Abstract
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to \(L^{\infty }\), which implies \(C^{1,\alpha }\) smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.
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1 Introduction
We consider \(\Omega \subset \mathbb {R}^N\) (\(N\ge 2\)) an open bounded domain with smooth boundary \(\partial \Omega \). A classical result in the theory of eigenvalue problems guarantees that the problem
possesses a nondecreasing sequence of eigenvalues and a sequence of corresponding eigenfunctions which define a Hilbert basis in \(L^2(\Omega )\) (see, [16]). Moreover, it is known that the first eigenvalue of problem (1.1) is characterized in the variational point of view by
Suppose that \(p>1\) is a given real number and consider the nonlinear eigenvalue problem with Neumann boundary condition
where \(\Delta _p u:=\text {div}(|\nabla u|^{p-2}\nabla u)\) stands for the p-Laplace operator and \(\lambda \in \mathbb {R}\). This problem was considered in [15], and using a direct method in calculus of variations (if \(p>2\)) or a mountain-pass argument (if \(p\in (\frac{2N}{N+2},2)\)) it was shown that the set of eigenvalues of problem (1.2) is exactly the interval \([0,\infty )\). Indeed, it is sufficient to find one positive eigenvalue, say \(-\Delta _p u=\lambda u.\) Then a continuous family of eigenvalues can be found by the reparametrization \(u=\alpha v,\) satisfying \(-\Delta _p v=\mu (\alpha )v\), with \(\mu (\alpha )=\frac{\lambda }{\alpha ^{p-2}}.\)
In this paper, we consider the so-called (p, 2)-Laplace operator (see, [18]) with Dirichlet boundary conditions. More precisely, we analyze the following nonlinear eigenvalue problem:
where \(p\in (1,\infty )\backslash \{2\}\) is a real number. We recall that if \(1<p<q,\) then \(L^q(\Omega )\subset L^p(\Omega )\) and as a consequence, one has \(W^{1,q}_0(\Omega )\subset W^{1,p}_0(\Omega ).\) We will say that \(\lambda \in \mathbb {R}\) is an eigenvalue of problem (1.3) if there exists \(u\in W^{1,p}_0(\Omega )\backslash \{0\}\) (if \(p>2\) ), \(u\in W^{1,2}_0(\Omega )\backslash \{0\}\) (if \(1<p<2\)) such that
for all \(v\in W^{1,p}_{0}(\Omega )\) (if \(p>2\)), \(v\in W^{1,2}_{0}(\Omega )\) (if \(1<p<2\)). In this case, such a pair \((u,\lambda )\) is called an Eigenpair, and \(\lambda \in \mathbb {R}\) is called an eigenvalue and \(u\in W^{1,p}_0(\Omega )\backslash \{0\}\) is an eigenfunction associated with \(\lambda .\) We say that \(\lambda \) is a “first eigenvalue”, if the corresponding eigenfunction u is positive or negative.
The operator \(-\Delta _p-\Delta \) appears in quantum field theory (see, [5]), where it arises in the mathematical description of propagation phenomena of solitary waves. We recall that a solitary wave is a wave which propagates without any temporal evolution in shape.
The operator \(-\Delta _p-\Delta \) is a special case of the so called (p, q)-Laplace operator given by \(-\Delta _p-\Delta _q\) which has been widely studied; for some results related to our studies, see, e.g., [6, 7, 10, 21, 25] .
The main purpose of this work was to study the nonlinear eigenvalue problem (1.3) when \(p>2,\) and \(1<p<2\), respectively . In particular, we show in section 2 that the set of the first eigenvalues is given by the interval \((\lambda _1^D,\infty )\), where \(\lambda _1^D\) is the first Dirichlet eigenvalue of the Laplacian. We show that the first eigenvalue of (1.3) can be obtained variationally, using a Nehari set for \(1<p<2\), and a minimization for \(p>2.\) Also in the same section, we recall some results of [15, 22, 23].
In Sect. 3, we prove that the eigenfunctions associated with \(\lambda \) belong to \(L^{\infty }(\Omega )\): the first eigenvalue \(\lambda _1^D\) of problem (1.3) is simple and the corresponding eigenfunctions are positive or negative. In addition, in Sect. 3.3 we show a homeomorphism property related to \(-\Delta _p-\Delta \).
In Sect. 4, we prove that \(\lambda _1^D\) is a bifurcation point for a branch of first eigenvalues from zero if \(p>2,\) and \(\lambda _1^D\) is a bifurcation point from infinity if \(p<2.\) Also the higher Dirichlet eigenvalues \(\lambda ^D_k\) are bifurcation points (from 0 if \(p>2,\) respectively, from infinity if \(1<p<2\) ), if the multiplicity of \(\lambda ^D_k\) is odd. Finally in Sect. 5, we prove by variational methods that if \(\lambda \in (\lambda ^D_k,\lambda ^D_{k+1}),\) then there exist at least k nonlinear eigenvalues using Krasnoselski’s genus. In what follows, we denote by \(\Vert .\Vert _{1,p}\) and \(\Vert .\Vert _2\) the norms on \(W^{1,p}_0(\Omega )\) and \(L^2(\Omega )\) defined, respectively, by
We recall the Poincaré inequality, i.e., there exists a positive constant \(C_p(\Omega )\) such that
2 The Spectrum of the Nonlinear Problem
We now begin with the discussion of the properties of the spectrum of the nonlinear eigenvalues problem (1.3).
Remark 2.1
Any \(\lambda \le 0\) is not an eigenvalue of problem (1.3).
Indeed, suppose by contradiction that \(\lambda =0\) is an eigenvalue of equation (1.3), then relation (1.4) with \(v=u_{0}\) gives
Consequently, \(|\nabla u_{0}|=0\); therefore, \(u_0\) is constant on \(\Omega \) and \(u_0=0\) on \(\Omega \). And this contradicts the fact that \(u_0\) is a nontrivial eigenfunction. Hence \(\lambda =0\) is not an eigenvalue of problem (1.3). Now it remains to show that any \(\lambda <0\) is not an eigenvalue of (1.3). Suppose by contradiction that \(\lambda <0\) is an eigenvalue of (1.3), with \(u_{\lambda }\in W^{1,p}_0(\Omega )\backslash \{0\}\) the corresponding eigenfunction. The relation (1.4) with \(v=u_{\lambda }\) implies
Which yields a contradiction and thus \(\lambda <0\) cannot be an eigenvalue of problem (1.3).
Lemma 2.2
Any \(\lambda \in (0,\lambda ^D_1]\) is not an eigenvalue of (1.3).
For the proof see also [15].
Proof
Let \(\lambda \in (0,\lambda ^D_1)\), i.e., \(\lambda ^D_1>\lambda .\) Let us assume by contradiction that there exists a \(\lambda \in (0,\lambda ^D_1)\) which is an eigenvalue of (1.3) with \(u_{\lambda }\in W^{1,2}_0(\Omega )\backslash \{0\},\) the corresponding eigenfunction. Letting \(v=u_{\lambda }\) in relation (1.4), we have on the one hand,
and on the other hand,
By subtracting both sides of (2.1) by \( \lambda \displaystyle {\int _{\Omega }}u_{\lambda }^2~\mathrm{d}x\), we obtain
Therefore, \((\lambda ^D_1-\lambda ) \displaystyle {\int _{\Omega }}u^2_{\lambda }~\mathrm{d}x\le 0,\) which is a contradiction. Hence, we conclude that \(\lambda \in (0,\lambda ^D_1)\) is not an eigenvalue of problem (1.3). In order to complete the proof of the Lemma 2.2 we shall show that \(\lambda =\lambda ^D_1\) is not an eigenvalue of (1.3).
By contradiction we assume that \(\lambda =\lambda ^D_1\) is an eigenvalue of (1.3). So there exists \(u_{\lambda ^D_1}\in W^{1,2}_0(\Omega )\backslash \{0\}\) such that relation (1.4) holds true. Letting \(v=u_{\lambda ^D_1}\) in relation (1.4), it follows that
But \(\lambda ^D_1 \displaystyle {\int _{\Omega }}u_{\lambda ^D_1}^2~\mathrm{d}x\le \displaystyle {\int _{\Omega }}|\nabla u_{\lambda ^D_1}|^2~\mathrm{d}x;\) therefore
Using relation (1.5), we have \(u_{\lambda ^D_1}=0,\) which is a contradiction since \(u_{\lambda ^D_1}\in W^{1,2}_0(\Omega )\backslash \{0\}.\) So \(\lambda =\lambda ^D_1\) is not an eigenvalue of (1.3). \(\square \)
Theorem 2.3
Assume \(p\in (1,2)\). Then the set of first eigenvalues of problem (1.3) is given by
Proof
Let \(\lambda \in (\lambda _1^D,\infty )\), and define the energy functional
One shows that \(J_{\lambda }\in C^1(W^{1,2}_0(\Omega ),\mathbb {R})\) (see, [18]) with its derivatives given by
Thus we note that \(\lambda \) is an eigenvalue of problem (1.3) if and only if \(J_{\lambda }\) possesses a nontrivial critical point. Considering \(J_{\lambda }(\rho e_1),\) where \(e_1\) is the \(L^2\)-normalized first eigenfunction of the Laplacian, we see that
Hence, we cannot establish the coercivity of \(J_{\lambda }\) on \(W^{1,2}_0(\Omega )\) for \(p\in (1,2)\), and consequently we cannot use a direct method in calculus of variations in order to determine a critical point of \(J_{\lambda }.\) To overcome this difficulty, the idea will be to analyze the functional \(J_{\lambda }\) on the so-called Nehari manifold defined by
Note that all non-trivial solutions of (1.3) lie on \(\mathcal {N}_{\lambda }.\) On \(\mathcal {N}_{\lambda }\) the functional \(J_{\lambda }\) takes the following form
We have seen in Lemma 2.2 that any \(\lambda \in (0,\lambda _1^D]\) is not an eigenvalue of problem (1.3); see also [15]. It remains to prove the following:
Claim: Every \(\lambda \in (\lambda ^D_1,\infty )\) is a first eigenvalue of problem (1.3). Indeed, we will split the proof of the claim into four steps follows:
-
Step 1.
Here we will show that \(\mathcal {N}_{\lambda }\ne \emptyset \) and every minimizing sequence for \(J_{\lambda }\) on \(\mathcal {N}_{\lambda }\) is bounded. Since \(\lambda >\lambda ^D_1\) there exists \(v_{\lambda }\in W^{1,2}_0(\Omega )\) such that
$$\begin{aligned} \int _{\Omega }\left| \nabla v_{\lambda }\right| ^2<\lambda \int _{\Omega }v^2_{\lambda }~\mathrm{d}x. \end{aligned}$$Then there exists \(t>0\) such that \(tv_{\lambda }\in \mathcal {N}_{\lambda } \Rightarrow \)
$$\begin{aligned}&\int _{\Omega }\left| \nabla \left( tv_{\lambda }\right) \right| ^2~\mathrm{d}x+\int _{\Omega }\left| \nabla \left( tv_{\lambda }\right) \right| ^p~\mathrm{d}x=\lambda \int _{\Omega }\left( tv_{\lambda }\right) ^2~\mathrm{d}x\Rightarrow \nonumber \\&t^2\int _{\Omega }\left| \nabla v_{\lambda }\right| ^2~\mathrm{d}x+t^p\int _{\Omega }\left| \nabla v_{\lambda }\right| ^p~\mathrm{d}x=t^2\lambda \int _{\Omega }v_{\lambda }^2~\mathrm{d}x \Rightarrow \nonumber \\&t=\left( \frac{\int _{\Omega }|\nabla v_{\lambda }|^p~\mathrm{d}x}{\lambda \int _{\Omega }v_{\lambda }^2~\mathrm{d}x- \int _{\Omega }|\nabla v_{\lambda }|^2~\mathrm{d}x}\right) ^{\frac{1}{2-p}}>0. \end{aligned}$$With such t we have \(tv_{\lambda }\in \mathcal {N}_{\lambda }\) and \(\mathcal {N}_{\lambda }\ne \emptyset .\) Note that for \(u\in B_r(v_{\lambda }),\) \(r>0\) small, the inequality \(\lambda \int _{\Omega }|u|^2\mathrm{d}x>\int _{\Omega }|\nabla u|^2\mathrm{d}x\) remains valid, and then \(t(u)u\in \mathcal {N}_{\lambda }\) for \(u\in B_r(v_{\lambda }).\) Since \(t(u)\in C^1\) we conclude that \(\mathcal {N}_{\lambda }\) is a \(C^1\)-manifold. Let \(\{u_k\}\subset \mathcal {N}_{\lambda }\) be a minimizing sequence of \(J_{\lambda }|_{\mathcal {N}_{\lambda }}\), i.e., \(J_{\lambda }(u_k)\rightarrow m=\displaystyle {\inf _{w\in \mathcal {N}_{\lambda }}}J_{\lambda }(w).\) Then
$$\begin{aligned}&\lambda \int _{\Omega }u^2_k~\mathrm{d}x{-}\int _{\Omega }|\nabla u_k|^2~\mathrm{d}x\nonumber \\&\quad =\int _{\Omega }|\nabla u_k|^p~\mathrm{d}x{\rightarrow }\left( \frac{2}{p}-1\right) ^{-1}m~\mathrm {as}\,k{\rightarrow }\infty . \end{aligned}$$(2.2)Assume by contradiction that \(\{u_k\}\) is not bounded, i.e., \(\displaystyle \int _{\Omega }|\nabla u_k|^2~\mathrm{d}x\rightarrow \infty \) as \(k\rightarrow \infty \). It follows that \(\displaystyle {\int _{\Omega }}u^2_k~\mathrm{d}x \rightarrow \infty \) as \(k\rightarrow \infty \), thanks to relation (2.2). We set \(v_k=\frac{u_k}{\Vert u_k\Vert _2}.\) Since \(\displaystyle {\int _{\Omega }}|\nabla u_k|^2~\mathrm{d}x<\lambda \displaystyle {\int _{\Omega }}u_k^2~\mathrm{d}x \), we deduce that \(\displaystyle {\int _{\Omega }}|\nabla v_k|^2~\mathrm{d}x<\lambda ,\) for each k and \(\Vert v_k\Vert _{1,2}<\sqrt{\lambda }.\) Hence \(\{v_k\}\subset W^{1,2}_0(\Omega )\) is bounded in \(W^{1,2}_0(\Omega ).\) Therefore, there exists \(v_0\in W^{1,2}_0(\Omega )\) such that \(v_k\rightharpoonup v_0\) in \(W^{1,2}_0(\Omega )\subset W^{1,p}_0(\Omega )\) and \(v_k\rightarrow v_0\) in \(L^2(\Omega ).\) Dividing relation (2.2) by \(\Vert u_k\Vert ^p_{2}\), we get
$$\begin{aligned} \int _{\Omega }|\nabla v_k|^p~\mathrm{d}x=\frac{\lambda \displaystyle {\int _{\Omega }}u^2_k~\mathrm{d}x- \displaystyle {\int _{\Omega }}|\nabla u_k|^2~\mathrm{d}x}{\Vert u_k\Vert ^p_{2}}\rightarrow 0~~\mathrm {as}\,k \rightarrow \infty , \end{aligned}$$since \(\lambda \displaystyle {\int _{\Omega }}u^2_k~\mathrm{d}x- \displaystyle {\int _{\Omega }}|\nabla u_k|^2~\mathrm{d}x\rightarrow \left( \frac{2}{p}-1\right) ^{-1}m<\infty \) and \(\Vert u_k\Vert ^p_{2}\rightarrow \infty \) as \(k\rightarrow \infty \). On the other hand, since \(v_k\rightharpoonup v_0\) in \(W^{1,p}_0(\Omega )\), we have \(\displaystyle {\int _{\Omega }}|\nabla v_0|^p~\mathrm{d}x\le \lim _{k\rightarrow \infty }\inf \displaystyle {\int _{\Omega }}|\nabla v_k|^p~\mathrm{d}x=0\) and consequently \(v_0=0\). It follows that \(v_k\rightarrow 0\) in \(L^2(\Omega ),\) which is a contradiction since \(\Vert v_k\Vert _{2}=1\). Hence, \(\{u_k\}\) is bounded in \(W^{1,2}_0(\Omega ).\)
-
Step 2.
\(m= \displaystyle {\inf _{w\in \mathcal {N}_{\lambda }}}J_{\lambda }(w)>0.\) Indeed, assume by contradiction that \(m=0\). Then, for \(\{u_k\}\) as in step 1, we have
$$\begin{aligned}&0<\lambda \int _{\Omega }u^2_k~\mathrm{d}x- \int _{\Omega }\left| \nabla u_k\right| ^2~\mathrm{d}x\nonumber \\&\quad =\int _{\Omega }\left| \nabla u_k\right| ^p~\mathrm{d}x\rightarrow 0, \text {as} k\rightarrow \infty . \end{aligned}$$(2.3)By Step 1, we deduce that \(\{u_k\}\) is bounded in \(W^{1,2}_0(\Omega ).\) Therefore there exists \(u_0\in W^{1,2}_0(\Omega )\) such that \(u_k \rightharpoonup u_0\) in \(W^{1,2}_0(\Omega )\) and \(W^{1,p}_0(\Omega )\) and \(u_k\rightarrow u_0\) in \(L^2(\Omega ).\) Thus \(\displaystyle {\int _{\Omega }}|\nabla u_0|^p\le \lim _{k\rightarrow \infty }\inf \displaystyle {\int _{\Omega }}|\nabla u_k|^p~\mathrm{d}x=0.\) And consequently \(u_0=0\), \(u_k \rightharpoonup 0\) in \(W^{1,2}_0(\Omega )\) and \(W^{1,p}_0(\Omega )\) and \(u_k\rightarrow 0\) in \(L^2(\Omega ).\) Writing again \(v_k=\frac{ u_k}{\Vert u_k\Vert _2}\) we have
$$\begin{aligned} 0<\frac{\lambda \displaystyle {\int _{\Omega }}u^2_k~\mathrm{d}x-\int _{\Omega }\left| \nabla u_k\right| ^2~\mathrm{d}x}{\Vert u_k\Vert ^2_{2}}=\Vert u_k\Vert ^{p-2}_{2} \displaystyle {\int _{\Omega }}\left| \nabla v_k\right| ^p~\mathrm{d}x, \end{aligned}$$therefore,
$$\begin{aligned} \int _{\Omega }\left| \nabla v_k\right| ^p~\mathrm{d}x= & {} \Vert u_k\Vert ^{2-p}_{2}\left( \frac{\lambda \Vert u_k\Vert ^2_2}{\Vert u_k\Vert ^2_{2}}-\frac{\displaystyle {\int _{\Omega }}\left| \nabla u_k\right| ^2~\mathrm{d}x}{\Vert u_k\Vert ^2_{2}}\right) \\= & {} \Vert u_k\Vert ^{2-p}_{2}\left( \lambda - \displaystyle {\int _{\Omega }}\left| \nabla v_k\right| ^2~\mathrm{d}x\right) \rightarrow 0~ \mathrm {as}\, k\rightarrow \infty , \end{aligned}$$since \(\Vert u_k\Vert _{2}\rightarrow 0\) and \(p\in (1,2)\), and \(\{v_k\}\) is bounded in \(W_0^{1,2}(\Omega ).\) Next since \(v_k \rightharpoonup v_0\), we deduce that \(\displaystyle {\int _{\Omega }}|\nabla v_0|^p~\mathrm{d}x\le \lim _{k\rightarrow \infty }\inf \displaystyle {\int _{\Omega }}|\nabla v_k|^p\mathrm{d}x=0\) and we have \(v_0=0.\) And it follows that \(v_k\rightarrow 0\) in \(L^2(\Omega )\) which is a contradiction since \(\Vert v_k\Vert _{2}=1\) for each k. Hence, m is positive.
-
Step 3.
There exists \(u_0\in \mathcal {N}_{\lambda }\) such that \(J_{\lambda }(u_0)=m.\) Let \(\{u_k\}\subset \mathcal {N}_{\lambda }\) be a minimizing sequence, i.e., \(J_{\lambda }(u_k)\rightarrow m\) as \(k\rightarrow \infty .\) Thanks to Step 1, we have that \(\{u_k\}\) is bounded in \(W_0^{1,2}(\Omega ).\) It follows that there exists \(u_0\in W_0^{1,2}(\Omega )\) such that \(u_k\rightharpoonup u_0\) in \(W_0^{1,2}(\Omega )\) and \(W_0^{1,p}(\Omega )\) and strongly in \(L^2(\Omega ).\) The results in the two steps above guarantee that \(J_{\lambda }(u_0)\le \displaystyle {\lim _{k\rightarrow \infty }}\inf J_{\lambda }(u_k)=m.\) Since for each k we have \(u_k\in \mathcal {N}_{\lambda }\), we have
$$\begin{aligned} \int _{\Omega }\left| \nabla u_k\right| ^2~\mathrm{d}x+\int _{\Omega }\left| \nabla u_k\right| ^p~\mathrm{d}x=\lambda \int _{\Omega }u^2_k~\mathrm{d}x~~~\text {for all}\, k. \end{aligned}$$(2.4)Assuming \(u_0\equiv 0\) on \(\Omega \) implies that \( \displaystyle {\int _{\Omega }}u^2_k~\mathrm{d}x\rightarrow 0\) as \(k\rightarrow \infty \), and by relation (2.4) we obtain that \(\displaystyle {\int _{\Omega }}|\nabla u_k|^2~\mathrm{d}x\rightarrow 0\) as \(k\rightarrow \infty .\) Combining this with the fact that \(u_k\) converges weakly to 0 in \(W_0^{1,2}(\Omega )\), we deduce that \(u_k\) converges strongly to 0 in \(W_0^{1,2}(\Omega )\) and consequently in \(W_0^{1,p}(\Omega )\). Hence we infer that
$$\begin{aligned} \lambda \int _{\Omega }u^2_k~\mathrm{d}x- \int _{\Omega }\left| \nabla u_k\right| ^2~\mathrm{d}x=\int _{\Omega }\left| \nabla u_k\right| ^p~\mathrm{d}x\rightarrow 0, \text {as} k\rightarrow \infty . \end{aligned}$$Next, using similar argument as the one used in the proof of Step 2, we will reach to a contradiction, which shows that \(u_0\not \equiv 0.\) Letting \(k\rightarrow \infty \) in relation (2.4), we deduce that
$$\begin{aligned} \int _{\Omega }|\nabla u_0|^2~\mathrm{d}x+\int _{\Omega }|\nabla u_0|^p~\mathrm{d}x\le \lambda \int _{\Omega }u_0^2~\mathrm{d}x. \end{aligned}$$If there is equality in the above relation, then \(u_0\in \mathcal {N}_{\lambda }\) and \(m\le J_{\lambda }(u_0)\). Assume by contradiction that
$$\begin{aligned} \int _{\Omega }|\nabla u|^2~\mathrm{d}x+\int _{\Omega }|\nabla u|^p~\mathrm{d}x<\lambda \int _{\Omega }u^2~\mathrm{d}x. \end{aligned}$$(2.5)Let \(t>0\) be such that \(tu_0\in \mathcal {N}_{\lambda },\) i.e.,
$$\begin{aligned} t=\left( \frac{\lambda \displaystyle {\int _{\Omega }}u_0^2~\mathrm{d}x- \displaystyle {\int _{\Omega }}\left| \nabla u_0\right| ^2~\mathrm{d}x}{\displaystyle {\int _{\Omega }}\left| \nabla u_0\right| ^p~\mathrm{d}x}\right) ^{\frac{1}{p-2}}. \end{aligned}$$We note that \(t\in (0,1)\) since \(1<t^{p-2}\) (thanks to (2.5)). Finally, since \(tu_0\in \mathcal {N}_{\lambda }\) with \(t\in (0,1)\) we have
$$\begin{aligned}&0<m\le J_{\lambda }(tu_0)\\&\quad =\left( \frac{2}{p}-1\right) \int _{\Omega }\left| \nabla (tu_0)\right| ^p~\mathrm{d}x=t^p\left( \frac{2}{p}-1\right) \int _{\Omega }\left| \nabla u_0\right| ^p~\mathrm{d}x\\&\quad =t^p J_{\lambda }(u_0)\\&\quad \le t^p\lim _{k\rightarrow \infty }\inf J_{\lambda }(u_k)=t^p m<m~\text {for}\, t\in (0,1), \end{aligned}$$and this is a contradiction which assures that relation (2.5) cannot hold and consequently we have \(u_0\in \mathcal {N}_{\lambda }\). Hence \(m\le J_{\lambda }(u_0)\) and \( m= J_{\lambda }(u_0)\).
-
Step 4.
We conclude the proof of the claim. Let \(u\in \mathcal {N}_{\lambda }\) be such that \(J_{\lambda }(u)=m\) (thanks to Step 3). Since \(u\in \mathcal {N}_{\lambda }\), we have
$$\begin{aligned} \int _{\Omega }|\nabla u|^2~\mathrm{d}x+\int _{\Omega }|\nabla u|^p~\mathrm{d}x=\lambda \int _{\Omega }u^2~\mathrm{d}x, \end{aligned}$$and
$$\begin{aligned} \int _{\Omega }|\nabla u|^2~\mathrm{d}x<\lambda \int _{\Omega }u^2~\mathrm{d}x. \end{aligned}$$Let \(v\in \partial B_1(0)\subset W_0^{1,2}(\Omega )\) and \(\varepsilon >0\) be very small such that \(u+\delta v\ne 0\) in \(\Omega \) for all \(\delta \in (-\varepsilon ,\varepsilon )\) and
$$\begin{aligned} \int _{\Omega }\left| \nabla (u+\delta v)\right| ^2~\mathrm{d}x<\lambda \int _{\Omega }\left( u+\delta v\right) ^2~\mathrm{d}x ; \end{aligned}$$this is equivalent to
$$\begin{aligned}&\qquad \lambda \int _{\Omega }u^2~\mathrm{d}x-\int _{\Omega }|\nabla u|^2~\mathrm{d}x>\delta \left( 2\int _{\Omega }\nabla u\cdot \nabla v~\mathrm{d}x- 2\lambda \int _{\Omega }u v~\mathrm{d}x\right) \\&\qquad \quad +\delta ^2\left( \int _{\Omega }|\nabla v|^2~\mathrm{d}x-\lambda \int _{\Omega }v^2~\mathrm{d}x\right) , \end{aligned}$$which holds true for \(\delta \) small enough since the left-hand side is positive while the function
$$\begin{aligned} h(\delta ):= & {} |\delta |\left| 2\int _{\Omega }\nabla u\cdot \nabla v~\mathrm{d}x- 2\lambda \int _{\Omega }u v~\mathrm{d}x\right| \\&+\,\delta ^2\left| \int _{\Omega }|\nabla v|^2~\mathrm{d}x-\lambda \int _{\Omega }v^2~\mathrm{d}x\right| \end{aligned}$$dominates the term from the right-hand side and \(h(\delta )\) is a continuous function (polynomial in \(\delta \)) which vanishes in \(\delta =0\). For each \(\delta \in (-\varepsilon ,\varepsilon ),\) let \(t(\delta )>0\) be given by
$$\begin{aligned} t(\delta )=\left( \frac{\lambda \displaystyle {\int _{\Omega }}\left( u+\delta v\right) ^2~\mathrm{d}x- \int _{\Omega }\left| \nabla (u+\delta v)\right| ^2~\mathrm{d}x}{\displaystyle {\int _{\Omega }}\left| \nabla (u+\delta v)\right| ^p~\mathrm{d}x}\right) ^{\frac{1}{p-2}}, \end{aligned}$$so that \(t(\delta )\cdot (u+\delta v)\in \mathcal {N}_{\lambda }.\) We have that \(t(\delta )\) is of class \(C^1(-\varepsilon ,\varepsilon )\) since \(t(\delta )\) is the composition of some functions of class \(C^1\). On the other hand, since \(u\in \mathcal {N}_{\lambda }\) we have \(t(0)=1.\) Define \(\iota :(-\varepsilon ,\varepsilon )\rightarrow \mathbb {R}\) by \(\iota (\delta )=J_{\lambda }(t(\delta )(u+\delta v))\) which is of class \(C^1(-\varepsilon ,\varepsilon )\) and has a minimum at \(\delta =0.\) We have
$$\begin{aligned}&\iota '(\delta )=\left[ t'(\delta )(u+\delta v)+v t(\delta )\right] J'_{\lambda }\left( t(\delta )(u+\delta v)\right) \Rightarrow \nonumber \\&0=\iota '(0)=J'_{\lambda }\left( t(0)(u)\right) \left[ t'(0)u+v t(0)\right] =\langle J'_{\lambda }(u),v\rangle \end{aligned}$$since \(t(0)=1\) and \(t'(0)=0.\) This shows that every \(\lambda \in (\lambda ^D_1,\infty )\) is an eigenvalue of problem (1.3).
\(\square \)
In the next theorem we consider the case \(p>2.\) For similar results for the Neumann case, (see, [22]).
Theorem 2.4
For \(p>2\), the set of first eigenvalues of problem (1.3) is given by \((\lambda ^D_1,\infty ).\)
The proof of Theorem 2.4 will follow as a direct consequence of the lemmas proved below:
Lemma 2.5
Let
Then \(\lambda _1(p)=\lambda ^D_1,\) for all \(p>2.\)
Proof
We clearly have \(\lambda _1(p)\ge \lambda ^D_1\) since a positive term is added. On the other hand, consider \(u_n=\frac{1}{n} e_1\) (where \(e_1\) is the first eigenfunction of \(-\Delta \)), we get
\(\square \)
Lemma 2.6
For each \(\lambda >0\), we have
Proof
Clearly,
On the one hand, using Poincaré’s inequality with \(p=2\), we have \(\displaystyle {\int _{\Omega }}u^2~\mathrm{d}x\le C_2(\Omega )\displaystyle {\int _{\Omega }}|\nabla u|^2~\mathrm{d}x, \forall u\in W^{1,p}_0(\Omega )\subset W^{1,2}_0(\Omega )\) and then applying the Hölder inequality to the right-hand side term of the previous estimate, we obtain
so \(\displaystyle {\int _{\Omega }}u^2~\mathrm{d}x\le D \Vert u\Vert ^2_{1,p},\) where \(D=C_2(\Omega ) |\Omega |^{\frac{p-2}{p}}.\) Therefore, for \(\lambda >0,\)
and the the right-hand side of (2.6) tends to \(\infty \), as \(\Vert u\Vert _{1,p}\rightarrow \infty \), since \(p>2.\) \(\square \)
Lemma 2.7
Every \(\lambda \in (\lambda ^D_1,\infty )\) is a first eigenvalue of problem (1.3).
Proof
For each \(\lambda >\lambda ^D_1\) define \(F_{\lambda }: W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) by
Standard arguments show that \(F_{\lambda }\in C^1(W^{1,p}_0(\Omega ),\mathbb {R})\) (see, [18]) with its derivative given by
for all \(u,\phi \in W^{1,p}_0(\Omega ).\) Estimate (2.6) shows that \(F_{\lambda }\) is coercive in \(W^{1,p}_0(\Omega ).\) On the other hand, \(F_{\lambda }\) is also weakly lower semi-continuous on \(W^{1,p}_0(\Omega )\) since \(F_{\lambda }\) is a continuous convex functional (see [4], Proposition 1.5.10 and Theorem 1.5.3) . Then we can apply a calculus of variations result, in order to obtain the existence of a global minimum point of \(F_{\lambda },\) denoted by \(\theta _{\lambda },\) i.e., \(F_{\lambda }(\theta _{\lambda })={\min _{W^{1,p}_0(\Omega )}}F_{\lambda }.\) Note that for any \(\lambda > \lambda ^D_1\) there exists \(u_{\lambda }\in W^{1,p}_0(\Omega )\) such that \(F_{\lambda }(u_{\lambda })< 0\) . Indeed, taking \(u_{\lambda }=re_1,\) we have
But then \(F_{\lambda }(\theta _{\lambda })\le F_{\lambda }(u_{\lambda })< 0\), which means that \(\theta _{\lambda }\in W^{1,p}_0(\Omega )\backslash \{0\}.\) On the other hand, we have \(\langle F'_{\lambda }(\theta _{\lambda }),\phi \rangle =0,\forall \phi \in W^{1,p}_0(\Omega )\) (\(\theta _{\lambda }\) is a critical point of \(F_{\lambda }\)) with \(\theta _{\lambda }\in W^{1,p}_0(\Omega )\backslash \{0\}\subset W^{1,2}_0(\Omega ) \backslash \{0\}.\) Consequently each \(\lambda > \lambda ^D_1\) is an eigenvalue of problem (1.3). \(\square \)
A similar result of Theorem 3.1 was proved in [17] in the case of the p-Laplacian.
3 Properties of Eigenfunctions and the Operator \(-\Delta _p-\Delta \)
3.1 Boundedness of the Eigenfunctions
We shall prove boundedness of eigenfunctions and use this fact to obtain \(C^{1,\alpha }\) smoothness of all eigenfunctions of the quasi-linear problem (1.3). The latter result is due to [17, Theorem 4.4], which originates from [13, 26].
Theorem 3.1
Let \((u,\lambda )\in W^{1,p}_0(\Omega )\times \mathbb {R}^{\star }_+\) be an eigensolution of the weak formulation (1.4). Then \(u\in L^{\infty }(\Omega ).\)
Proof
By Morrey’s embedding theorem it suffices to consider the case \(p\le N.\) Let us assume first that \(u>0.\) For \(M\ge 0\) define \(w_M(x)=\min \{u(x),M\}\). Letting
we have \(g\in C(\mathbb {R})\) piecewise smooth function with \(g(0)=0.\) Since \(u\in W^{1,p}_0(\Omega )\) and \(g'\in L^{\infty }(\Omega ),\) then \(g\circ u\in W^{1,p}_0(\Omega )\) and \(w_M\in W^{1,p}_0(\Omega )\cap L^{\infty }(\Omega )\) (see, Theorem B.3 in [17]). For \(k>0,\) define \(\varphi =w^{kp+1}_M\), then \(\nabla \varphi =(kp+1)\nabla w_Mw^{kp}_M\) and \(\varphi \in W^{1,p}_0(\Omega )\cap L^{\infty }(\Omega ).\)
Using \(\varphi \) as a test function in (1.4), one obtains
On the other hand, using the fact that \(w^{kp+1}_M\le u^{kp+1}\), it follows that
We have \(\nabla (w^{k+1}_M)=(k+1)\nabla w_M w^k_M\Rightarrow |\nabla w^{k+1}_M|^p=(k+1)^pw^{kp}_M|\nabla w_M|^p.\) Since the integrals on the left are zero on \(\{x : u(x)>M\}\) we can take \(u=w_M\) in the previous inequality, and it follows that
Replacing \(|\nabla w_M|^p w^{kp}_M\) by \(\frac{1}{(k+1)^p}|\nabla w^{k+1}_M|^p\), we have
which implies that
and then
By Sobolev’s embedding theorem, there is a constant \(c_1>0\) such that
where \(p^{\star }\) is the Sobolev critical exponent. Consequently, we have
and, therefore,
But by (3.2),
and we note that we can find a constant \(c_2>0\) such that
\(\left( \lambda \frac{(k+1)^p}{kp+1}\right) ^{\frac{1}{p\sqrt{k+1}}}\le c_2\), independently of k and consequently,
Letting \(M\rightarrow \infty \), Fatou’s lemma implies
Choosing \(k_1,\) such that \((k_1+1)p=p^{\star },\) then \(\Vert u\Vert _{(k_1+1)p^{\star }}\le c_1^{\frac{1}{k_1+1}}c_2^{\frac{1}{\sqrt{k_1+1}}}\Vert u\Vert _{p^{\star }}\). Next we choose \(k_2\) such that \((k_2+1)p=(k_1+1)p^{\star };\) then taking \(k_2=k\) in inequality (3.8), it follows that
By induction we obtain
where the sequence \(\{k_n\}\) is chosen such that \((k_n+1)p=(k_{n-1}+1)p^{\star }, k_0=0.\) One gets \(k_n+1=(\frac{p^{\star }}{p})^n.\) As \(\frac{p}{p^{\star }}<1,\) there is \(C>0\) (which depends on \(c_1\) and \(c_2\)) such that for any \(n=1,2,\ldots \)
with \(r_n=(k_n+1)p^{\star }\rightarrow \infty \) as \(n\rightarrow \infty .\) We note that (3.11) follows by iterating the previous inequality (3.10). We will indirectly show that \(u\in L^{\infty }(\Omega ).\) Suppose \(u\not \in L^{\infty }(\Omega ),\) then there exists \(\varepsilon >0\) and a set A of positive measure in \(\Omega \) such that \(|u(x)|>C \Vert u\Vert _{p^{\star }}+\varepsilon =K,\) for all \(x\in A.\) We then have,
which contradicts (3.11). If u changes sign, we consider \(u=u^+ -u^-\) where
We have \(u^+,u^-\in W^{1,p}_0(\Omega ).\) For each \(M{>}0\) define \(w_M{=}\min \{u^+(x),M\}\) and take again \(\varphi =w^{kp+1}_M\) as a test function in (1.4). Proceeding the same way as above we conclude that \(u^+\in L^{\infty }(\Omega )\). Similarly, we have \(u^-\in L^{\infty }(\Omega )\). Therefore, \(u=u^+ -u^-\) is in \(L^{\infty }(\Omega ).\) \(\square \)
3.2 Simplicity of the Eigenvalues
We prove an auxiliary result which will imply uniqueness of the first eigenfunction.
Let
for all \((u,v)\in D_I,\) where
and
Proposition 3.2
For all \((u,v)\in D_I\), we have \(I(u,v)\ge 0.\) Furthermore, \(I(u,v)=0\) if and only if there exists \(\alpha \in \mathbb {R}^{\star }_+\) such that \(u=\alpha v.\)
Proof
We first show that \(I(u,v)\ge 0.\) We recall that (if \(2<p<\infty \))
and (if \(1<p<2\))
Let us consider \(\beta =\frac{u^p-v^p}{u^{p-1}}\), \(\eta =\frac{v^p-u^p}{v^{p-1}}\), \(\xi =\frac{u^2-v^2}{u}\) and \(\zeta =\frac{v^2-u^2}{v}\) as test functions in (1.4) for any \(p>1\). Straightforward computations give
Therefore,
and
By symmetry we have
and
Thus
So
where
and
for all \(t=\frac{v}{u}>0, R=\nabla u, S=\nabla v\in \mathbb {R}^N\) and \(r=|\nabla u|, s=|\nabla v|\in \mathbb {R}^+.\) We clearly have that F is non-negative. Now let us show that G is non-negative. Indeed, we observe that
and \(G(t,s,0)=0\Rightarrow s=0.\) If \(r\ne 0\), by setting \(z=\frac{s}{tr}\) we obtain
and G can be written as
with \(f(z)=z^p-pz+(p-1),\) \(g(z)=(p-1)z^p-pz^{p-1}+1,\) \(h(z)=k(z)=z^2-2z+1\) \(\forall p>1 .\) We can see that f, g, h and k are non-negative. Hence G is non-negative and thus \(I(u,v)\ge 0\) for all \((u,v)\in D_I.\) In addition since f, g, h and k vanish if and only if \(z=1\), then \(G(t,s,r)=0\) if and only if \(s=tr.\) Consequently, if \(I(u,v)=0\) then we have
almost everywhere in \(\Omega .\) This is equivalent to \(\left( u\nabla v-v\nabla u\right) ^2=0,\) which implies that \(u=\alpha v\) with \(\alpha \in \mathbb {R}^{\star }_+.\) \(\square \)
Theorem 3.3
The first eigenvalues \(\lambda \) of Eq. (1.3) are simple, i.e., if u and v are two positive first eigenfunctions associated to \(\lambda ,\) then \(u=v.\)
Proof
By Proposition 3.2, we have \(u=\alpha v.\) Inserting this into the equation (1.3) implies that \(\alpha =1.\) \(\square \)
3.3 Invertibility of the Operator \(-\Delta _p-\Delta \)
To simplify some notations, here we set \(X=W^{1,p}_0(\Omega )\) and its dual \(X^{\star }=W^{-1,p'}(\Omega ),\) where \(\frac{1}{p}+\frac{1}{p'}=1.\)
For the proof of the following lemma, we refer to [19]:
Lemma 3.4
Let \(p>2\). Then there exist two positive constants \(c_1, c_2\) such that, for all \(x_1,x_2\in \mathbb {R}^n,\) we have the following:
-
(i)
\((x_2-x_1)\cdot (|x_2|^{p-2}x_2-|x_1|^{p-2}x_1)\ge c_1|x_2-x_1|^p\)
-
(ii)
\(\left| |x_2|^{p-2}x_2-|x_1|^{p-2}x_1\right| \le c_2(|x_2|+|x_1|)^{p-2}|x_2-x_1|\)
Proposition 3.5
For \(p>2,\) the operator \(-\Delta _p-\Delta \) is a global homeomorphism.
The proof is based on the previous Lemma 3.4.
Proof
Define the nonlinear operator \(A:X\rightarrow X^{\star }\) by
\(\langle Au,v\rangle =\displaystyle {\int _{\Omega }}\nabla u\cdot \nabla v~\mathrm{d}x+\displaystyle {\int _{\Omega }}|\nabla u|^{p-2}\nabla u\cdot \nabla v~\mathrm{d}x\) for all \(u,v\in X.\)
To show that \(-\Delta _p-\Delta \) is a homeomorphism, it is enough to show that A is a continuous strongly monotone operator, (see [9, Corollary 2.5.10]).
For \(p>2,\) for all \(u, v\in X\), by (i), we get
Thus A is a strongly monotone operator.
We claim that A is a continuous operator from X to \(X^{\star }.\) Indeed, assume that \(u_n\rightarrow u\) in X. We have to show that \(\Vert Au_n-Au\Vert _{X^{\star }}\rightarrow 0\) as \(n\rightarrow \infty .\) Indeed, using (ii) and Hölder’s inequality and the Sobolev embedding theorem, one has
Thus \(\Vert Au_n-Au\Vert _{X^{\star }}\rightarrow 0\), as \(n\rightarrow +\infty ,\) and hence A is a homeomorphism. \(\square \)
4 Bifurcation of Eigenvalues
In the next subsection we show that for Eq. (1.3) there is a branch of first eigenvalues bifurcating from \((\lambda _1^D, 0)\in \mathbb {R}^+\times W^{1,p}_0(\Omega ).\)
4.1 Bifurcation from Zero: The Case \(p>2\)
By Proposition 3.5, Eq. (1.3) is equivalent to
We set
\(~u{\in } L^2(\Omega )\subset W^{-1,p'}(\Omega )~\text {and}~\lambda {>}0.\) By \(\Sigma {=}\{(\lambda ,u)\in \mathbb {R}^+\times W^{1,p}_0(\Omega )/~u\ne 0~,S_{\lambda }(u)=0 \}\), we denote the set of nontrivial solutions of (4.1).
A bifurcation point for (4.1) is a number \(\lambda ^{\star }\in \mathbb {R}^+\) such that \((\lambda ^{\star },0)\) belongs to the closure of \(\Sigma .\) This is equivalent to say that, in any neighborhood of \((\lambda ^{\star },0)\) in \(\mathbb {R}^+\times W^{1,p}_0(\Omega )\), there exists a nontrivial solution of \(S_{\lambda }(u)=0\).
Our goal is to apply the Krasnoselski bifurcation theorem [see, [1]].
Theorem 4.1
(Krasnoselski, 1964)
Let X be a Banach space and let \(T\in C^1(X,X)\) be a compact operator such that \(T(0)=0\) and \(T'(0)=0.\) Moreover, let \(A\in \mathcal {L}(X)\) also be compact. Then every characteristic value \(\lambda ^{*}\) of A with odd (algebraic) multiplicity is a bifurcation point for \(u=\lambda Au+T(u).\)
We state our bifurcation result.
Theorem 4.2
Let \(p>2.\) Then every eigenvalue \(\lambda _k^D\) with odd multiplicity is a bifurcation point in \(\mathbb {R}^+\times W^{1,p}_0(\Omega )\) of \(S_{\lambda }(u)=0,\) in the sense that in any neighbourhood of \((\lambda ^D_k,0)\) in \(\mathbb {R}^+\times W^{1,p}_0(\Omega )\) there exists a nontrivial solution of \(S_{\lambda }(u)=0\).
Proof
We write the equation \(S_{\lambda }(u)=0\) as
where \(Au=(-\Delta )^{-1}u\) and \(T_{\lambda }(u)=[(-\Delta _p-\Delta )^{-1}-(-\Delta )^{-1}](\lambda u)\), where we consider
and \((-\Delta )^{-1} : L^2(\Omega )\subset W^{-1,2}(\Omega )\rightarrow W^{1,2}_0(\Omega )\subset \subset L^2(\Omega ).\)
For \(p>2,\) the mapping
is compact thanks to Rellich–Kondrachov theorem. We clearly have \(A\in \mathcal {L}(L^2(\Omega ))\) and \(T_{\lambda }(0)=0\). Now we have to show that
-
(1)
\(T_{\lambda }\in C^1\).
-
(2)
\(T_{\lambda }'(0)=0\).
In order to show (1) and (2), it suffices to show that
-
(a)
\(-\Delta _p-\Delta : W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )\) is continuously differentiable in a neighborhood \(u\in W^{1,p}_0(\Omega )\).
-
(b)
\((-\Delta _p-\Delta )^{-1}\) is a continuous inverse operator.
According to Proposition 3.5, \(-\Delta _p-\Delta \) is a homeomorphism; hence \((-\Delta _p-\Delta )^{-1}\) is continuous and this shows (b). We also recall that in section 3.2, we have shown that \(\lambda ^D_1\) is simple.
Let us show (a). We claim that \(-\Delta _p : W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )\) is Gâteaux differentiable. Indeed, for \(\varphi \in W^{1,p}_0(\Omega )\) we have
Define
and let \((u_n)_{n\ge 0}\subset W^{1,p}_0(\Omega ).\) Assume that \(u_n\rightarrow u,\) as \(n\rightarrow \infty \) in \(W^{1,p}_0(\Omega ).\) We have
Therefore,
By assumption, we can assume that, up to subsequences,
- \((*)\):
-
\(\nabla u_n\rightarrow \nabla u\) in \(\left( L^p(\Omega )\right) ^N\) as \(n\rightarrow \infty \) and
- \((**)\):
-
\(\nabla u_n(x)\rightarrow \nabla u(x)\) almost everywhere as \(n\rightarrow \infty .\)
Then \( |\nabla u_n|^{p-4}\langle \nabla u_n,\nabla v\rangle \langle \nabla u_n,\nabla \varphi \rangle \rightarrow |\nabla u|^{p-4}\langle \nabla u,\nabla v\rangle \langle \nabla u,\nabla \varphi \rangle \) as \(n \rightarrow \infty \) and consequently \(\langle B(u_n)v,\varphi \rangle \rightarrow \langle B(u )v,\varphi \rangle \)as \(n\rightarrow \infty .\) Thus, we find that \(-\Delta _p-\Delta \in C^1\) and thanks to the Inverse function theorem \((-\Delta _p-\Delta )^{-1}\) is differentiable in a neighborhood of \(u\in W^{1,p}_0(\Omega )\). Therefore, according to the Krasnoselski bifurcation Theorem, we obtain that \(\lambda _k^D\) is a bifurcation point at zero. \(\square \)
4.2 Bifurcation from Infinity: The Case \(1<p<2\)
We recall the nonlinear eigenvalue problem we are investigating
Under a solution of (4.3) (for \(1<p<2\)), we understand a pair \((\lambda ,u)\in \mathbb {R}^+_{\star }\times W^{1,2}_0(\Omega )\) satisfying the integral equality
Definition 4.3
Let \(\lambda \in \mathbb {R}.\) We say that the pair \((\lambda ,\infty )\) is a bifurcation point from infinity for problem (4.3) if there exists a sequence of pairs \(\{(\lambda _n,u_n)\}_{n=1}^{\infty }\subset \mathbb {R}\times W^{1,p}_0(\Omega )\) such that Eq. (4.4) holds and \((\lambda _n,\Vert u_n\Vert _{1,2})\rightarrow (\lambda ,\infty ).\)
We now state the main theorem concerning the bifurcation from infinity.
Theorem 4.4
The pair \((\lambda ^D_1,\infty )\) is a bifurcation point from infinity for the problem (4.3).
For \(u\in W^{1,2}_0(\Omega ),~u\ne 0,\) we set \(v=u/\Vert u\Vert _{1,2}^{2-\frac{1}{2}p}\). We have \(\Vert v\Vert _{1,2}=\frac{1}{\Vert u\Vert _{1,2}^{1-\frac{1}{2}p}}\) and
Introducing this change of variable in (4.4), we find that
But, on the other hand, we have
Consequently, it follows that Eq. (4.5) is equivalent to
This leads to the following nonlinear eigenvalue problem (for \(1<p<2\)):
The proof of Theorem 4.4 follows immediately from the following remark, and the proof that \((\lambda _1^D,0)\) is a bifurcation of (4.7).
Remark 4.5
With this transformation, we have that the pair \((\lambda ^D_1,\infty )\) is a bifurcation point for the problem (4.3) if and only if the pair \((\lambda ^D_1,0)\) is a bifurcation point for the problem (4.7).
Let us consider a small ball \(B_r(0) :=\{~w~\in W^{1,2}_0(\Omega )/~~~\Vert w\Vert _{1,2}< r~\},\) and consider the operator
Proposition 4.6
Let \(1<p<2.\) There exists \(r>0\) such that the mapping
\(T : B_r(0)\subset W^{1,2}_0(\Omega )\rightarrow W^{-1,2}(\Omega )\) is invertible, with a continuous inverse.
Proof
In order to prove that the operator T is invertible with a continuous inverse, we again rely on [9, Corollary 2.5.10]. We show that there exists \(\delta >0\) such that
with \(r>0\) sufficiently small.
Indeed, using that \(-\Delta _p\) is strongly monotone on \(W^{1,p}_0(\Omega )\) on the one hand and the Hölder inequality on the other hand, we have
Now, we obtain by the Mean Value Theorem that there exists \(\theta \in [0,1]\) such that
Hence, continuing with the estimate of Eq. (4.8), we get
and thus the claim, for \(r>0\) small enough.
Hence, the operator T is strongly monotone on \(B_r(0)\) and it is continuous, and hence the claim follows. \(\square \)
Clearly the mappings
are also local homeomorphisms for \(1<p<2\) with \(\gamma =4-p>0\). Consider now the homotopy maps
Then we can find a \(\rho >0\) such that the ball
and
are compact mappings. Set now
Notice that \(\tilde{S}_{\lambda }\) is a compact perturbation of the identity in \(L^2(\Omega ).\) We have \(0\notin H([0,1]\times \partial B_r(0) ).\) So it makes sense to consider the Leray–Schauder topological degree of \(H(\tau ,\cdot )\) on \(B_r(0).\) And by the property of the invariance by homotopy, one has
Theorem 4.7
The pair \((\lambda ^D_1,0)\) is a bifurcation point in \(\mathbb {R}^+\times L^2(\Omega )\) of \(\tilde{S}_{\lambda }(u)=0\), for \(1<p<2.\)
Proof
Suppose by contradiction that \((\lambda ^D_1,0)\) is not a bifurcation for \(\tilde{S}_{\lambda }.\) Then, there exist \(\delta _0>0\) such that for all \(r\in (0,\delta _0)\) and \(\varepsilon \in (0,\delta _0),\)
Taking into account that (4.10) holds, it follows that it make sense to consider the Leray–Schauder topological degree \(\deg (\tilde{S}_{\lambda }, B_r(0), 0)\) of \(\tilde{S}_{\lambda }\) on \(B_r(0).\)
We observe that
Proving (4.11) guarantee the well posedness of \(\deg (I{-}(\lambda ^D_1\pm \varepsilon )H(\tau ,\cdot ), B_r(0), 0)\) for any \(\tau \in [0,1].\)
Indeed, by contradiction suppose that there exists \(v\in \partial B_r(0)\subset L^2(\Omega )\) such that
\(v-\left( \lambda _1^D-\varepsilon \right) H(\tau ,v)=0,\) for some \(\tau \in [0,1].\)
One concludes that then \(v\in W^{1,2}_0(\Omega ),\) and then that
However, we get the contradiction,
By the contradiction assumption, we have
By homotopy using (4.9), we have
Now, using (4.13) and (4.12), we find that
Furthermore, since \(\lambda ^D_1\) is a simple eigenvalue of \(-\Delta ,\) it is well known [see [1]] that
In order to get contradiction (to relation (4.14)), it is enough to show that
\(r>0\) sufficiently small. We have to show that
Suppose by contradiction that there is \(r_n\rightarrow 0\), \(\tau _n\in [0,1]\) and \(u_n\in \partial B_{r_n}(0)\) such that
or equivalently
Dividing the Eq. (4.17) by \(\Vert u_n\Vert _{1,2}\), we obtain
and by setting \(v_n=\frac{u_n}{\Vert u_n\Vert _{1,2}},\) it follows that
But since \(\Vert v_n\Vert _{1,2}=1,\) we have \(v_n\rightharpoonup v\) in \(W^{1,2}_0(\Omega )\) and \(v_n\rightarrow v\) in \(L^2(\Omega ).\) Furthermore, the first term in the left-hand side of Eq. (4.18) tends to zero in \(W^{-1,p'}(\Omega )\) as \(r_n\rightarrow 0\) and hence in \(W^{-1,2}(\Omega ).\) Equation (4.17) then implies that \(v_n\rightarrow v\) strongly in \(W^{1,2}_0(\Omega )\) since \(-\Delta : W^{1,2}_0(\Omega )\rightarrow W^{-1,2}(\Omega )\) is a homeomorphism and thus v with \(\Vert v\Vert _{1,2}=1\) solves \(-\Delta v=(\lambda _1^D+\varepsilon )v\), which is impossible because \(\lambda _1^D+\varepsilon \) is not the first eigenvalue of \(-\Delta \) on \(W^{1,2}_0(\Omega )\) for \(\varepsilon >0.\)
Therefore, by homotopy it follows that
Now, thanks to (4.15), we find that
which contradicts Eq. (4.14). \(\square \)
Theorem 4.8
The pair \((\lambda ^D_k,0)\) (\(k>1\)) is a bifurcation point of \(\tilde{S}_{\lambda }(u)=0\), for \(1<p<2\) if \(\lambda ^D_k\) is of odd multiplicity.
Proof
Suppose by contradiction that \((\lambda ^D_k,0)\) is not a bifurcation for \(\tilde{S}_{\lambda }.\) Then, there exist \(\delta _0>0\) such that for all \(r\in (0,\delta _0)\) and \(\varepsilon \in (0,\delta _0),\)
Taking into account that (4.19) holds, it follows that it make sense to consider the Leray–Schauder topological degree \(\deg (\tilde{S}_{\lambda }, B_r(0), 0)\) of \(\tilde{S}_{\lambda }\) on \(B_r(0).\)
We show that
Proving (4.20) garantees the well posedness of \(\deg (I-(\lambda ^D_k\pm \varepsilon )H(\tau ,\cdot ), B_r(0), 0)\) for any \(\tau \in [0,1].\) Indeed, consider the projections \(P^-\) and \(P^+\) onto the spaces \(\text {span}\{e_1,\dots ,e_{k-1}\}\) and \(\text {span}\{e_k,e_{k+1},\dots \} \), respectively, where \(e_1\dots ,e_k,e_{k+1},\dots \)denote the eigenfunctions associated with the Dirichlet problem (1.1).
Suppose by contradiction that relation (4.20) does not hold. Then there exists \(v\in \partial B_r(0)\subset L^2(\Omega )\) such that \(v-(\lambda ^D_k-\varepsilon )H(\tau ,v)=0,\) for some \(\tau \in [0,1].\) This is equivalent of having
Replacing v by \(P^+v+P^-v,\) and multiplying equation (4.21) by \(P^+v-P^-v\) in the both sides, we obtain
But
and using the Hölder inequality, the embedding \(W^{1,2}_0(\Omega )\subset W^{1,p}_0(\Omega )\) and the fact that \(P^+v\) and \(P^-v\) do not vanish simultaneously, there is some positive constant \(C'>0\) such that \(\Vert P^+v-P^-v\Vert _{1,2}\le C'(\Vert P^+v\Vert ^2_{1,2}+\Vert P^-v\Vert ^2_{1,2})=C'\Vert P^+v-P^-v\Vert ^2_{1,2},~~\text {since}~\left( P^+v,P^-v\right) _{1,2}=0,\) we have
On the other hand, thanks to the Poincaré inequality as well as the variational characterization of eigenvalues we find
and
we can bound from below these two inequalities together by \(\Vert \nabla P^+v\Vert ^2_{2}+\Vert \nabla P^-v\Vert ^2_{2}.\)
Finally, we have
for r taken small enough. This shows that (4.20) holds.
By the contradiction assumption, we have
By homotopy using (4.20), we have
where \(\beta \) is the sum of algebraic multiplicities of the eigenvalues \(\lambda _k^D-\varepsilon < \lambda .\) Similarly, if \(\beta '\) denotes the sum of the algebraic multiplicities of the characteristic values of \((-\Delta )^{-1}\) such that \(\lambda >\lambda ^D_k+\varepsilon ,\) then
But since \([\lambda ^D_k-\varepsilon ,\lambda ^D_k+\varepsilon ]\) contains only the eigenvalue \(\lambda ^D_k,\) it follows that \(\beta '=\beta +\alpha ,\) where \(\alpha \) denotes the algebraic multiplicity of \(\lambda ^D_k.\) Consequently, we have
since \(\lambda ^D_k\) is with odd multiplicity. This contradicts (4.22). \(\square \)
5 Multiple Solutions
In this section we prove multiplicity results by distinguishing again the two cases \(1<p<2\) and \(p>2\). We recall the following definition which will be used in this section. Let X be a Banach space and \(\Omega \subset X\) an open bounded domain which is symmetric with respect to the origin of X, that is, \(u\in \Omega \Rightarrow -u\in \Omega .\) Let \(\Gamma \) be the class of all the symmetric subsets \(A\subseteq X\backslash \{0\}\) which are closed in \(X\backslash \{0\}.\)
Definition 5.1
(Krasnoselski genus) Let \(A\in \Gamma \). The genus of A is the least integer \(p\in \mathbb {N}^*\) such that there exists \(\Phi : A\rightarrow \mathbb {R}^p\) continuous, odd and such that \(\Phi (x)\ne 0\) for all \(x\in A.\) The genus of A is usually denoted by \(\gamma (A)\).
Theorem 5.2
Let \(1<p<2\) or \(2<p<\infty \), and suppose that \(\lambda \in (\lambda ^D_k,\lambda ^D_{k+1})\) for any Then Eq. (1.3) has at least k pairs of nontrivial solutions.
Proof
Case 1: \(1<p<2\). In this case we will avail of [ [1], Proposition 10.8]. We consider the energy functional \(I_{\lambda }: W^{1,2}_0(\Omega )\backslash \{0\}\rightarrow \mathbb {R}\) associated with the problem (1.3) defined by
The functional \(I_{\lambda }\) is not bounded from below on \(W^{1,2}_0(\Omega )\), so we consider again the natural constraint set, the Nehari manifold on which we minimize the functional \(I_{\lambda }.\) The Nehari manifold is given by
On \(\mathcal {N}_{\lambda },\) we have \(I_{\lambda }(u)=(\frac{2}{p}-1)\displaystyle {\int _{\Omega }}|\nabla u|^p~\mathrm{d}x>0.\) We clearly have that \(I_{\lambda }\) is even and bounded from below on \(\mathcal {N}_{\lambda }.\)
Now, let us show that every (PS) sequence for \(I_{\lambda }\) has a converging subsequence on \(\mathcal {N}_{\lambda }.\) Let \((u_n)_n\) be a (PS) sequence, i.e., \(|I_{\lambda }(u_n)|\le C\), for all n, for some \(C>0\) and \(I'_{\lambda }(u_n)\rightarrow 0\) in \(W^{-1,2}(\Omega )\) as \(n\rightarrow +\infty .\) We first show that the sequence \((u_n)_n\) is bounded on \(\mathcal {N}_{\lambda }\). Suppose by contradiction that this is not true, then \(\displaystyle {\int _{\Omega }}|\nabla u_n|^2~\mathrm{d}x\rightarrow +\infty \) as \(n\rightarrow +\infty .\) Since \(I_{\lambda }(u_n)=(\frac{2}{p}-1)\displaystyle {\int }_{\Omega }|\nabla u_n|^p~\mathrm{d}x\) we have \(\displaystyle {\int }_{\Omega }|\nabla u_n|^p~\mathrm{d}x\le c.\) On \(\mathcal {N}_{\lambda }\), we have
and hence \(\displaystyle {\int _{\Omega }}u_n^2~\mathrm{d}x\rightarrow +\infty .\) Let \(v_n=\frac{u_n}{\Vert u_n\Vert _2};\) then \(\displaystyle {\int _{\Omega }}|\nabla v_n|^2~\mathrm{d}x\le \lambda \) and hence \(v_n\) is bounded in \(W^{1,2}_0(\Omega ).\) Therefore, there exists \(v_0\in W^{1,2}_0(\Omega )\) such that \(v_n\rightharpoonup v_0\) in \(W^{1,2}_0(\Omega )\) and \(v_n\rightarrow v_0\) in \(L^2(\Omega ).\) Dividing (5.1) by \(\Vert u_n\Vert ^p_2,\) we have
since \(\lambda \displaystyle {\int _{\Omega }}u_n^2~\mathrm{d}x-\int _{\Omega }|\nabla u_n|^2~\mathrm{d}x=(\frac{2}{p}-1)^{-1}I_{\lambda }(u_n)\), \(|I_{\lambda }(u_n)|\le C\) and \(\Vert u_n\Vert ^p_2\rightarrow +\infty .\) Now, since \(v_n\rightharpoonup v_0\) in \(W^{1,2}_0(\Omega )\subset W^{1,p}_0(\Omega ),\) we infer that
and consequently \(v_0=0.\) So \(v_n\rightarrow 0\) in \(L^2(\Omega )\) and this is a contradiction since \(\Vert v_n\Vert _2=1.\) So \((u_n)_n\) is bounded on \(\mathcal {N}_{\lambda }\).
Next, we show that \(u_n\) converges strongly to u in \(W^{1,2}_0(\Omega ).\)
To do this, we will use the following vector inequality for \(1<p<2\)
for all \(x_1,x_2\in \mathbb {R}^N\) and for some \(C'>0, \) (see [19]).
We have \(\displaystyle {\int _{\Omega }}u_n^2~\mathrm{d}x\rightarrow \displaystyle {\int _{\Omega }}u^2~\mathrm{d}x\) and since \(I_{\lambda }'(u_n)\rightarrow 0\) in \(W^{-1,2}(\Omega ),\) \(u_n\rightharpoonup u\) in \(W^{1,2}_0(\Omega ),\) we also have \(I_{\lambda }'(u_n)(u_n-u)\rightarrow 0\) and \(I_{\lambda }'(u)(u_n-u)\rightarrow 0\) as \(n\rightarrow +\infty .\) On the other hand, one has
Therefore, \(\Vert u_n-u\Vert _{1,2}\rightarrow 0\) as \(n\rightarrow +\infty \) and \(u_n\) converges strongly to u in \(W^{1,2}_0(\Omega ).\)
Let \(\Sigma '=\{A\subset \mathcal {N}_{\lambda }:~A~\text {closed}~\text {and}~-A=A\}\) and \(\Gamma _j=\{A\in \Sigma ':~\gamma (A)\ge j\},\) where \(\gamma (A)\) denotes the Krasnoselski’s genus. We show that \(\Gamma _j\ne \emptyset .\)
Set \(E_j=\text {span}\{e_i,~~i=1,\dots ,j\},\) where \(e_i\) are the eigenfunctions associated with the problem (1.1). Let \(\lambda \in (\lambda ^D_j,\lambda ^D_{j+1}),\) and consider \(v\in S_{j}:=\{ v\in E_j:~\int _{\Omega }|v|^2~\mathrm{d}x=1\}.\) Then set
Then \(\lambda \int _{\Omega }v^2~\mathrm{d}x-\int _{\Omega }|\nabla v|^2~\mathrm{d}x\ge \lambda \int _{\Omega }v^2~\mathrm{d}x-\sum \limits _{i=1}^{j}\int _{\Omega }\lambda _i| e_i|^2~\mathrm{d}x\ge (\lambda -\lambda _j)\int _{\Omega }|v|^2~\mathrm{d}x > 0\). Hence, \(\rho (v)v\in \mathcal {N}_{\lambda },\) and then \(\rho (S_j)\in \Sigma ',\) and \(\gamma (\rho (S_j))=\gamma (S_j)=j\) for \(1\le j\le k,\) for any .
It is then standard (see [1], Proposition 10.8) to conclude that
yields k pairs of nontrivial critical points for \(I_{\lambda },\) which gives rise to k nontrivial solutions of problem (1.3).
Case 2: \( p>2\).
In this case, we will rely on the following theorem:
Theorem (Clark, [11]) .
Let X be a Banach space and \(G\in C^1(X,\mathbb {R})\) satisfying the Palais–Smale condition with \(G(0)=0.\) Let \(\Gamma _k =\{~A\in \Sigma ~:~\gamma (A)\ge k~\}\) with \(\Sigma = \{~A\subset X~;~A=-A~\text {and}~A~\text {closed}~ \}.\) If \(c_k=\inf \nolimits _{A\in \Gamma _k}\sup \nolimits _{u\in A}G(u)\in (-\infty , 0),\) then \(c_k\) is a critical value.
Let us consider the \(C^1\) energy functional \(I_{\lambda }: W^{1,p}_0(\Omega )\rightarrow \mathbb {R}\) defined as
We want to show that
is a critical point for \(I_{\lambda }\), where \(\Sigma '=\{A\subseteq S_j\},\) where \(S_j=\{v\in E_j~:~\int _{\Omega }|v|^2 \mathrm{d}x=1\}.\)
We clearly have that \(I_{\lambda }(u)\) is an even functional for all \(u\in W^{1,p}_0(\Omega )\), and also \(I_{\lambda }(u)\) is bounded from below on \(W^{1,p}_0(\Omega )\) since \(I_{\lambda }(u)\ge C\Vert u\Vert _{1,p}^p-C'\Vert u\Vert ^2_{1,p}\).
We show that \(I_{\lambda }(u)\) satisfies the (PS) condition. Let \(\{u_n\}\) be a Palais–Smale sequence, i.e., \(|I_{\lambda }(u_n)|\le M\) for all n, \(M>0\) and \(I_{\lambda }'(u_n)\rightarrow 0\) in \(W^{-1,p'}(\Omega )\) as \(n\rightarrow \infty .\) We first show that \(\{u_n\}\) is bounded in \(W^{1,p}_0(\Omega ).\) We have
and so \(\{u_n\}\) is bounded in \(W^{1,p}_0(\Omega ).\)
Therefore, \(u\in W^{1,p}_0(\Omega )\) exists such that, up to subsequences that we will denote by \((u_n)_n\) we have \(u_n\rightharpoonup u\) in \(W^{1,p}_0(\Omega )\) and \(u_n\rightarrow u\) in \(L^2(\Omega ).\)
We will use the following inequality for \(v_1,v_2\in \mathbb {R}^N:\) there exists \(R>0\) such that
for \(p>2\) (see [19]). Then we obtain
Therefore, \(\Vert u_n-u\Vert _{1,p}\rightarrow 0\) as \(n\rightarrow +\infty \), and so \(u_n\) converges to u in \(W^{1,p}_0(\Omega ).\)
Next, we show that there exists sets \(A_j\) of genus \(j=1,\dots ,k\) such that \(\sup \limits _{u\in A_j}I_{\lambda }(u)<0.\)
Consider \(E_j=\text {span}\{e_i,~i=1,\dots ,j\}\) and \(S_j=\{v\in E_j:~\int _{\Omega }|v|^2~\mathrm{d}x=1\}.\) For any \(s\in (0,1)\), we define the set \(A_j(s):=s(S_j\cap E_j)\) and so \(\gamma (A_j(s))=j\) for \(j=1,\dots ,k.\) We have, for any \(s\in (0,1)\)
for \(s>0\) sufficiently small, since \(\displaystyle {\int _{\Omega }}|\nabla v|^p~\mathrm{d}x\le c_j\), where \(c_j\) denotes some positive constant.
Finally, we conclude that \(\sigma _{\lambda ,j}\) \((j=1,\dots ,k)\) are critical values thanks to Clark’s Theorem. \(\square \)
References
Ambrosetti, A., Malchiodi, A.: Nonlinear analysis and semilinear elliptic problems 2007. In: Cambridge Studies in Advanced Mathematics 104 (2007)
Allaire, G.: Numerical Analysis and Optimization. Oxford University, OUP Oxford (2007)
Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. Math. 305(16), 725–728 (1987)
Badiale, M., & Serra, E. (2010). Semilinear Elliptic Equations for Beginners: Existence Results via the Variational Approach. Springer Science & Business Media
Benci, V., D’Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154(4), 297–324 (2000)
Bobkov, V., Tanaka, M.: On positive solutions for \((p, q)\)-Laplace equations with two parameters. Calc. Var. Partial. Differ. Equ. 54, 3277–330 (2015)
Bobkov, V., Tanaka, M.: Remarks on minimizers for \((p, q)\)-Laplace equations with two parameters. Commun. Pure Appl. Anal. 17(3), 1219–1253 (2018)
Brezis, H., & Brézis, H. (2011). Functional analysis, Sobolev spaces and partial differential equations (Vol. 2, No. 3, p. 5). New York: Springer
Chang, K.C.: Methods in Nonlinear Analysis. Springer-Verlag, Berlin (2005)
Cingolani, S., Degiovanni, M.: Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)
Clark, D.C.: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Di Benedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)
Evans, L.C.: Partial Differential Equations, 2nd edn. Department of Mathematics, University of California, Berkeley (2010)
Fǎrcǎseanu, M., Mihǎilescu, M., Stancu-Dumitru, D.: On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition. Nonlinear Anal. 116, 19–25 (2015)
Henrot, A.: Extremum problems for eigenvalues of elliptic operators. Springer Science & Business Media, 2006
Lê, A.: Eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64, 1057–1099 (2006)
Liang, Z., Han, X., Li, A.: Some properties and applications related to the (2, p)-Laplacian operator, al. Bound. Value Probl. 2016, 58 (2016)
Lindqvist, P. (2017). Notes on the p-Laplace equation (No. 161). University of Jyväskylä
Lindqvist, P.: Addendum: on the equation \(\text{ div }(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc 116, 583–584 (1992). (Proceedings of the Amer. Math. Soc. 109(1990)157-164)
Marano, S., Mosconi, S.: Some recent results on the Dirichlet problem for \((p, q)\)-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 11(2), 279–291 (2018)
Mihǎilescu, M.: An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Commun. Pure Appl. Anal. 10, 701–708 (2011). https://doi.org/10.3934/cpaa.2011.10.701
Mihǎilescu, M., Rǎdulescu, V.: Continuous spectrum for a class of nonhomogeneous differential operators. Manuscr. Math. 125, 157–167 (2008)
Rabinowitz, P.H.: On bifurcation from infinity. J. Differ. Equ. 14, 462–475 (1973)
Uniqueness of a positive solution and existence of a sign-changing solution for (p, q)-Laplace equation. J. Nonlinear Funct. Anal, 2014, vol. 2014, p. 1–15
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Torre, F., Ruf, B.: Multiplicity of solutions for a superlinear p-Laplacian equation. Nonlinear Anal. Theory Methods Appl. 73, 2132–2147 (2010)
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Zongo, E.WB., Ruf, B. Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators. Mediterr. J. Math. 19, 99 (2022). https://doi.org/10.1007/s00009-022-02015-4
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DOI: https://doi.org/10.1007/s00009-022-02015-4