Abstract
We are interested in the following Dirichlet problem:
on a bounded domain \(\Omega \subset \mathbb {R}^N\) with \(0 \in \Omega \). We assume that the nonlinear part is superlinear on some closed subset \(K \subset \Omega \) and asymptotically linear on \(\Omega \setminus K\). We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u. Moreover, we study also the multiplicity of solutions to the associated normalized problem.
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1 Introduction
We are interested in the problem
where \(\lambda , \mu \in \mathbb {R}\) are real parameters, \(f : \Omega \times \mathbb {R}\rightarrow \mathbb {R}\), \(\Omega \subset \mathbb {R}^N\) is a bounded domain in \(\mathbb {R}^N\) with \(0 \in \Omega \).
Semilinear problems of general form
appear when one looks for stationary states of time-dependent problems, including the heat equation \(\frac{\partial u}{\partial t} - \Delta u = h(x,u)\) or the wave equation \(\frac{\partial ^2 u}{ \partial t^2} - \Delta u = h(x,u)\). In nonlinear optics, the nonlinear Schrödinger equation is studied
and looking for standing waves \(\Psi (t,x) = e^{i\lambda t} u(x)\) leads then to a semilinear problem.
The time-dependent equation [Eq. (1.2)] appears in physical models in the case of bounded domains \(\Omega \) ([11, 12, 24]), as well as in the case \(\Omega = \mathbb {R}^N\) ([8, 18]). Two points of view of solutions to (1.1) are possible; either \(\lambda \) may be prescribed or may be considered as a part of the unknown. In the latter case, a natural additional condition is the prescribed mass \(\int _\Omega u^2 \, dx\). In the paper, we will consider both cases; namely, we will look for solutions for the unconstrained problem (1.1) as well as the constrained one; see (1.3).
Equation (1.1) (and systems of such equations) on bounded domains has been studied in the presence of bounded potentials [4] and singular at the origin [13]; see also [10, 14, 15] for the case of unbounded \(\Omega \). Its constrained counterpart without the potential has been studied, e.g., in [19, 20], where (1.3) was studied with \(f(x,u)=|u|^{p-2}u\), \(\nu =\mu =0\) in the mass-subcritical, mass-critical, and mass-supercritical cases. In this paper, we are interested in the presence of a potential
which is singular in \(\Omega \) as well as on the whole boundary \(\partial \Omega \). We mention here that Schrödinger operators were studied with potentials being singular at the point on the boundary [7], as well as with potentials being singular on the whole boundary [16, 17]. We assume that \(\Omega \) is a domain satisfying the following condition:
-
(C)
\({-}\Delta d \ge 0\) in \(\Omega \), in the sense of distributions, where \(d(x) := \textrm{dist}(x, \mathbb {R}^N \setminus \Omega )\).
This condition allows us to study the singular potential by means of Hardy-type inequalities (Sect. 2). As we will see in Sect. 2 (see Proposition 2.1), any convex domain \(\Omega \) satisfies (C).
We impose the following condition on parameters appearing in the problem:
-
(N)
\(\mu ,\nu \ge 0\), \(\frac{\mu }{(N-2)^2} + \nu < \frac{1}{4}\), \(N \ge 3\).
In what follows, \(\lesssim \) denotes the inequality up to a multiplicative constant. Moreover, C denotes a generic constant which may vary from one line to another.
On the nonlinear part of Eq. (1.1), we propose the following assumptions.
-
(F1)
\(f : \Omega \times \mathbb {R}\rightarrow \mathbb {R}\) is a Carathéodory function and there is \(2< p < 2^*\), such that
$$\begin{aligned} |f(x,u)| \lesssim 1+|u|^{p-1}, \quad u \in \mathbb {R}, \ x \in \Omega ; \end{aligned}$$ -
(F2)
\(f(x,u) = o(u)\) uniformly in \(x \in \Omega \) as \(u \rightarrow 0\);
-
(F3)
\(F(x,u)/|u|^2 \rightarrow +\infty \) as \(|u| \rightarrow +\infty \) for \(x \in K\), where \(F(x,u) := \int _0^u f(x,s) \, ds\) and \(K \subset \Omega \) is a closed set with \(| \textrm{int}\, K | > 0\);
-
(F4)
f(x, u)/|u| is nondecreasing on \((-\infty , 0)\) and on \((0,\infty )\);
-
(F5)
\(\lim _{|u|\rightarrow +\infty } \frac{f(x,u)}{u} = \Theta (x)\) uniformly in \(x \in \Omega \setminus K\), where \(\Theta \in L^\infty (\Omega \setminus K)\).
A simple example satisfying all foregoing conditions is the following:
where \(\Gamma \in L^\infty (K)\) is a nonnegative function. Note that the foregoing example is a combination of nonlinearities appearing in nonlinear optics ([8, 11, 18]). Namely, for materials with the Kerr effect, one has \(f(x,u) = |u|^{p-2}u\) with \(p=4\), and in the saturation effect, the nonlinearity is asymptotically linear and \(f(x,u) = \frac{|u|^2}{1+|u|^2} u\). Therefore, we can deal with media being composites of materials with different polarizations.
To show the boundedness of minimizing sequences to the problem (1.1), we impose the following abstract condition:
-
(A)
\(-\lambda \) is not an eigenvalue of \(-\Delta - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)\) with Dirichlet boundary conditions on \(L^2(\Omega \setminus K)\).
As we will see in Sect. 2, (A) is satisfied if, e.g., \(\lambda \ge |\Theta |_\infty \) (cf. Theorem 2.2).
Theorem 1.1
Suppose that (C), (N), (F1)–(F5), (A) are satisfied and \(\lambda \ge 0\). Then, there is a nontrivial weak solution to (1.1) with the energy level c satisfying (3.1).
Theorem 1.2
Suppose that (C), (N), (F1)–(F5), (A) hold, \(\lambda \ge 0\), and f is odd in \(u \in \mathbb {R}\). Then, there is infinitely many weak solutions to (1.1).
In the last section, we also study the normalized problem
where \(\rho \) is fixed and \((\lambda , u) \in \mathbb {R}\times H^1_0(\Omega )\) is an unknown. Then, we obtain the following multiplicity result in the so-called mass-subcritical case.
Theorem 1.3
Suppose that (C), (N), (F1) hold with \(p < 2_* := 2+\frac{4}{N}\), and f is odd in \(u \in \mathbb {R}\). Then, there is infinitely many weak solutions to (1.3).
2 The domain \(\Omega \)nd the singular Schrödinger operator
We recall that if \(A \subset \mathbb {R}^N\) is a closed, nonempty set, we can define the distance function \(\textrm{dist}(\cdot , A) : \mathbb {R}^N \rightarrow [0,\infty )\) by
We collect the following properties of the distance function:
-
(i)
\(|\textrm{dist}(x, A) - \textrm{dist}(x',A) \le |x-x'|\) for all \(x,x' \in \mathbb {R}^N\),
-
(ii)
if \(x \in \mathbb {R}^N \setminus A\), then \(\textrm{dist}(\cdot , A)\) is differentiable at x if and only if there is unique \(x \in A\), such that \(\textrm{dist}(x,A) = |x-y|\) and then \(\nabla \textrm{dist}(x, A) = \frac{x-y}{|x-y|}\).
Now, we consider \(A := \mathbb {R}^N \setminus \Omega \). It is clear that A is a closed subset of \(\mathbb {R}^N\). We recall that we denote \(d(x) = \textrm{dist}(x, \mathbb {R}^N \setminus \Omega )\). Observe that, due to Rademacher’s theorem ([26, Theorem 2.2.1]) and (i), d is differentiable almost everywhere and, from (ii) \(|\nabla d| = 1\), almost everywhere. We remind that the assumption (C) says that
holds in the sense of distributions. We note the following fact.
Proposition 2.1
If \(\Omega \subset \mathbb {R}^N\) is a convex domain in \(\mathbb {R}^N\), then \(d \big |_\Omega \) is concave and satisfies (C).
Proof
First note that \(d \big |_\Omega : \Omega \rightarrow [0,\infty )\) is a concave function. Indeed, fix any \(x,y \in \Omega \) and \(\alpha \in [0,1]\). Let \(z = \alpha x + (1-\alpha )y\). Choose \(z_0 \in \partial \Omega \), such that \(d(z) = |z-z_0|\). Let \(T_{z_0} := z_0 + \textrm{span} \{ z-z_0 \}^\perp \) be an affine subspace of \(\mathbb {R}^N\) orthogonal to \(z-z_0\) containing \(z_0\). Define \(x_0, y_0\) as orthogonal projections of x, y, respectively, onto \(T_{z_0}\). Then
which completes the proof of concavity. Moreover, since d is concave on \(\Omega \), from [9, Theorem 6.8], there is a nonnegative Radon measure \(\mu \) on \(\Omega \) satisfying
namely
Clearly, for \(\varphi \ge 0\), we get
and condition (C) holds. \(\square \)
To study singular terms in (1.1), we recall the following Hardy-type inequalities. If \(u \in H^1_0 (\Omega )\), where \(\Omega \) is a domain in \(\mathbb {R}^N\) with finite Lebesgue measure and \(0 \in \Omega \), then (see [6])
Now, let \(\Omega \subset \mathbb {R}^N\) be a bounded domain satisfying (C). Then, for \(u \in H^1_0(\Omega )\), the following Hardy inequality involving the distance function holds (see [2]):
We consider the operator \({\mathcal {A}} := -\Delta - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)\) on \(L^2 (\Omega \setminus K)\) with Dirichlet boundary conditions. Then, the domain is \({{\mathcal {D}}}({{\mathcal {A}}}) := H^2 (\Omega \setminus K) \cap H^1_0 (\Omega \setminus K)\).
Theorem 2.2
The operator \({{\mathcal {A}}}: {{\mathcal {D}}}(A) \subset L^2 (\Omega \setminus K) \rightarrow L^2 (\Omega \setminus K)\) is elliptic, self-adjoint on \(L^2 (\Omega \setminus K)\) and has compact resolvents. Moreover, the spectrum \(\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )\) and consists of eigenvalues \(-|\Theta |_\infty < \lambda _1 \le \lambda _2 \le \ldots \) with \(\lambda _n \rightarrow +\infty \) as \(n \rightarrow +\infty \).
Proof
It is well known that \({{\mathcal {D}}}({{\mathcal {A}}})\) is closed in \(L^2(\Omega \setminus K)\). It is easy to check that \({{\mathcal {A}}}\) is self-adjoint. The compactness of its resolvents easily follows from the Rellich–Kondrachov theorem. Hence, its spectrum consists of eigenvalues \(\lambda _n\) with \(\lambda _n \rightarrow +\infty \). To see that \(\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )\), suppose that \(\lambda \) is an eigenvalue of \({{\mathcal {A}}}\) with an associated eigenfunction \(u \in H^1_0(\Omega \setminus K)\). We can treat u as a function in \(H^1_0(\Omega )\) being zero on K. Then, using (2.1) and (2.2)
Hence, \(\lambda \ge -|\Theta |_\infty \). To show that \(\lambda \ne -|\Theta |_\infty \), suppose by contradiction that there is \(u \in H^1_0 (\Omega \setminus K)\) with
Thus
However, using (2.1) and (2.2), we get
Hence, \(\int _{\Omega \setminus K} |\nabla u|^2 \, dx = 0\) and \(u = 0\), which is a contradiction. Hence, \(\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )\). \(\square \)
3 Variational setting and critical point theory
Suppose that \((E, \Vert \cdot \Vert )\) is a Hilbert space and \({{\mathcal {J}}}: E \rightarrow \mathbb {R}\) is a nonlinear functional of the general form
where \({{\mathcal {I}}}\) is of \({{\mathcal {C}}}^1\) class and \({{\mathcal {I}}}(0)=0\). We introduce the so-called Nehari manifold
Observe that \({{\mathcal {I}}}'(u)(u) > 0\) on \({{\mathcal {N}}}\). Indeed
To utilize the mountain pass approach, we consider the following space of paths:
for the radius \(r > 0\) given in (J1) below, and the following mountain pass level:
Moreover, we set
where Q is a vector subspace of E given in (J2) below.
We propose an abstract theorem which is a combination of [3, Theorem 5.1] and [5, Theorem 2.1]. The proof is a straightforward modification of proofs of mentioned theorems; however, we include it here for the reader’s convenience.
Theorem 3.1
Suppose that
-
(J1)
there is \(r > 0\), such that
$$\begin{aligned} a := \inf _{\Vert u\Vert =r} {{\mathcal {J}}}(u) > 0; \end{aligned}$$ -
(J2)
there is a closed vector subspace \(Q \subset E\), such that \(\frac{{{\mathcal {I}}}(t_n u_n)}{t_n^2} \rightarrow +\infty \) for \(t_n \rightarrow +\infty \), \(u_n \in Q\) and \(u_n \rightarrow u \ne 0\);
-
(J3)
for all \(t > 0\) and \(u \in {{\mathcal {N}}}\), there holds
$$\begin{aligned} \frac{t^2-1}{2} {{\mathcal {I}}}'(u)(u) - {{\mathcal {I}}}(tu) + {{\mathcal {I}}}(u) \le 0. \end{aligned}$$
Then, \(\Gamma _Q \ne \emptyset \), \({{\mathcal {N}}}\cap Q \ne \emptyset \) and
Moreover, there is a Cerami sequence for \({{\mathcal {J}}}\) on the level c, i.e., a sequence \(\{ u_n \}_n \subset E\), such that
Proof
Observe that there exists \(v \in Q \setminus \{0\}\) with \(\Vert v\Vert > r\), such that \({{\mathcal {J}}}(v) < 0\). Indeed, fix \(u \in Q \setminus \{0\}\) and from (J2), there follows that
and we may take \(v := t u\) for sufficiently large \(t > 0\). In particular, the family of paths \(\Gamma _Q\) is nonempty. Moreover, \({{\mathcal {J}}}(tu) \rightarrow 0\) as \(t \rightarrow 0^+\), and for \(t = \frac{r}{\Vert u\Vert } > 0\), we get \({{\mathcal {J}}}(tu) > 0\). Hence, taking (3.2) into account, \((0,+\infty ) \ni t \mapsto {{\mathcal {J}}}(tu) \in \mathbb {R}\) has a local maximum, which is a critical point of \({{\mathcal {J}}}(tu)\) and \(tu \in {{\mathcal {N}}}\). Hence, \({{\mathcal {N}}}\cap Q \ne \emptyset \). Suppose that \(u \in {{\mathcal {N}}}\cap Q\). Then, from (J3)
and therefore, u is a maximizer (not necessarily unique) of \({{\mathcal {J}}}\) on \(\mathbb {R}_+ u := \{ su \ : \ s > 0 \}\). Hence, for any \(u \in {{\mathcal {N}}}\cap Q\), there are \(0 < t_{\min } (u) \le 1 \le t_{\max }(u)\), such that \(t u \in {{\mathcal {N}}}\cap Q\) for any \(t \in [t_{\min }(u), t_{\max } (u)]\) and
is constant. Moreover, \({{\mathcal {J}}}'(tu)(u) > 0\) for \(t \in (0, t_{\min }(u))\) and \({{\mathcal {J}}}'(tu)(u) < 0\) for \(t \in (t_{\max } (u), +\infty )\), \(Q \setminus {{\mathcal {N}}}\) consists of two connected components and any path \(\gamma \in \Gamma _Q\) intersects \({{\mathcal {N}}}\cap Q\). Thus
Since
there follows, under (J1), that:
The existence of a Cerami sequence follows from the mountain pass theorem. \(\square \)
To study the multiplicity of solutions, we will recall the symmetric mountain pass theorem. We consider the following condition:
-
(J4)
there exists a sequence of subspaces \(\widetilde{E}_1 \subset \widetilde{E_2} \subset \ldots \subset E\), such that \(\dim \widetilde{E}_k=k\) for every \(k \ge 1\) and there is a radius \(R_k\), such that \(\sup _{u \in \widetilde{E}_k,\ \Vert u\Vert \ge R_k} {{\mathcal {J}}}\le 0\).
Then, the following theorem holds.
Theorem 3.2
([1, Corolarry 2.9], [21, Theorem 9.12]) Suppose that \({{\mathcal {J}}}\), as above, is even and satisfies (J1), (J4) and a Palais–Smale condition (namely, any Palais–Smale sequence for \({{\mathcal {J}}}\) contains a convergent subsequence). Then, \({{\mathcal {J}}}\) has an unbounded sequence of critical values.
We work in the usual Sobolev space \(H^1_0(\Omega )\) being the completion of \({{\mathcal {C}}}_0^\infty (\Omega )\) with respect to the norm
Define the bilinear form \(B : H^1_0(\Omega ) \times H^1_0 (\Omega ) \rightarrow \mathbb {R}\) by
Lemma 3.3
B defines an inner product on \(H^1_0 (\Omega )\). Moreover, the associated norm is equivalent with the usual one.
Proof
To check that B is positive-definite, we utilize (2.1), (2.2), and (N) to get
and the statement follows from the Poincaré inequality. Moreover, from
there follows that B generates a norm on \(H^1_0(\Omega )\) equivalent to the standard one. \(\square \)
Let \(\Vert \cdot \Vert \) denote the norm generated by B, namely
Then, we can define the energy functional \({{\mathcal {J}}}: H^1_0 (\Omega ) \rightarrow \mathbb {R}\) by
where \(G(x,u) := \int _0^u g(x,s) \, ds\) and F is given in (F3). It is well known that under (F1), (F2) the functional is of \({{\mathcal {C}}}^1\) class and
Hence, its critical points are weak solutions to (1.1).
4 Verification of (J1)–(J4)
Observe that (F1), (F2) imply that for every \(\varepsilon > 0\), one can find \(C_\varepsilon > 0\), such that
There follows also a similar inequality for F, namely:
We note also that if, in addition, (F4) holds, then \(F(x,u) \ge 0\). Moreover, we recall that the functional \({{\mathcal {J}}}\) is defined by (3.3).
Lemma 4.1
Suppose that (C), (N), (F1)–(F5) hold. Then, \({{\mathcal {J}}}\) satisfies (J1)–(J3) in Theorem 3.1 and (J4) in Theorem 3.2.
-
(J1)
Using (4.1) and Sobolev embeddings, we obtain
$$\begin{aligned} \int _\Omega F(x,u)\,dx\le \varepsilon |u|_2^2+C_\varepsilon |u|_p^p\lesssim \varepsilon \Vert u\Vert ^2+C_\epsilon \Vert u\Vert ^p. \end{aligned}$$Hence, can choose \(\varepsilon >0\) and \(r>0\), such that
$$\begin{aligned} \int _\Omega F(x,u)\,dx\le \frac{1}{4}\Vert u\Vert ^2 \end{aligned}$$for all \(\Vert u\Vert \le r\). Then, we get
$$\begin{aligned} {\mathcal {J}}(u)&= \frac{1}{2}\Vert u\Vert ^2-\int _\Omega F(x,u)\,dx \ge \frac{1}{4}\Vert u\Vert ^2=\frac{r^2}{4}>0 \end{aligned}$$for all \(\Vert u\Vert =r.\)
-
(J2)
Let \(Q := H^1_0 (\textrm{int}\, K)\). Let \(t_n\rightarrow +\infty \), \(u_n \in Q\) and \(u_n\rightarrow u\ne 0\). Then, from Fatou’s lemma and (F3)
$$\begin{aligned} \frac{{\mathcal {I}}(t_nu_n)}{t_n^2}=\frac{\int _K F(x,t_nu_n)\,dx}{t_n^2} \rightarrow +\infty \quad \text{ as } n \rightarrow +\infty . \end{aligned}$$ -
(J3)
Fix \(u \in {{\mathcal {N}}}\). Define
$$\begin{aligned} (0,\infty ) \ni t \mapsto \varphi (t):=\frac{t^2-1}{2}{\mathcal {I}}'(u)(u)-{\mathcal {I}}(tu)+{\mathcal {I}}(u)\in \mathbb {R}. \end{aligned}$$Note that \(\varphi (1)=0\). Moreover
$$\begin{aligned} \varphi '(t)&=t {\mathcal {I}}'(u)(u)-{\mathcal {I}}'(tu)(u)=\int _\Omega f(x,u)tu\,dx-\int _\Omega f(x,tu)u\,dx. \end{aligned}$$Suppose that \(t\in (0,1)\). Then, (F4) implies that for a.e. \(x \in \Omega \), \(f(x,tu)u \le tf(x,u)u\) and, therefore, \(\varphi '(t) \ge 0\). Similarly, \(\varphi '(t)\le 0\) for \(t>1\), which implies that \(\varphi (t)\le \varphi (0)=0\) for all \(t>0\).
-
(J4)
Let \(\widetilde{E} \subset H^1_0(\textrm{int}\,K) \subset H^1_0(\Omega )\) be a finite-dimensional subspace. Note that on \(\widetilde{E}\), all norms are equivalent. Suppose, by contradiction, that there is a sequence \((u_n) \subset \widetilde{E}\), such that \(\Vert u_n\Vert \rightarrow +\infty \) and \({{\mathcal {J}}}(u_n) > 0\). Let \(w_n(x) := u_n(x) / \Vert u_n\Vert \). It is clear that \(\Vert w_n\Vert =1\) and, since \(\widetilde{E}\) is finite-dimensional, there is \(w \in \widetilde{E} \setminus \{0\}\), such that \(\Vert w_n-w\Vert \rightarrow 0\). In particular \(|\textrm{supp}\,(w) \cap K| > 0\). Then, for a.e. \(x \in \textrm{supp}\,(w) \cap K\), we have that
$$\begin{aligned} u_n(x)^2 = \Vert u_n\Vert ^2 w_n(x)^2 \rightarrow +\infty . \end{aligned}$$Hence, by the Fatou’s lemma and (F3)
$$\begin{aligned} 0 < \frac{{{\mathcal {J}}}(u_n)}{\Vert u_n\Vert ^2}&= \frac{1}{2} - \int _{\Omega } \frac{F(x,u_n)}{\Vert u_n\Vert ^2} \, dx \\ {}&\le \frac{1}{2} - \int _{\textrm{supp}\,(w) \cap K} \frac{F(x,u_n)}{u_n^2} w_n^2 \, dx \rightarrow -\infty , \end{aligned}$$which is a contradiction.
5 Cerami sequences and proofs of main theorems
Lemma 5.1
Any Cerami sequence for \({{\mathcal {J}}}\) is bounded.
Proof
Suppose that \(\Vert u_n\Vert \rightarrow +\infty \) up to a subsequence. We define \(v_n = \frac{u_n}{\Vert u_n\Vert }\). Then, \(\Vert v_n\Vert =1\) and \(v_n \rightharpoonup v_0\) in \(H^1_0(\Omega )\). From compact Sobolev embeddings, \(v_n \rightarrow v_0\) in \(L^2(\Omega )\), in \(L^p (\Omega )\) and almost everywhere.
We consider three cases.
-
Suppose that \(v_0 = 0\). Define \(F_\varepsilon (x,u) := F(x,u) + \frac{\varepsilon }{p} |u|_p^p\). Then, for every \(\varepsilon > 0\), the following condition holds:
$$\begin{aligned} \frac{F_\varepsilon (x,u)}{u^2} \rightarrow +\infty \quad \text{ as } \ |u|\rightarrow +\infty \ \text{ uniformly } \text{ in } \Omega . \end{aligned}$$Then, usual arguments (cf. [23, Lemma 2.2]) show that
$$\begin{aligned} {{\mathcal {J}}}_\varepsilon (u) \ge {{\mathcal {J}}}_\varepsilon (tu) - \frac{t^2-1}{2} {{\mathcal {J}}}_\varepsilon '(u)(u), \quad u \in H^1_0 (\Omega ), \end{aligned}$$where \({{\mathcal {J}}}_\varepsilon (u) := {{\mathcal {J}}}(u) - \frac{\varepsilon }{p} |u|_p^p\). Taking \(\varepsilon \rightarrow 0^+\), we get the following inequality:
$$\begin{aligned} {{\mathcal {J}}}(u) \ge {{\mathcal {J}}}(tu) - \frac{t^2-1}{2} {{\mathcal {J}}}'(u)(u). \end{aligned}$$Taking \(t := \frac{t}{\Vert u_n\Vert }\) and \(u := u_n\) above, we obtain that
$$\begin{aligned} {{\mathcal {J}}}(u_n) \ge {{\mathcal {J}}}\left( \frac{t}{\Vert u_n\Vert }u_n\right) - \frac{\frac{t^2}{\Vert u_n\Vert ^2}-1}{2} {{\mathcal {J}}}'(u_n)(u_n) = {{\mathcal {J}}}(t v_n) + o(1). \end{aligned}$$Hence
$$\begin{aligned} {{\mathcal {J}}}(u_n) \ge \frac{t^2}{2} - \int _\Omega F(x, tv_n) \, dx + o(1). \end{aligned}$$Moreover
$$\begin{aligned} \left| \int _\Omega F(x,tv_n) \, dx \right| \le \varepsilon t^2 \int _\Omega |v_n|^2 \, dx + C_\varepsilon t^p \int _\Omega |v_n|^p \, dx \rightarrow 0 \quad \text{ as } n \rightarrow \infty . \end{aligned}$$Thus
$$\begin{aligned} c + o(1) = {{\mathcal {J}}}(u_n) \ge \frac{t^2}{2} + o(1) \end{aligned}$$for any \(t > 0\)—a contradiction.
-
Now, we suppose that \(v_0 \ne 0\) and \(|\textrm{supp}\,v_0 \cap K | > 0\). Then
$$\begin{aligned} o(1)=\frac{{{\mathcal {J}}}(u_n)}{\Vert u_n\Vert ^2}&= \frac{1}{2} - \int _\Omega \frac{F(x,u_n)}{\Vert u_n\Vert ^2} \, dx = \frac{1}{2} - \int _\Omega \frac{F(x,u_n)}{u_n^2} v_n^2 \, dx \\ {}&\le \frac{1}{2} - \int _{\textrm{supp}\,v_0 \cap K} \frac{F(x,u_n)}{u_n^2} v_n^2 \, dx \rightarrow -\infty , \end{aligned}$$a contradiction.
-
Suppose that \(v_0 \ne 0\) and \(|\textrm{supp}\,v_0 \cap K| = 0\). Then, \(\textrm{supp}\,v_0 \subset \Omega \setminus K\). Fix \(\varphi \in {{\mathcal {C}}}_0^\infty (\Omega \setminus K)\) and note that
$$\begin{aligned} o(1) = {{\mathcal {J}}}'(u_n)(\varphi ) = \langle u_n, \varphi \rangle - \int _{\Omega } f(x,u_n) \varphi \, dx. \end{aligned}$$Observe that
$$\begin{aligned} \int _{\Omega } f(x,u_n)\varphi \, dx&= \Vert u_n\Vert \int _{\Omega } \frac{f(x,u_n)}{u_n} v_n \varphi \, dx \\ {}&= \Vert u_n\Vert \left( \int _{\textrm{supp}\,\varphi \cap \textrm{supp}\,v} \frac{f(x,u_n)}{u_n} v_n \varphi \, dx + o(1) \right) . \end{aligned}$$Observe that for almost every \(x \in \textrm{supp}\,\varphi \cap \textrm{supp}\,v\), we obtain that \({|u_n(x)| = |v_n(x)| \Vert u_n\Vert \rightarrow +\infty }\). From (F5)
$$\begin{aligned} \frac{f(x,u_n(x))}{u_n(x)} v_n(x) \varphi (x) \rightarrow \Theta (x) v_0(x) \varphi (x) \end{aligned}$$pointwise, a.e. on \(\textrm{supp}\,\varphi \cap \textrm{supp}\,v\). Combining (F4) and (F5), we also get that
$$\begin{aligned} \left| \frac{f(x,u_n(x))}{u_n(x)} \right| ^2 \le |\Theta |_\infty ^2 \end{aligned}$$and \(\frac{f(\cdot ,u_n)}{u_n} \rightarrow \Theta \) in \(L^2(\textrm{supp}\,\varphi \cap \textrm{supp}\,v)\). Thus, from Lebesgue dominated convergence theorem and the Hölder inequality, we get
$$\begin{aligned} \int _{\textrm{supp}\,\varphi \cap \textrm{supp}\,v} \frac{f(x,u_n)}{u_n} v_n \varphi \, dx \rightarrow \int _{\Omega } \Theta (x) v \varphi \, dx. \end{aligned}$$Hence
$$\begin{aligned} \langle v, \varphi \rangle = \int _\Omega \Theta (x) v \varphi \, dx. \end{aligned}$$In particular, 0 is an eigenvalue of the operator \(-\Delta + \lambda - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)\) with Dirichlet boundary conditions on \(\Omega \setminus K\), which contradicts (A).
\(\square \)
Proof of Theorem 1.1
Since Cerami sequence \(u_n\) is bounded, we have following convergences (up to a subsequence):
Hence, for any \(\varphi \in {{\mathcal {C}}}^\infty _0(\Omega )\)
because obviously weak convergence of \(u_n\) implies that
and we will use the Vitali convergence theorem to prove that
Hence, we need to check the uniform integrability of the following family \(\left\{ \left( f(x,u_n)-f(x,u_0)\right) \varphi \right\} _n\). Using (F1) and Lemma 5.1 we obtain that for any measurable set \(E \subset \Omega \)
Then, for any \(\varepsilon > 0\), we can choose \(\delta > 0\) small enough that
for \(|E|<\delta \). Hence \({{\mathcal {J}}}'(u_n)(\varphi )\rightarrow {{\mathcal {J}}}'(u_0)(\varphi )\), and \({{\mathcal {J}}}'(u_0)=0\). \(\square \)
Proof of Theorem 1.2
The statement follows directly from Theorem 3.2 and Lemma 4.1. \(\square \)
6 Multiple solutions to the mass-subcritical normalized problem
In what follows we are interested in the normalized problem (1.3), where \(\lambda \) is not prescribed anymore and is the part of the unknown \((\lambda , u) \in \mathbb {R}\times H^1_0(\Omega )\). Then, solutions are critical point of the energy functional
restricted to the \(L^2\)-sphere in \(H^1_0(\Omega )\)
and \(\lambda \) arises as a Lagrange multiplier.
We recall the well-known Gagliardo-Nirenberg inequality
where \(\delta _p := N \left( \frac{1}{2} - \frac{1}{p} \right) \) and \(C_{p,N} > 0\) is the optimal constant.
Lemma 6.1
\({{\mathcal {J}}}_0\) is coercive and bounded from below on \({{\mathcal {S}}}\).
Proof
Using (F1), (2.1), (2.2), and (6.1), we obtain
where
Thus, \({{\mathcal {J}}}_0\) is coercive and bounded from below on \({{\mathcal {S}}}\). \(\square \)
Lemma 6.2
\({{\mathcal {J}}}_0\) satisfies the Palais-Smale condition on \({{\mathcal {S}}}\), i.e., any Palais–Smale sequence for \({{\mathcal {J}}}_0 |_{{\mathcal {S}}}\) has a convergent subsequence.
Proof
Let \((u_n) \subset {{\mathcal {S}}}\) be a Palais–Smale sequence for \({{\mathcal {J}}}_0 |_{{\mathcal {S}}}\). Then, Lemma 6.1 implies that \((u_n)\) is bounded in \(H_0^1 (\Omega )\). Hence, we may assume that (up to a subsequence)
Moreover
for some \(\lambda _n \in \mathbb {R}\). In particular
Note that
Observe that from (F1)
Therefore, \((\lambda _n) \subset \mathbb {R}\) is bounded, and (up to a subsequence) \(\lambda _n \rightarrow \lambda _0\). Therefore, up to a subsequence
It is clear that \(\langle u_n, u\rangle \rightarrow \Vert u\Vert ^2\) and that
Moreover, from (F1)
Hence, \(\Vert u_n\Vert \rightarrow \Vert u\Vert \) and, therefore, \(u_n \rightarrow u\) in \(H^1_0(\Omega )\). \(\square \)
Proof of Theorem 1.3
From [22, Theorem II.5.7], we obtain that \({{\mathcal {J}}}_0\) has at least \({\hat{\gamma }}({{\mathcal {S}}})\) critical points, where
and \(\gamma \) denotes the Krasnoselskii’s genus for symmetric and compact sets. We will show that \({\hat{\gamma }}({{\mathcal {S}}}) = +\infty \). Indeed, fix \(k \in \mathbb {N}\). It is sufficient to construct a symmetric and compact set \(K \subset {{\mathcal {S}}}\) with \(\gamma (K) = k\). Choose functions \(w_1, w_2, \ldots , w_k \in {{\mathcal {C}}}_0^\infty (\Omega ) \cap {{\mathcal {S}}}\) with pairwise disjoint supports, namely, \(w_i w_j = 0\) for \(i \ne j\). Now, we set
It is clear that \(K \subset {{\mathcal {S}}}\) is symmetric and compact. We will show that \(\gamma (K) = k\). In what follows \(S^{m-1}\) denotes the \((m-1)\)-dimensional sphere in \(\mathbb {R}^m\) of radius 1 centered at the origin. Note that \(h : K \rightarrow S^{k-1}\) given by
is a homeomorphism, which is odd. Hence, \(\gamma (K) \le k\). Suppose by contradiction that \(\gamma (K) < k\). Then, there is a continuous and odd function \(\widetilde{h} : K \rightarrow S^{\gamma (K) -1}\). However, \(\widetilde{h} \circ h^{-1} : S^{k-1} \rightarrow S^{\gamma (K)-1}\) is an odd, continuous map, which contradicts the Borsuk–Ulam theorem [22, Proposition II.5.2], [25, Theorem D.17]. Hence, \(\gamma (K) = k\). \(\square \)
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Acknowledgements
Adam Konysz was partly supported by the Excellence Initiative-Research University at Nicolaus Copernicus University (Grants4NCUStudents 90-SIDUB.6102.18.2021.G4NCUS2). Moreover, some parts of this work were completed during his short research stays at Institute of Mathematics, Polish Academy of Sciences; he wishes to thank for the warm hospitality. Bartosz Bieganowski and Adam Konysz were also partly supported by National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817).
Funding
Adam Konysz was partly supported by the Excellence Initiative-Research University at Nicolaus Copernicus University (Grants4NCUStudents 90-SIDUB.6102.18.2021.G4NCUS2). Bartosz Bieganowski and Adam Konysz were also partly supported by National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817).
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Bieganowski, B., Konysz, A. Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain. J. Fixed Point Theory Appl. 25, 83 (2023). https://doi.org/10.1007/s11784-023-01085-5
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DOI: https://doi.org/10.1007/s11784-023-01085-5