Set \(v_0=m(\vert u\vert ). \) From step 2, \(\rho (v_0, M^*)\le 1\) and then
$$\begin{aligned} \rho (u,M)\le \rho (u,M)+\rho (v_0, M^*)= \int _\Omega uv_0\, dx \le \Vert u\Vert _M. \end{aligned}$$
Now we prove Holder’s inequality. From step 2 applied to \(M^*\) and \(\frac{v}{\Vert v\Vert _{M^*}}\) we have \(\rho \Big (\frac{v}{\Vert v\Vert _{M^*}},M^*\Big )\le 1,\) so then
and Holder’s inequality follows.
The proof of (iii) is trivial. \(\square \)
Finally, we give the following compact embedding result:
Theorem B.12
Let M be a N-function satisfying the \(\Delta _2\)-condition (B.1) and let \(\{u_n\}_{n\in {\mathbb {N}}}\) be a sequence in \({{\mathcal {L}}}(\Omega )\) such that
Then
Thus, the convergence in M-mean implies the converge with respect to the \(\Vert \cdot \Vert _M\) norm.
Proof
Let \(\varepsilon >0\) and take \(m\in {\mathbb {N}}\) such that \(\frac{1}{2^{m-1}}<\varepsilon .\) Using condition (B.1) we also have
Let \(n_0\in {\mathbb {N}}\) be such that for all \(n\ge n_0\) we have
From step 1 of the proof in the previous proposition we have that for all \(n\ge n_0\)
which implies that
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Cuesta, M., Pardo, R. Positive Solutions for Slightly Subcritical Elliptic Problems Via Orlicz Spaces. Milan J. Math. 90, 229–255 (2022). https://doi.org/10.1007/s00032-022-00354-1
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DOI: https://doi.org/10.1007/s00032-022-00354-1